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http://iaeme.com/Home/journal/IJMET 414 [email protected]
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
© IAEME Publication Scopus Indexed
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
Department of Mechanical Engineering,
Chadalawada Ramanamma Engineering College, Tirupati, India
T. Monica
Department of Mechanical Engineering,
MLR Institute of Technology, Hyderabad, India
U. Pranavi
Department of Mechanical Engineering,
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
http://iaeme.com/Home/journal/IJMET 415 [email protected]
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
http://iaeme.com/Home/journal/IJMET 416 [email protected]
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
S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi
http://iaeme.com/Home/journal/IJMET 417 [email protected]
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.
Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order Shear
Deformation Theory
http://iaeme.com/Home/journal/IJMET 418 [email protected]
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)
S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi
http://iaeme.com/Home/journal/IJMET 419 [email protected]
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}{
Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order Shear
Deformation Theory
http://iaeme.com/Home/journal/IJMET 420 [email protected]
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%
S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi
http://iaeme.com/Home/journal/IJMET 421 [email protected]
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
Bose and Reddy (1998)
Matsunaga (2000)
Kant and Manjunatha (1988)
Noor (1973)
3.2753
3.2679
3.2664
3.2678
3.2841
20
Present Model
Bose and Reddy (1998)
Matsunaga (2000)
Kant and Manjunatha (1988)
Noor (1973)
3.7604
3.7011
3.6967
3.7005
3.8241
30
Present Model
Bose and Reddy (1998)
Matsunaga (2000)
Kant and Manjunatha (1988)
Noor (1973)
3.9812
3.9456
3.9362
3.9438
4.1089
40
Present Model
Bose and Reddy (1998)
Matsunaga (2000)
Kant and Manjunatha (1988)
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)
Xiang and Wang (2009)
Phan and Reddy (1985)
Noor (1973)
3.3262
3.3196
3.3684
3.3087
3.2841
20
Present Model
Liew et.al, (2003)
Xiang and Wang (2009)
Phan and Reddy (1985)
Noor (1973)
3.8395
3.8272
3.8684
3.8105
3.8241
30
Present Model
Liew et.al, (2003)
Xiang and Wang (2009)
Phan and Reddy (1985)
Noor (1973)
4.1376
4.1308
4.1664
4.1088
4.1088
40
Present Model
Liew et.al, (2003)
Xiang and Wang (2009)
Phan and Reddy (1985)
Noor (1973)
4.3381
4.3420
4.3752
4.3148
4.3008
Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order Shear
Deformation Theory
http://iaeme.com/Home/journal/IJMET 422 [email protected]
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
-
Table 4 Non-Dimensionalized fundamental frequencies for symmetric cross-ply laminated square
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
S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi
http://iaeme.com/Home/journal/IJMET 423 [email protected]
Table 6 Non-Dimensionalized fundamental frequencies for symmetric cross-ply laminated square
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)
Reddy and Phan HSPDT (1985)
10.754
10.682
10.673
10.989
10 Present Model
Desai et.al, (2003)
Cho et.al, (1991)
Reddy and Phan HSPDT (1985)
15.154
15.069
15.066
15.270
20 Present Model
Desai et.al, (2003)
Cho et.al, (1991)
Reddy and Phan HSPDT (1985)
17.624
17.636
17.535
17.668
25 Present Model
Desai et.al, (2003)
Cho et.al, (1991)
Reddy and Phan HSPDT (1985)
18.061
18.067
18.054
18.050
50 Present Model
Desai et.al, (2003)
Cho et.al, (1991)
Reddy and Phan HSPDT (1985)
18.644
18.670
18.670
18.606
100 Present Model
Desai et.al, (2003)
Cho et.al, (1991)
Reddy and Phan HSPDT (1985)
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
Bose and Reddy (1998)
Matsunaga (2000)
Kant and Manjunatha(1988)
Noor (1973)
Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order Shear
Deformation Theory
http://iaeme.com/Home/journal/IJMET 424 [email protected]
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)
Xiang and Wang (2009)
Phan and Reddy (1985)
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)
S. Lokesh, Dr. CH. Lakshmi Tulasi, T. Monica and U. Pranavi
http://iaeme.com/Home/journal/IJMET 425 [email protected]
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)
Vibration Analysis of Laminated Composite Plates Using Layerwise Higher Order Shear
Deformation Theory
http://iaeme.com/Home/journal/IJMET 426 [email protected]
• 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.
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