VIBRATION ANALYSIS O F LAMINATED COMPOSITE ......S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U....

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 8, Issue 7, July 2017, pp. 414–427, Article ID: IJMET_08_07_048

Available online at http://iaeme.com/Home/issue/IJMET?Volume=8&Issue=7

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

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VIBRATION ANALYSIS OF LAMINATED

COMPOSITE PLATES USING LAYERWISE

HIGHER ORDER SHEAR DEFORMATION

THEORY

S. Lokesh

Department of Mechanical Engineering,

Institute of Aeronautical Engineering, Hyderabad, India

Dr. CH. Lakshmi Tulasi


Chadalawada Ramanamma Engineering College, Tirupati, India

T. Monica


MLR Institute of Technology, Hyderabad, India

U. Pranavi


Vardhaman College of Engineering, Hyderabad, India

ABSTRACT

This paper represents the investigation on the response of a symmetric composite

laminated plate. As we know that vibration and composite material are two main

growing research topics now a days. Almost all the structural components subjected

to dynamic loading in their working life and vibration affects working life of the

structure. Layerwise Higher Order Shear Deformation Theory (HSDT) is used to

predict of the free vibration characteristics of laminated composite plates. The displacements of each layer are expressed in terms of Layerwise HSDT functions of

the thickness. The displacement field of present theory contains nine unknowns, as in

the higher order shear deformation theory Navier’s solution method is used for

finding the analytical solutions. Non-dimensional fundamental frequencies of simply

supported cross-ply and anti-symmetric angle-ply laminated composite plates have

been obtained by using Layerwise HSDT. It is shown that the present Layerwise HSDT

can provide accurate results. The accuracy of the present theory is ascertained by

comparing it with various available results in the literature. The results show that the

present model performs better than all the existing higher order shear deformation

theories.

S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi


Key words: Laminated composite plate, Vibration Analysis, layerwise HSDT, Cross-

ply, Angle-ply.

Cite this Article: S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi.

Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order

Shear Deformation Theory. International Journal of Mechanical Engineering and

Technology, 8(7), 2017, pp. 414–427.

http://iaeme.com/Home/issue/IJMET?Volume=8&Issue=7

1. INTRODUCTION

Laminated composites with continuous fibres are widely being used in various engineering

fields such as aeronautical and aerospace industry, marine, aviation, civil, sport, as well as in

other fields of modern technology and other applications. They are preferred due to their

characteristics like high stiffness to weight ratio, excellent fatigue strength, high energy

absorption, self damping capacity, and good resistance to corrosive agents, capable of being

engineered according to requirements. It is challenging task to find the accurate prediction of

the response characteristics of composite structures. Hence, it is necessary to analyze the free

vibration characteristics of laminated composite plates. Several theories have been developed

to analyze laminated composite plates. An exhaustive survey on the literature regarding the

vibration characteristics of laminated composite plates has been carried out.

Yu (1962), studied the propagation of plane harmonic waves in sandwich plates where no

limitation was imposed upon the magnitude of the ratios between the thickness, material

densities and elastic constants of the core and the facings. He applied Mindlin's bending

theory of plates to all layers of the sandwich and obtained extremely compacted equations of

motion. And he further investigated on vibration of sandwich plates including viscous

damping and large deflections. He accommodated the theory of transverse shear deformation

and rotary inertia effects, which is important while dealing with conventional sandwiches.

Noor AK (1973), presented complete list of FSDTs and HSDTs for the static, free

vibration and buckling analysis of laminate composites. He presented exact three dimensional

elasticity solutions for the free vibration of isotropic, orthotropic and anisotropic composite

laminate plates which serve as benchmark solutions for comparison by many researchers.

Reddy J.N (1984), proposed higher-order shear deformation theory of laminated

composite plates which contains the same dependent unknowns as in the first-order shear

deformation theory of Whitney and Pagano but accounts for parabolic distribution of the

transverse shear strains through the thickness of the plate. He obtained exact closed-form

solutions of symmetric cross-ply laminates and compared the results with three-dimensional

elasticity solutions and first-order shear deformation theory solutions.

Mallikarjuna, Kant T (1989), presented a simple Co finite element formulation and

solutions using a set of higher order displacement models for the free vibration analysis of

general laminated composite and sandwich plates. He also presented solutions for the free

vibrational analysis of general laminated composite and sandwich beams.

Kansa (1990), introduced the concept of solving partial differential equations (PDEs) by

an unsymmetric RBF collocation method based upon the multiquadratics interpolation

functions. He used alternative methods to the finite element methods for the analysis of plates,

such as the meshless methods based on radial basis functions, which is attractive due to the

absence of a mesh and the ease of collocation methods.

Noiser et al. (1993), studied Reddy's layerwise theory which is combined with a wave

propagation approach enabled with all the conventional boundary conditions. Using this

method they investigated the effect of shell parameters on natural frequencies under various

Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order Shear

Deformation Theory


boundary conditions. One of the major advantages of the layer-wise theory is the possibility it

provides for analyzing thick laminates and also, interlamina stresses (in forced vibrations)

with high accuracy.

E. Carrera (1998), presented the dynamic analysis of multilayered plates using layer-wise

mixed theories with respect to existing two-dimensional theories. They have employed

Reissner’s mixed variational equation to derive the differential equations, in terms of the

introduced stress and displacement variables, that govern the dynamic equilibrium and

compatibility of each layer.

Ganapathi and Makhecha (2001), proposed an improved ZIGT for free vibration analysis

of laminated composite plates in which the C0 interpolation functions are only required

during their finite element implementation. Compared to the previous ZIGTs, it is more

convenient to develop the simple conforming quadrilateral elements. They presented an eight-

node quadrilateral element based on the proposed ZIGT by incorporating the terms associated

with the consistent mass matrix, for the numerical study of the free vibration behaviours of

laminated composite and sandwich plates.

Matsunaga (2002) developed a higher-order theory based on a complete power series

expansion of the displacement field in the thickness coordinate. He presented closed-form

solutions for the vibration of simply supported cross-ply laminated plates and demonstrated

that, for expansion orders higher than three, a noticeable improvement is obtained in

comparison with TSDT.

A.R. Setoodeh, G. Karami (2003), proposed a generalized layer-wise laminated plate

theory based on a three-dimensional approach for static, vibration, and buckling analysis of

fibre reinforced laminated composite plates.

Liu ML, To CWS (2003), developed the computational models for the free vibration and

damping analysis based on the FSDT and HSDT, relatively few models were developed based

on the Layerwise theories. The computational model developed based on the layer wise

theories include the 18-node, three-dimensional higher-order mixed model for free vibration

analysis of multi-layered thick composite plates, in which the continuity of the transverse

stress and the displacement fields were enforced through the thickness of laminated composite

plate.

Latheswary et.al, (2004), investigated the static and free vibration analysis of moderately

thick laminated composite plates using a 4-node finite element formulation based on higher-

order shear deformation theory, and the transient analysis of layered anisotropic plates using a

shear deformable 9-noded Lagrangian element-based on first-order shear deformation theory.

Akhras G, Li W (2005), studied free linear vibration behaviour of laminated composite

rectangular plates is by using moving least squares differential quadrature procedure, based on

the first order shear deformation. He developed a spline finite strip method for static and free

linear vibration analysis of composite square plates using Reddy’s higher-order shear

deformation theory.

Wu Z, Chen WZ (2006), extended the global–local higher-order theory to predict natural

frequencies of laminated composite and sandwich plates. These theories can predict more

accurate natural frequencies of laminated composite and sandwich plate, and the number of

unknowns involved in these models is independent of the number of layers.

M.Cetkovic, Dj. Vuksanovic (2008), used generalize layerwise theory (GLPT) of Reddy

to study bending, vibration and buckling of laminated composite and sandwich plates. The

theory assumes transverse variation of the in-plane displacement components in terms of one-

dimensional linear Lagrangian finite elements. Transverse shear stresses satisfy Hook’s law,

http://appliedmechanics.asmedigitalcollection.asme.org.sci-hub.org/solr/searchresults.aspx?author=E.+Carrera&q=E.+Carrera



3D equilibrium equations, inter laminar continuity and traction free boundary conditions and

have quadratic variation within each layer of the laminate.

Ferreira et.al, (2008), used layer wise theory based on Mindlin’s first-order shear

deformation theory in each layer to analyze the vibration and static responses of composite

and sandwich plates. Flexural analysis based on multiquadric radial basis function is also

done. The RBFs and wavelet collocation are applied for the static and vibration analysis of

laminated composite and sandwich plates. The results compared are more efficient.

Zhang and Wang (2009) presented a layer wise B-spline finite strip method with

consideration of delamination kinematics to study the vibration and buckling behaviour of

delaminated composite laminates.

Wook and Reddy (2010) developed a finite element model based on LWT of Reddy for

the analysis of delamination in cross-ply laminated beams which was able to capture accurate

local stress fields and the strain energy release rates.

Nguyen-Thoi et.al, (2013) proposed the integrated strain smoothing technique into the

FEM to create a series of smoothed FEM (SFEM). They further investigated S-FEM models

and applied to various problems such as plates and shells, piezoelectricity, fracture mechanics,

visco-elastoplasticity, limit and shakedown, and some other applications etc., and formulated

a edge-based smoothed stabilized discrete shear gap method based on the first-order shear

deformation theory (FSDT) for static, and free vibration analysis of isotropic Mindlin plates

by incorporating the ES-FEM with the original DSG3 element.

J.L Mantari, C. Guedes Soares (2013) developed a new higher order shear deformation

theory for elastic, composite and sandwich plates and shells. He introduced a generalized 5

degrees of freedom HSDT to study the bending and free vibration of plates and shells and

presented layerwise finite element formulation of the developed higher-order shear

deformation theory for the flexure of thick multilayered plates. He developed his work by

finding an analytical solution to the static analysis of functionally graded plates (FGPs) by

using a new trigonometric higher-order theory in which the stretching effect had been

included.

Marjanovic et.al, (2013), presented the structural analysis of laminated composite and

sandwich plates and observed the different forms of damage. They observed that

Delamination is the most common type of damage for laminated composite plates. They

found it is of the great importance that the bond between the face sheets and soft-core in

sandwich plate remain intact for the panel to perform on the appropriate level, so the presence

of delamination is of the great danger for the sandwich plates. Due to the presence of these,

often microscopic, structural defects, loading capacity of the plate is reduced severely.

After the thorough literature survey it is found that many models are developed to study

the characteristics of laminated composite plates. They are based on different assumptions

concerning the strain, stress, displacement fields inside the plate. In the present work an

attempt is made to find the Vibration characteristics of a laminated composite plate

completely by means of analytical procedure using the Layerwise HSDT.

2. VIBRATION ANALYSIS OF LAMINATED COMPOSITE PLATES

BASED ON LAYERWISE HSDT

The composite materials have found wide use in many weight sensitive structures such as air

craft, and missile structural components because of their high absorbing capacity for

vibrations. To use them efficiently good understanding of structural and dynamical behaviour

and also an accurate knowledge of the deformation characteristics, stress distribution and

natural frequencies under various load conditions are needed.


Deformation Theory


3. NAVIER SOLUTION USING HIGHER-ORDER DISPLACEMENT

MODEL BASED ON LAYERWISE THEORY

In the Navier method the displacements are expanded in a double Fourier series in terms of

unknown parameters. The choice of the trigonometric functions in the series is restricted to

those which satisfy the boundary conditions of the problem. Substitution of the displacement

expansions in the governing equations result in an invertible set of algebraic equations among

the parameters of the displacement expansion.

The simply supported boundary conditions for the higher-order shear deformation theory

are:

At edges x = 0 and x = a

v0 = 0, wo = 0, y = 0, Mx = 0, v0* = 0, y* = 0, Mx* = 0, Nx = 0, Nx* = 0, (1)(a)

At edges y = 0 and y = b

u0= 0, wo = 0, x = 0, My = 0, u0* = 0, x* = 0, My* = 0, Ny = 0, Ny* = 0, (1)(b)

The simply supported boundary conditions shown in Eq. (1) are considered for solutions

of laminated composite plates using displacement model. The boundary conditions in Eq. 4.1

are satisfied as:

yxtUtyxu mnnm

sincos)(),,(11

0

=

=

=

(2) (a)

yxtVtyxv mnnm

cossin)(),,(11

0

=

=

=

(2)(b)

yxtWtyxw mnnm

sinsin)(),,(11

0

=

=

=

(2)(c)

yxtXtyx mnnm

x sincos)(),,(11

=

=

=

(2)(d)

yxtYtyx mnnm

y cossin)(),,(11

=

=

=

(2)(e)

yxtUtyxu mnnm

sincos)(),,( *

11

*

0

=

=

=

(2)(f)

yxtVtyxV mnnm

o cossin)(),,(*

11

*

=

=

=

(2)(g)

yxtXtyx mnnm

x sincos)(),,(*

11

*

=

=

=

(2)(h)

The mechanical loads are also expanded in double Fourier sine series as:

yxtQtyxq mnnm

sinsin)(),,(11

=

=

=

(2)(k)

Where

Qmn (z, t) = dxdyyxtyxqab

ba

sinsin),,(4

00

….. (2)(l)



Where = a

m

and = b

n

Navier solution exists only if the following terms are zero

656361565452454442363432252321161412

65646261565346433534323126231613

65646261565346433534323126231613

65646261565346433534323126231613

L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L L,

,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,

L

DDDDDDDDDDDDDDDD

BBBBBBBBBBBBBBBB

AAAAAAAAAAAAAAAA

For such laminated the co-efficients (Umn, Vmn, Wmn, Xmn, Ymn, **** ,,, mnmnmnmn YXVU ) of

the Navier solution can be calculated from

[ S11 S12 S13 S14 S15 S16 S17 S18 S19

S21 S22 S23 S24 S25 S26 S27 S28 S29

S31 S32 S33 S34 S35 S36 S37 S38 S39

S41 S42 S43 S44 S45 S46 S47 S48 S49

S51 S52 S53 S54 S55 S56 S57 S58 S59

S61 S62 S63 S64 S65 S66 S67 S68 S69

S71 S72 S73 S74 S75 S76 S77 S78 S79

S81 S82 S83 S84 S85 S86 S87 S88 S99

S91 S92 S93 S94 S95 S96 S97 S98 S99 ]

{

𝑈𝑚

𝑉𝑚

𝑊𝑚

𝑋𝑚

𝑌𝑚

𝑈𝑚∗

𝑉𝑚∗

𝑋𝑚∗

𝑌𝑚∗ }

+

[ 𝑚11 00𝑚410𝑚610𝑚810

0𝑚2200

𝑚52 0𝑚720𝑚92

00𝑚33000000

𝑚1400𝑚440𝑚640𝑚840

0𝑚2500𝑚550𝑚750𝑚95

𝑚1600𝑚460𝑚660𝑚860

0𝑚2700𝑚570𝑚770𝑚97

𝑚1800𝑚480𝑚680𝑚880

0𝑚2900𝑚590𝑚790𝑚99]

{

𝑈𝑚𝑉𝑚𝑊𝑚𝑋𝑚𝑌𝑚𝑈𝑚∗

𝑉𝑚∗

𝑋𝑚∗

𝑌𝑚∗ }

=

{

00𝑄𝑚000000 }

..... (3)

For free vibration Eq. 4 reduces to the Eigen value problem as

}0{}]){[]([ 2 =− MS (4)

Where = (Umn, Vmn, Wmn, Xmn, Ymn,)t

For a non trivial solution, , the determent of the coefficient matrix in Eq. 5 should

be zero, which yields the characteristic equation :

([S] – [M]) = 0 (5)

Where = 2 is the Eigen value.

0}{


Deformation Theory


The elements Sij (i=1,2….9 and j=1,2…9) are given and solutions for each m, n gives Umn,

Vmn, Wmn, Xmn, Ymn, **** ,,, mnmnmnmn YXVU , which are used to compute xooo wvu ,,, ,

****,,,, yxooy vu

4. RESULTS AND DISCUSSION

After the convergence study the accuracy of the developed theory is validated with available

theories. Effect of different parameters on the vibration behaviour of laminated composite

plate is discussed. The effect of Side-to-thickness ratio, aspect ratio and modulus ratio of

laminated composite plates with non-dimensional fundamental frequency are studied.

Material: Graphite Epoxy

Young's Modulus: E1=25Gpa, E2=1Gpa

Shear Modulus: G12=G13=0.5Gpa, G23=0.2Gpa

Poisson's Ratio: 12 = 23 = 13 = 0.25

The numerical results obtained from the vibration analysis are tabulated in the Table 1 to 6

• The Non-Dimensional fundamental frequencies for three layered cross-ply laminated square plate with different modulus ratios are tabulated in Table 1. It is evident from Table 1 that the

present method gives better results for higher modulus ratios. It is observed that the global

average error is 3.8%

• The Non-Dimensional fundamental frequencies for symmetric cross-ply laminated square plate with different modulus ratios are tabulated in Table 2. The results obtained by proposed

theory agree reasonably well with the results obtained by FSDT (Liew et.al,) (2003), HSDT

(Phan and Reddy) (1985), ELS (Noor) (1973). It is observed that the maximum percentage

error is 4%

• The Non-Dimensional fundamental frequencies for three layered cross-ply laminated square plate with different side to thickness ratios are tabulated in Table 3. This shows that the

present results are in good agreement and gives better results at higher thickness ratios. And

for moderate thickness ratios the global average error increases to 5%

• The Non-Dimensional fundamental frequencies for symmetric cross-ply laminated square plate with different side to thickness ratios are tabulated in Table 4. The results obtained by

proposed theory are in good agreement with the results obtained by Carrera (1998). The Non-

• Dimensional fundamental frequencies increase with increase in thickness ratios.

• The Non-Dimensional fundamental frequencies for anti-symmetric angle-ply laminated square plate with different side to thickness ratios are tabulated in Table 5. This shows that present

results are in good agreement and gives better results at higher thickness.

• The Non-Dimensional fundamental frequencies for symmetric cross-ply laminated square plate with different aspect ratios are tabulated in Table 6. The present results are in acceptable

limit. The maximum percentage error noticed is 5%



Table 1 Non-Dimensionalized fundamental frequencies for three layered cross-ply laminated square

plate with different modulus ratios

E1/E2

Source Frequencies (ω)

3

Present Model

Bose and Reddy (1998)

Matsunaga (2000)

Kant and Manjunatha (1988)

Noor (1973)

2.6375

2.6286

2.6276

2.6285

2.6474

10

Present Model


Matsunaga (2000)


Noor (1973)

3.2753

3.2679

3.2664

3.2678

3.2841

20

Present Model


Matsunaga (2000)


Noor (1973)

3.7604

3.7011

3.6967

3.7005

3.8241

30

Present Model


Matsunaga (2000)


Noor (1973)

3.9812

3.9456

3.9362

3.9438

4.1089

40

Present Model


Matsunaga (2000)


Noor (1973)

4.1978

4.1150

4.0951

4.1074

4.3006

Table 2 Non-Dimensionalized fundamental frequencies for symmetric cross-ply laminated square

plate with different modulus ratios

E1/E2 Source Frequencies (ω)

3

Present Model

Liew et.al, (2003)

Xiang and Wang (2009)

Phan and Reddy (1985)

Noor (1973)

2.6482

-

-

2.6238

2.6726

10

Present Model

Liew et.al, (2003)



Noor (1973)

3.3262

3.3196

3.3684

3.3087

3.2841

20

Present Model

Liew et.al, (2003)



Noor (1973)

3.8395

3.8272

3.8684

3.8105

3.8241

30

Present Model

Liew et.al, (2003)



Noor (1973)

4.1376

4.1308

4.1664

4.1088

4.1088

40

Present Model

Liew et.al, (2003)



Noor (1973)

4.3381

4.3420

4.3752

4.3148

4.3008


Deformation Theory


Table 3 Non-Dimensionalized fundamental frequencies for three layered cross-ply laminated square

plate with different side to thickness ratios

a/h

Source E1/E2

3 10 20 30 40

5 Present Model

Chalak et.al,(2013)

Aagaah et.al,(2006)

8.2823

8.0309

8.9350

8.9150

8.6481

9.1730

9.6861

9.1773

10.1950

3.6712

3.5549

-

3.3415

3.1796

-

10 Present Model

Chalak et.al,(2013)

Aagaah et.al,(2006)

11.8529

11.5158

12.1900

12.8642

12.3054

14.8400

15.1214

13.9952

17.3650

4.8513

4.6588

-

4.5312

4.2430

-

20 Present Model

Chalak et.al,(2013)

Aagaah et.al,(2006)

14.012

13.9638

14.0040

15.986

14.7874

17.1880

19.8512

19.0078

23.2390

5.3512

5.2296

-

4.9412

4.7679

-

50 Present Model

Chalak et.al,(2013)

Aagaah et.al,(2006)

14.9826

15.0688

14.9060

17.2415

15.9016

18.6130

22.5482

22.4593

30.1200

5.8614

5.4465

-

5.1542

4.954

-

100 Present Model

Chalak et.al,(2013)

Aagaah et.al,(2006)

15.2210

15.2529

15.0410

17.930

16.0871

18.7910

23.745

23.1693

33.8740

5.6214

5.4822

-

5.1241

4.9892

-


plate with different side to thickness ratios

a/h Source E1/E2 5 Present Model

Carrera (1998)

10.9874

10.8413

10 Present Model

Carrera (1998)

15.298

15.150

20 Present Model

Carrera (1998)

17.784

17.626

50 Present Model

Carrera (1998)

18.921

18.600

100 Present Model

Carrera (1998)

18.981

18.753

Table 5 Non-Dimensionalized fundamental frequencies for anti- symmetric angle-ply laminated

square plate with different side to thickness ratios

a/h Source Frequencies (ω)

5 Present Model

Matsunga (2001)

Reddy (1984)

12.8265

12.6810

12.9719

10 Present Model

Matsunga (2001)

Reddy (1984)

19.1756

19.0832

19.2659

20 Present Model

Matsunga (2001)

Reddy (1984)

23.2016

23.1645

23.2388

50 Present Model

Matsunga (2001)

Reddy (1984)

24.7476

24.5906

24.9046

100 Present Model

Matsunga (2001)

Reddy (1984)

24.6727

24.1711

25.1744




plate with different aspect ratios

Aspect ratios Source Frequencies (ω)

2 Present Model

Desai et.al, (2003)

Cho et.al, (1991)

Reddy and Phan HSPDT (1985)

5.612

5.315

5.923

5.576

5 Present Model

Desai et.al, (2003)

Cho et.al, (1991)


10.754

10.682

10.673

10.989

10 Present Model

Desai et.al, (2003)

Cho et.al, (1991)


15.154

15.069

15.066

15.270

20 Present Model

Desai et.al, (2003)

Cho et.al, (1991)


17.624

17.636

17.535

17.668

25 Present Model

Desai et.al, (2003)

Cho et.al, (1991)


18.061

18.067

18.054

18.050

50 Present Model

Desai et.al, (2003)

Cho et.al, (1991)


18.644

18.670

18.670

18.606

100 Present Model

Desai et.al, (2003)

Cho et.al, (1991)


18.812

18.835

18.835

18.755

Figure 1 Non-Dimensionalized fundamental frequency (ω) Vs Modulus ratio (E1/E2) for three layered

cross-ply laminated square plate

2.5

2.75

3

3.25

3.5

3.75

4

4.25

4.5

4.75

5

0 10 20 30 40 50

No

n-D

ime

nsi

on

al f

un

dam

en

tal

fre

qu

en

cie

s (ω

)

Modulus Ratio (E1/E2)

Present Model


Matsunaga (2000)

Kant and Manjunatha(1988)

Noor (1973)


Deformation Theory


Figure 2 Non-Dimensionalized fundamental frequency (ω) Vs Modulus ratio (E1/E2) for symmetric

cross-ply laminated square plate

Figure 3 Non-Dimensionalized fundamental frequency (ω) Vs side to thickness ratio (a/h) for three

layered cross-ply laminated square plate

Figure 4 Non-Dimensionalized fundamental frequency (ω) Vs side to thickness ratio (a/h) for

symmetric cross-ply laminated square plate

2.5

2.75

3

3.25

3.5

3.75

4

4.25

4.5

4.75

5

0 10 20 30 40 50

Non

-Dim

ensio

nal

fundam

en

tal fr

equen

cie

s

(ω)

Modulus Ratio (E1/E2)

Present Model

Liew et.al, (2003)



Noor (1973)

0

5

10

15

20

25

0 20 40 60 80 100 120

Non-D

imensio

nal

Fundam

enta

l fr

equency (ω

)

Side to thickness ratio a/h

E1/E2=3

E1/E2=10

E1/E2=20

E1/E2=30

E1/E2=40

10

12

14

16

18

20

0 20 40 60 80 100 120

Non-D

imensio

nal Fundam

enta

l fr

equencyω

Side to thickness ratio a/h

Present Model

Carrera (1998)



Figure 5 Non-Dimensionalized fundamental frequency (ω) Vs side to thickness ratio (a/h) for anti-

symmetric angle-ply laminated square plate

Figure 6 Non-Dimensionalized fundamental frequency (ω) Vs Aspect ratio (a/b) for symmetric cross-

ply laminated square plate

5. CONCLUSIONS

An investigation on the response of a symmetric composite laminated plate is conducted.

Layerwise HSDT is used to predict of the free vibration characteristics of laminated

composite plates.

The displacements of each layer are expressed in terms of Layerwise HSDT functions of

the thickness. The displacement field of present theory contains nine unknowns, as in the

higher order shear deformation theory Navier’s solution method is used for finding the

analytical solutions. Non-dimensional fundamental frequencies of simply supported cross-ply

and anti-symmetric angle-ply laminated composite plates have been obtained by using

Layerwise HSDT. It is shown that the present Layerwise HSDT can provide accurate results.

The accuracy of the present theory is ascertained by comparing it with various available

results in the literature. The results show that the present model performs better than all the

existing higher order shear deformation theories.

From the study following conclusions are drawn.

456789

1011121314151617181920

0 20 40 60 80 100 120No

n-D

ime

nsi

on

al f

un

dam

en

tal

fre

qu

en

cie

s (

ω)

Aspect Ratio (a/b)

Prsent Model

Desai et.al, (2003)

Cho et.al (1991)

Reddy and Phan HSPDT(1985)

10

12.5

15

17.5

20

22.5

25

27.5

30

0 1 2 3 4 5 6

No

n-D

ime

nsi

on

al F

un

dam

en

tal

fre

qu

en

cy (

ω)

Side to thickness ratio (a/h)

Present Model

Matsunga (2001)

Reddy (1984)


Deformation Theory


• Non dimensional fundamental frequencies are increasing with the increase of laminate plate modulus ratio.

• The effect of shear deformation on natural frequencies decreases with the increasing side to thickness ratio.

• The aspect ratio increases, the non-dimensional fundamental frequency increases. This is because of increase of stiffness of the plate.

• The results predicted by present model are almost identical for all modes of vibration of thin to thick plates.

• Present theory can accurately predict static and dynamic behaviour of laminated composite plates for a greater range of problems with few elements.

REFERENCES

[1] Aagaah MR, Mahinfalah M, Jazar GN. Natural frequencies of laminated composite plates using third order shear deformation theory. Compos Struct; 72:273–9, 2006.

[2] Akhras G, Li W. Static and free vibration analysis of composite plates using spline finite strips with higher-order shear deformation. Compos: Part B; 36:496–503, 2005.

[3] A.R. Setoodeh, G. Karami. Static, free vibration and buckling analysis of anisotropic thick laminated composite plates on distributed and point elastic supports using a 3-D layer-

wise FEM Engineering Structures 26 211–220, 2003.

[4] Bose P, Reddy JN. Analysis of composite plates using various plate theories. Part 2: finite element model and numerical results. Struct Eng Mech;6:727–46, 1998.

[5] Carrera E. Layer-wise mixed models for accurate vibrations analysis of multilayered plates. J Appl Mech;65:820±8, 1998.

[6] Chalak HD, Chakrabarti A, Iqbal MA, Sheikh AH. Free vibration analysis of laminated soft core sandwich plates. J Vib Acoust;135:011013, 2013.

[7] Cho KN, Bert CW, Striz AG. Free vibration of laminated rectangular plates analyzed by higher order individual-layer theory. J Sound Vib;145(3):429–42, 1991.

[8] Desai YM, Ramtekkar GS, Shah AH. Dynamic analysis of laminated composite plates using a layer-wise mixed finite element model. J Compos Struct;59:237–49, 2003.

[9] E. Carrera Historical review of zig-zag theories for multilayered plates and shells, Appl. Mech. Rev., vol. 56, pp. 287–308, 2003.

[10] E.J. Kansa, Multiquadric—A scattered data approximation scheme with applications to computational fluid dynamics. I: Surface approximations and partial derivative estimates,

Comput. Math. Applic., vol. 19, no. 8/9, pp. 127–145, 1990.

[11] Ferreira AJM, Fasshauer GE, Batra RC, Rodrigues JD. Static deformations and vibration analysis of composites and sandwich plates using a layerwise theory and RBF-PS

discretizations with optimal shape parameter Compos Struct;86;328-43, 2008.

[12] Ganapathi M, Makhecha DP. Free vibration analysis of multi-layered composite laminates based on an accurate higher-order theory. Compos Part B Eng;32:535–43, 2001.

[13] J.L. Mantari and C. Guedes Soares “Generalized layerwise HSDT Tand finite element formulation for symmetric laminated and sandwich composite plates”, Composite

Structures, 2013.

[14] Kant T, Manjunatha BS. An un-symmetric FRC laminate C3 finite element model with 12 degrees of freedom per node. Eng Comput;5:300–8, 1988 .

[15] Latheswary S, Valsarajan KV, Sadasiva Rao YVK. Dynamic response of moderately thick composite plates. J Sound Vib;270(1–2):417–26, 2004.



[16] Liew KM, Huang YQ, Reddy JN. Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method. Comput

Method Appl M;192:2203-22, 2003.

[17] Liu ML, To CWS. Free vibration analysis of laminated composite shell structures using hybrid strain based layerwise finite elements. Finite Elem Anal Des;40(1):83–120, 2003.

[18] Mallikarjuna, Kant T. Free vibration of symmetrical laminated plates using a higher order theory with finite element technique.Int J Numer Meth Eng;28:1875-89, 1989.

[19] Marjanovic M, Vuksanovic Dj. Linear analysis of single delamination in laminated composite plate using layerwise plate theory. In: Maksimovic S, Igic T, Trisovic N,

editors. Proceedings of the 4th international congress of serbian society of mechanics.

Belgrade, Serbia: Serbian Society of Mechanics; p. 443–8, 2013.

[20] Matsunaga H. Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos Struct;48:231–44, 2000.

[21] Matsunaga H. Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses. Int J Mech Sci;43:1925–44, 2001.

[22] Matsunaga, H.: Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses. Thin Wall Struct.40, 557–571, 2002.

[23] M.Cetkovic´, Dj. Vuksanovic Bending, free vibrations and buckling of laminated composite and sandwich plates using a layerwise displacement model Composite

Structures 88 219–227, 2009.

[24] Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Hoang S, Nguyen-Xuan H.An edge-based smoothed three-node Mindlin plate element (ES-MIN3) for static and free vibration

analyses of plates. KSCE J Civil Eng 2013. in press

[25] Noiser A, Kapania RK, Reddy JN. Free vibration analysis of laminated plates using a layerwise theory. AIAA J;31(2):2335–46, 1993.

[26] Noor A.K Free vibrations of multilayered composite plates. AIAA J;11:1038-9, 1973.

[27] Reddy JN. A simple higher-order theory for laminated composite plates. J ApplMech;51:745–52, 1984.

[28] Reddy JN, Phan ND. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J Sound Vib;98(2):157–70, 1985.

[29] Phan ND, Reddy JN. Analyses of laminated composite plates using a higher order deformation theory. Int J Numer Methods Eng;21(12):2201–19, 1985.

[30] Wang S, Zhang Y. Vibration analysis of rectangular composite laminated plates using layerwise B-spline finite strip method. Compos Struct;68(3):349–58, 2005.

[31] Wook Jin Na and J. N. Reddy, "Multiscale Analysis of Transverse Cracking in Cross-Ply Laminated Beams using the Layerwise Theory," Journal of Solid Mechanics, Vol. 2, No.

1, pp. 1-18, 2010.

[32] Wu Z, Chen WJ. Free vibration of laminated composite and sandwich plates using global–local higher-order theory. J Sound Vib;298:333–49, 2006.

[33] Yu, Y. Y., Damping of flexural vibrations of sandwich plates. J. Aerospace Sci., 29 790-803, 1962.

[34] Prof. Dr. Pankaj Sharma and Dr. M. P. Singh, Experimental and Theortical Study of The Thermal Performance of Heat Pipe Heat Exchanger. International Journal of Mechanical

Engineering and Technology, 7(3), 2016, pp. 112–128

[35] Sameer G. Patel and Gaurang R Vesmawala, Experimental Study on Concrete Box Girder Bridge under Traffic Induced Vibration. International Journal of Civil Engineering and

Technology, 8(1), 2017, pp. 504–511.

VIBRATION ANALYSIS O F LAMINATED COMPOSITE ......S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U....

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