Free vibration analysis of moderately thick antisymmetric cross-ply laminated rectangular plates with elastic edge constraints

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Free vibration analysis of moderately thick antisymmetric

angle-ply laminated rectangular plates with elastic edge constraints

Journal: Mechanics of Advanced Materials and Structures

Manuscript ID: UMCM-2011-0101

Manuscript Type: Manuscript

Date Submitted by the Author:

18-Jun-2011

Complete List of Authors: Sharma, Avadesh; Maulana Azad National Institute of Technology, Bhopal, Applied Mechanics Mittal, Narain; Maulana Azad National Institute of Technology, Bhopal, Applied Mechanics Sharma, Ashish; ShriRam College of Engineering & Management, Banmore, Mechanical Engineering,

Keywords: Free vibration, Mindlin plates, Antisymmetric, Angle-ply, Elastic edges

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1 Introduction

Laminated composite plates are being used in structural elements needed in various technology-

critical areas like aeronautics, space, automobile etc., primarily due to their low weight to

strength ratio. During the past few decades, several researchers have been presenting solutions to

bending, buckling and vibration problems of laminated plates. These many efforts have been

necessitated by complexities arising due to various reasons like geometry, boundary conditions

and material properties.

Boundary conditions often lead to some difficult problem situations. Most of the research papers,

including the most recent ones ([9] - [17]), present solutions for only classical boundary

conditions. These classical boundary conditions have, corresponding to every degree of freedom,

either the corresponding force (natural boundary conditions) or the displacement (essential

boundary condition) as a prescribed quantity [37]. The more challenging and realistic boundary

condition is the one which involves some suitable relationship between a displacement

component and the corresponding force.

These more realistic edge conditions are also being investigated by several researchers

([18] - [29]) with the help of a mathematical model known in literature as ‘elastic edges’.

The first known results of free vibration analysis of symmetrically laminated cross-ply

rectangular plates with edges having uniform elastic restraints translational as well as rotational

were presented by Liew et al. [18]. Shu and Wang [19] applied generalized differential

quadrature method for the vibration analysis of thin isotropic plates with mixed and nonuniform

boundary conditions.

Zhou [20] applied the Rayleigh-Ritz method along with static Timoshenko beam functions for

obtaining the natural frequencies of isotropic Mindlin rectangular plates. Friswell & Wang [21]

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applied finite element analysis to calculate the minimum support stiffness and optimum support

location required to raise the fundamental natural frequency of thin isotropic plates. Nallim &

Grocci [22] applied orthogonal polynomials along with Ritz method to present the free vibration

results of angle-ply symmetrically laminated composite plates with elastically restrained edges

based on Classical Laminated Plate Theory (CLPT). Ashour [23] did the vibration analysis of

isotropic plates having variable thickness in one direction with edges elastically restrained

against both rotation and translation using the finite strip transition matrix technique. Karami et

al. [24] studied the natural frequencies of moderately thick symmetric laminated plates with

elastically restrained edges using the Differential Quadrature Method (DQM). Ohya et al. [25]

presented the natural frequencies and mode shapes of the rectangular isotropic Mindlin plates

with internal columns resting on uniform elastic edge supports using the superposition method.

They achieved the compatibility between the plate and the column by requiring that the column

and plate rotations be equal.

Using one and two dimensional Fourier series expansions for the implicit spatial discretization,

Li et al. [26] presented an exact series solution for the transverse vibration of isotropic thin

rectangular plates with general elastic boundary supports. Li and Yu [27] developed an empirical

formula based on the analytical results obtained from the Rayleigh-Ritz method for predicting

natural frequencies of a thin orthotropic rectangular plate with uniformly restrained edges. Zhang

and Li [28] studied the vibration of thin isotropic rectangular plates with arbitrary non-uniform

elastic edge restraints, again, using two dimensional Fourier series expansions. Hsu [29]

presented the free vibration analysis of orthotropic rectangular plates resting on nonlinear elastic

foundations and having linearly elastic edge supports using DQM.

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Several methods have been used to study such types of problems. DQM is one of the newer

techniques being developed to study the problems whose mathematical model is a set of

differential equation(s) - linear or nonlinear, ordinary or partial. Shu [36] presents a good study

of the differential quadrature technique and its various applications in engineering problems like

the ones of Navier - Stokes equation, structural analysis and chemical engineering. Shu and

Richards [30] applied the generalized differential quadrature method to solve two-dimensional

incompressible Navier-Stokes equations. As already stated in a few of the references above, in

the context of analysis of plates, the DQM is becoming one of the commonly used techniques to

study different types of problems [31] - [35]. Karami and Malekzadeh [33] did the static and

stability analysis of arbitrary straight-sided quadrilateral thin plates using DQM. Wang and

Wang [34] studied the free vibration of thin sector plates by a new version of differential

quadrature method.

The present work attempts to extend the work of Karami et al. [24]. In comparison to the

vibration analysis of moderately thick symmetrically laminated plates described by three field

variables (w, φ x, φ y), the vibration analysis of moderately thick antisymmetric laminated plates

is described by five field variables (u, v, w, φ x, φ y), as described below. This is so because of the

fact that in the case of antisymmetric laminated plates, there is coupling between the in-plane

degrees of freedom (u, v) and the other three degrees of freedom primarily intended for modeling

the bending behavior of plates [37].

This work, thus, aims to study the free vibration problem of antisymmetrically laminated angle-

ply plates which appears to have not been studied as yet. The spatial discretization of the

governing five partial differential equations in the five field variables of the Mindlin plate theory

is done using the well established DQM. The developed formulation is validated by extensive

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convergence and comparison studies. The variation of natural frequencies with the variation of

edge stiffnesses is studied with respect to the other important parameters like thickness ratio,

aspect ratio, and moduli ratio and lamination schemes. These results are presented through a

number of graphical plots and tables.

2 Mathematical Formulation

Figure 1 shows the geometry of a laminated rectangular plate made up of orthotropic layers.

Considering the first order shear deformation theory, the displacement fields are expressed as

follows [37].

=

+=

+=

),,(),,,(*

),,(),,(),,,(*

),,(),,(),,,(*

tyxwtzyxw

tyxztyxvtzyxv

tyxztyxutzyxu

y

x

φ

φ

(1)

Equations of motion in terms of stress resultants and non-dimensional coordinates can thus be

derived using the principle of virtual work or the equilibrium considerations as, [37],

+=

∂

∂+

∂

∂2

2

12

2

0dt

dI

dt

udI

y

N

x

Nx

uv

xyxφ

α (2)

+=

∂

∂+

∂

∂

2

2

12

2

0dt

dI

dt

udI

y

N

x

Ny

uv

yxyφ

α (3)

2

2

0)(dt

wdItq

y

Q

x

Q yx =+∂

∂+

∂

∂ (4)

2

2

22

2

1dt

dI

dt

udIQ

y

M

x

Mx

uvz

xyxφ

α +=−∂

∂+

∂

∂ (5)

2

2

22

2

1dt

dI

dt

vdIQ

y

M

x

My

uvy

yxyφ

α +=−∂

∂+

∂

∂ (6)

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Here, N’s, M’s and Q’s are in-plane force resultants, moment resultants and shear force resultants

[37] as given in Figure 1. Here, )(tq (= 0, for the free vibration case considered in this paper), is

the transverse uniformly distributed load on the plate.

The inertias are defined as follows.

( )∑∫=

+=

n

k

z

z

kak

k

dzzzpIII

1

2

210),,1(,),,( (7)

Since every laminated plate considered in this work has symmetry in terms of density (ρ)

about the mid-plane (z = 0), the inertia component I1 is always zero. In equations 2 & 3, by

putting the parameter αuv = 0 or 1, the presence of in-plane inertia in the formulation is

controlled. Using the strain-displacement equations and constitutive equations given by Reddy

[37], the equations 2- 6 can be expressed in terms of the five displacement components (u, v, w,

φ x, φ y) defined in equation 1.

Method of solution

The spatial derivatives of a general function ψ(x, y) at the (i, j)th

point in the mesh (having

M divisions in x-direction and N divisions in y- direction) can be approximated using the DQM

as [31]:

∑∑= =

+

Ψ=∂∂

∂ M

k

N

lkl

q

yjl

p

xik

ij

qp

qp

CCyx 1 1

)()(ψ (8)

Where,

• )( p

xikC is the weighting coefficient of the value of ψ at the k

th point in x-direction for p

th

order derivative with respect to x calculated at the ith

point in x-direction.

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• )(q

yjlC is the weighting coefficient of the value ofψ at the l

th point in y-direction for q

th

order derivative with respect to y calculated at the jth

point in y-direction.

• kl

Ψ is the value of ψ at the (k, l)th

point in the mesh.

The weighting coefficients )( p

xikC and )(q

yjlC are evaluated using the recurrence relations given in

[30] using the cosine law for the generation of grid points as given in [33].

In the linear free vibration problem of a laminated plate, each of the five degrees of freedom at

each of the nodes of the mesh can be assumed to be varying sinusoidally with respect to time,

excluding the constrained degrees of freedom at the nodes located on boundary. Thus for

example, the displacement component w at the (i, j)th

point is assumed as,

tWw ijij ωsin= (9)

The elastic edge conditions applied in this work are as follows.

Along the edges x = 0, a:

0,0,0,0,0 =+=+=+=+=+yjxyxjxwjxvjxyujx

yx

kMkMwkQvkNukN φφ φφ (10)

Along the edges y = 0, b:

0,0,0,0,0 =+=+=+=+=+yjyxjxywjyvjyujxy

yx

kMkMwkQvkNukN φφ φφ (11)

Here, as shown in Figure 2, kij (i = u, v, w, x

φ , y

φ & j = 1, 2, 3, 4) are stiffness associated at the

respective edges (j = 1 for x = 0, j = 2 for y = 0, j = 3 for x = a & j = 4 for y = b) with the

corresponding degrees of freedom.

Finally, using equation 8 & 9, a linear algebraic equation is obtained corresponding to each of

the five degrees of freedom at each node from the equations of motion 2 - 6 or from the

boundary conditions 10 & 11 depending on the location of that node being in the interior or on

the boundary of the rectangular domain.

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All these algebraic equations are assembled to get the following matrix eigenvalue problem.

[ ] [ ] 02 =

Φ

Φ

+

Φ

Φ

−

y

x

y

x

W

V

U

KW

V

U

Mω (12)

Here, for e.g., the vector U contains the amplitudes for the displacement component u as follows,

=

−

MN

NM

U

U

U

U

U

)1(

12

11

....

....

....

(13)

The eigenvalue problem given in equation 12 is then solved using the linear software library -

GNU GSL [38].

3. Results and Discussions

The present study gives the free vibration results of moderately thick antisymmetric

angle-ply laminated rectangular plates made up of orthotropic layers. The boundary conditions

considered here are various combinations of elastic edge conditions given in equations 10 & 11.

The effects of boundary stiffnesses, moduli ratio, lamination scheme, thickness ratio and aspect

ratio are studied.

The present study is first validated by carrying out convergence study with respect to

mesh dimensions (M x N) and by comparison with the results available in the literature. The

default parameters of the laminated plates are as follows.

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,5.0,6.0,25.0,40xy yyzyxzxyyx

EGEGGEE ===== ν (14)

(15)

The moduli and Poisson’s ratio given in equation 14 and elsewhere are for a hypothetical layer

having θ = 0o and are accordingly modified [37] depending on the actual value of θ.

Angle-ply laminated plates are considered with different fiber orientations alternating between (θ

& − θ), starting at θ from layer at (z = −h/2).

The edge stiffnesses skij

given in equations 10 & 11 are prescribed with the help of non-

dimensionalized stiffness parameters skij

defined below.

For i = u, v, w:

4,2/3,1/22

3

22

3 ==== jforDakkandjforDbkkijijijij

(16)

For i =x

φ ,y

φ :

4,2/3,1/2222

==== jforDakkandjforDbkkijijijij

(17)

The boundary condition corresponding to each of the five degree of freedom at each of the four

edges can be continuously varied from the classical natural boundary condition to the classical

essential boundary condition by varying the appropriate ij

k from zero to a very large positive

value (= 1e12). Thus, for example, on the edge (x = 0 or a), as jx

kφ varies from zero to a large

positive value, the edge condition varies from the classical simply supported case to the clamped

case, with appropriate values of other stiffnesses at that edge.

The convergence study is as given in Table 1, for which,

1,30,6,2.0,1 0 =====uv

nahba αθ

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=

210210

021021

21212121

121121121121

121121121121

ee

ee

eeee

eeee

eeee

k ij (18)

The boundary conditions given in equation 18 are used as reference for later studies and the

varying skij

(mostly only in the last three rows) are mentioned appropriately. It can be seen in

these tables that convergence up to to the fourth significant digits is achieved at the mesh size of

(9 x 9).

In Table 2, the values of edge stiffnesses (ref. equation 18) are zero in the first two rows,

and, are varied as shown only in the last three rows since the studies presented by Karami et al.

[24] are limited to the three degrees of freedom problems of symmetrically laminated plates. The

comparisons show good agreement with most of the differences being less than 10 % and the

maximum difference being 4 %.

To validate the applicability of free edge conditions using the present formulation, some

results are generated and compared in Table 3 with the results given on page number 428 of [37].

As described earlier with the help of equations 10 & 11, the classical boundary conditions of

simply supported (S), Clamped (C) and Free (F) edges are easily applied by assigning the sk ij

the values comprising only of zeros and a large positive number (=1e12).

Comparison with the results given and reported by Viswanathan and Kim [14] is given

in Table 4. These results are for the fundamental frequencies of four-layered simply supported

antisymmetric angle-ply laminated plates. The present results agree very nicely with the results

reported earlier, particularly with those given by Bert and Chen [3].

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The effect of variation of the stiffness parameter ijk , while varying only the non-zero

ones in the last three rows of the equation 18, is plotted in Figure 3 for two values of the

thickness ratio h/a. It can be observed that the variations in frequencies become almost negligible

starting from the value of 104 for the stiffness parameter ijk .

Effect of variation of the stiffness parameter over the edges (x = a, y = b) only while keeping

the other two edges practically clamped (refer equation 18) is given in figure 4. The same effect

with respect to the edges (y = 0, y = b) is given the figure 5. Again, the variations in all the

three frequencies become almost negligible stating from the value of 104

for the stiffness

parameterij

k .

Effect of variation of the stiffness parameter over the edges (y = 0, y = b) only while

keeping the other two edges free (refer equation 18) is given in Figure 6. The variation pattern of

the frequencies is same as earlier. However, these results exemplify the robustness of the present

methodology with respect to the application of the free edge conditions.

The variation of first eight natural frequencies with the thickness ratio and the number of layers

is depicted in Table 5. No noticeable variation occurs beyond four numbers of layers for both the

thickness ratios considered here.

Figures 7 & 8 depict the effect of variation of the stiffness parameters (similar to the variation in

Figure 3) on the fundamental frequency of laminated square and rectangular plates of different

moduli ratio (Ex / Ey) and lamination schemes (n). Clearly, with (n = 5), the lamination scheme

leads to a symmetrically laminated plate - (30/−30)30(−30/30). Similarly, with (n = 6) the

lamination scheme leads to an antisymmetrically laminated plate - (30/−30)3. In both these

figures, the edge conditions considerably effect the spread among the fundamental frequencies of

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all the six plates. However, this spread is negligible for ( ijk < 10), and, later on the spread

becomes constant beyond ( ijk > 104).

4 Conclusions

Free vibration problem of moderately thick antisymmetric angle-ply laminated rectangular plates

having elastically supported edges has been solved using the differential quadrature method.

Both translational as well as rotational edge restraints have been considered. The formulation

facilitates for the first time a simple and efficient solution in the sense that a two dimensional

grid of 9 × 9 was found satisfactory for all the cases dealt herein. The methodology developed

proved robust and efficient for all combinations of edge conditions varying continuously from

the classical clamped to the classical free edge conditions. The method also gives accurate results

for two opposite edges being free and the other two having elastic edge conditions. Extensive

parametric studies with respect to the effect on natural frequencies have been carried out and the

results have been plotted for future reference.

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Advances in Engineering Software 39 (2008) 497-504.

[36] Chang Shu, Differential Quadrature and Its Applications in Engineering, Springer-Verlag

London Limited 2000.

[37] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis,

CRC Press, New York, 2004.

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[38] http://www.gnu.org/software/gsl/

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Table 1: Convergence study for an antisymmetric plate - (-30 o /30

o) 3,

yxEE / =30

yiEb /)/( 22

ρπω M=N

i=1 2 3 4 5 6 7 8

7

9

11

13

15

17

10.3167 13.4199 18.0681 19.5253 21.2997 26.0461 27.6114 28.6017

10.3161 13.3879 18.0291 19.4225 21.246 25.9398 27.6088 28.4968

10.316 13.3883 18.0302 19.4282 21.2465 25.9449 27.6277 28.2515

10.316 13.3882 18.0302 19.4281 21.2464 25.9448 27.627 28.2679

10.316 13.3882 18.0302 19.4281 21.2463 25.9448 27.627 28.267

10.316 13.3882 18.0302 19.4281 21.2463 25.9448 27.627 28.267

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Table 2: Comparison of non-dimensionalized frequencies yi

Eb /)/( 22 ρπω with the results

given by [24] for symmetrically laminated plates - 0),//( =−uv

αθθθ

θ ,

ijκ

Eq. 31

1.0/ =ah 2.0/ =ah

i=1 2 3 4 i=1 2 3 4

30o 10

2 [24] 0.9618 1.4370 1.8436 2.0940 0.8386 1.2226 1.4631 1.7094

† 0.9272 1.3028 1.5582 1.9111 0.8188 1.1712 1.3402 1.6554

104 1.9971 2.8675 3.7536 4.0357 1.2388 1.8494 2.1862 2.5768

1.8673 2.7465 3.7086 3.9194 1.2080 1.8164 2.1566 2.5393

108 2.0330 2.9205 3.8526 4.1240 1.2460 1.8610 2.2018 2.5956

1.9009 2.8020 3.8078 4.0132 1.2151 1.8285 2.1737 2.5588

45o 10

2 1.4064 2.0328 2.4011 2.7658 1.0743 1.5980 1.8078 2.1660

1.3346 1.9273 2.2088 2.6330 1.0489 1.5677 1.7185 2.1275

104 2.0173 3.1204 3.7002 4.3089 1.2605 1.9506 2.1661 2.6523

1.8737 2.9859 3.6421 4.1743 1.2286 1.9214 2.1305 2.6186

108 2.0280 3.1421 3.7270 4.3435 1.2628 1.9552 2.1707 2.6588

1.8840 3.0075 3.6694 4.2103 1.2309 1.9260 2.1357 2.6253

† Present results

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Table 3: Comparison of the non-dimensional fundamental frequencies yi

Ehb /)/( 2 ρω of

antisymmetric angle-ply laminated )/( θθ − square plates (h/a=0.1) with results given on page

no. 428 of [37].

BC Source 30o 45

o 60

0

SCSC Present 14.40 15.63 16.57 [37] 14.41 15.63 16.57

SFSF Present 6.95 4.77 3.33 [37] 6.95 4.76 3.33

SFSS Present 8.47 7.14 5.88 [37] 8.45 7.13 5.87

SFSC Present 8.65 7.53 6.70 [37] 8.67 7.52 6.70

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Table 4: Comparison of the non-dimensional fundamental frequencies yi

Eha /)/( 2ρω of

antisymmetric angle-ply laminated (45o/-45

o /45

o/-45

o) rectangular plats with the result given in

Table 2 of [14].

a/b=

(h/a) Source 0.2 1.0 2.0

0.1 V 8.767 18.359 34.305 G 4.930 18.060 31.280 R 8.724 18.609 34.247 B 8.664 18.46 34.87 P 8.664 18.463 34.874 0.02 V 12.001 24.348 52.677 G 9.840 23.910 53.680 R 9.816 24.343 53.989 B 9.51 23.24 52.59 P 9.507 23.237 52.288

V: [14], G: [8] B: [3], R: [4] P:Present

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Table 5: Effect of variation of thickness ratio (h/a) and number of layers (n) on the first eight

non dimensional natural frequencies22

22/)/( Dhb

iρπω .

h/a n i=1 2 3 4 5 6 7 8

0.05 2 1.75673 2.49768 2.86093 3.96023 4.23616 6.15947 6.32354 6.94205

4 1.82356 2.57433 3.21925 4.40313 4.75815 7.4695 7.94713 8.2708

6 1.82926 2.58228 3.26449 4.45945 4.8225 7.62146 8.16228 8.44546

8 1.83184 2.586 3.28606 4.48641 4.85313 7.69467 8.26406 8.52934

0.2 2 1.5055 2.18138 2.40692 3.20542 3.23897 3.97584 4.16127 4.586

4 1.55675 2.25138 2.57831 3.36592 3.47652 4.28786 4.34168 4.86433

6 1.56105 2.25801 2.59496 3.38027 3.50312 4.31195 4.35708 4.88649

8 1.56299 2.26104 2.60257 3.38677 3.51557 4.32266 4.364 4.89636

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Figure 1: Geometry of the problem. 120x77mm (300 x 300 DPI)

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Figure 2: Stiffness parameters for the application of elastic edge conditions as given in equations 10 & 11.

97x77mm (300 x 300 DPI)

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Figure 3: Effect of variation of the stiffness parameter (varying only the non-zero ones in the last three rows of the equation 18).

177x138mm (600 x 600 DPI)

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Figure 4: Effect of variation of the stiffness parameter over the edges (x = a, y = b) only while keeping the other two edges practically clamped (refer equation 18).

138x108mm (300 x 300 DPI)

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Figure 5: Effect of variation of the stiffness parameter over the edges (y = 0, y = b) only while keeping the other two edges practically clamped (refer equation 18).

138x108mm (300 x 300 DPI)

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Figure 6: Effect of variation of the stiffness parameter over the edges (y = 0, y = b) only while keeping the other two edges free (refer equation 18).

174x138mm (600 x 600 DPI)

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Figure 7:Effect of variation of the stiffness parameters on the fundamental frequency of laminated square plates of different moduli ratio(Ex/Ey) and

lamination schemes (n) 179x138mm (600 x 600 DPI)

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Figure 8: Effect of variation of the stiffness parameters on the fundamental frequency of laminated rectangular plates (a/b = 2) of different moduli ratio (Ex/Ey) and lamination schemes (n).

179x138mm (600 x 600 DPI)

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