Vibration Analysis of Anisotropic Simply Supported Plates by Using

Erasmo Carrera
Professor
Aeronautics and Space Engineering Department,
Politecnico di Torino,
Torino, Italy
Fiorenzo Adolfo Fazzolari1
School of Engineering and Mathematical Sciences,
City University London,
Northampton Square, London EC1V 0HB
United Kingdom
e-mail: [email protected]
Luciano Demasi
Assistant Professor
Department of Aerospace Engineering
and Engineering Mechanics,
San Diego State University,
San Diego, CA 92182-1308
1
Vibration Analysis of Anisotropic
Simply Supported Plates by
Using Variable Kinematic and
Rayleigh-Ritz Method
This work deals with accurate free-vibration analysis of anisotropic, simply supported
plates of square planform. Refined plate theories, which include layer-wise, equivalent
single layer and zig-zag models, with increasing number of displacement variables are
take into account. Linear up to fourth N-order expansion, in the thickness layer-plate
direction have been implemented for the introduced displacement field. Rayleigh-Ritz
method based on principle of virtual displacement is derived in the framework of Carrera’s unified formulation. Regular symmetric angle-ply and cross-ply laminates are
addressed. Convergence studies are made in order to demonstrate that accurate results
are obtained by using a set of trigonometric functions. The effects of the various parameters (material, number of layers, and fiber orientation) upon the frequencies and mode
shapes are discussed. Numerical results are compared with available results in literature.
[DOI: 10.1115/1.4004680]
Introduction
Structures composed of composite materials offer lower weight,
higher stiffness and strength then those composed of most metallic
materials. That coupled with advances in manufacturing of composite materials and structures, giving them a competitive edge
when compared with normal engineering materials and lead to
their extensive use. Composite plates and shells components now
constitute a large percentage of aerospace and submarine structures. They are increasingly used in areas like automotive engineering, and other applications. The use of the laminated
composite plate in many engineering applications has been
expanding rapidly in the past three decades. This resulted in considerably more research and interest in their dynamic behavior.
Indeed, a monograph by Leissa [1] that reviewed plate vibration
researched up to that point and included about 1000 references,
listed only a few articles that touched composite plates.
The first accurate treatment of plates can be attributed to Germain [2] and Lagrange [3] early in the 19th century. A good historical review of the development can be referred to in the books
of Soedel [4] and Timoshenko [5]. This theory is now referred to
as the classical plate theory (CPT). It uses the pure bending concept of plates in the development of the equations, where normals
to the middle surface remain straight and normal. It is valid for a
small deformation of thin plates. The inclusion of shear deformation in the fundamental equations of plates is due to Reissner [6]
and Mindlin [7]. Theories that account for shear deformation are
now referred to as thick plate theories or shear deformation plate
theories (SDPT).
Hearmon [8] presented what could be the first study on composite plates. Among the first to work on composite plates was Smith
[9]. A consistent theory for symmetrically laminated plates was
presented by Reissner and Stavski [10]. There is evidence data
that some Russian scientists may indeed have considered the problem earlier. Ambartsumian [11] and Lekhnitskii [12] probably
published the first book in the area of composite plates and shells.
Nevertheless, the original contribution by Lekhniskii has been
1
Corresponding author.
Contributed by the Technical Committee on Vibration and Sound of ASME for
publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received
November 16, 2010; final manuscript received May 30, 2011; published online November 28, 2011. Assoc. Editor: Massimo Ruzzene.
Journal of Vibration and Acoustics
almost ignored in the subsequent literature, even though the first
method able to describe the zig-zag effect and interlaminar continuous transverse stresses was provided by Lekhniskii in Ref. [13].
On the contrary, the works of Ren [14] and [15], were taken into
account. In two papers Ren has, in fact, extended Lekhnitskii’s
theory to orthotropic and anisotropic plates. A further pioneering
analysis was presented by Yu [16], where in plane zig-zag effect
and transverse shear were both fulfilled in correspondence to the
two interfaces of sandwich plate. Ashton and Whitney [17] presented fundamental equations of laminated plates. Vinson and
Sierakowski [18] presented analysis of composite beams, plates,
and shells, while Whitney [19] presented various structural analyses; including free vibration of laminated anisotropic plate, and in
Ref. [20] applied an extended Abartasumian theory to generally
anisotropic and symmetrical and nonsymmetrical plates. The subject was also picked up by Mohan and Kingsbury [21], as well as
Noor [22] among the many researchers in the field. In addition to
the previous articles and books, more recent literature on composite plate vibrations research, such as Qatu [23–28], and Leissa and
Narita [29], and can be found in various conference proceedings
and journals. A complete historical review of so called zig-zag
theories for laminated structures has been provided by Carrera in
Ref. [30].
During the last two decades, a variable kinematics 2D models
approach with hierarchical capabilities, for composite laminated
plates and shells, presented in many papers has been widely developed by Carrera. The primary work contribution is provided in
Ref. [31], where a generalization, proposing a systematic use of
reissner mixed variational theorem (RMVT) [32] as a tool to furnish a class of two dimensional theories for multilayered plates
analysis was presented. Attention was focused on approximated
solution techniques, and the resulting governing equation was
written as a system of algebraic equations. A weak form of
Hook’s law was also introduced into this work to reduce the
mixed cases to the displacement ones. Further details above hierarchical theories was provided in Ref. [33], where an overview of
finite elements, that have been developed for multilayered, anisotropic composite plates and shells, and [34] where assessment and
benchmarking were performed in order to validate the 2D hierarchical models developed is also available. Application of what
is reported in Refs. [31] and [35] to derive governing equation in
strong forms have been given in several other papers [36–42],
C 2011 by ASME
Copyright V
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both Navier-type close form and Finite element solutions were
given. Finite element formulation was also extensively developed
in Refs. [43,44], where different loadings, as well as boundary
conditions,
were treated. The variable kinematics modeling technique
based on Carrera’s unified formulation (CUF) offers a systematic
procedure to obtain refined structural models by considering the
order of the theory as a free parameter of the formulation. In the
framework of those axiomatic approaches which can be developed
on the basis of variational statements, CUF attention has been restricted to principle of virtual displacement (PVD) and RMVT
applications (see Ref. [45]).
The accuracy of CUF has been successfully demonstrated to
range from classical 2D models to quasi-3D descriptions for buckling, bending and free-vibration analyses. Carrera’s unified formulation applications have been restricted to closed form ‘exact’
solutions of Navier-type, and Finite element solutions. Closed
form solutions are restricted to simple geometries, simply supported boundary conditions and orthotropic behavior, while FE
solutions are difficult to obtain for those cases in which higher
modes and/or wave propagation description is required. The use
of CUF in conjunction to other approximated solution technique
would be therefore of interest for a more complete description of
vibration response of laminated plates. Some attempt has been
recently made by Ferreira, who have applied Radial Basis methods [46] by CUF, and R.J. Banerjee and M. Boscolo have
employed dynamic stiffness method (DSM) [47]. On that line in
this paper, for the first time, Rayleigh-Ritz formulation has been
embedded in the framework of CUF in order to perform freevibration analysis of anisotropic, simply supported laminated
plates. By virtue of this procedure, it has been possible overcome
the limits which are typical in finite element method (FEM), as far
as dynamic analysis at high frequencies is concerned. Indeed in a
more general case the accuracy of FEM is very low, for a given
number of degrees of freedom, because each basis function is a
polynomial of low degree. On the contrary, the present methodology use global basis functions such as polynomials (or trigono-
Fig. 1
061017-2 / Vol. 133, DECEMBER 2011
metric functions) of high degree which are nonzero, except at
isolated points, over the entire computational domain. When fast
iterative matrix solvers are used, the present methodology can be
much more efficient than FEM. Plate geometry and notation for
displacements, stresses and strains as well as Hooke’s law are
shown in Sec. 2. The variable kinematics modeling technique
based on CUF is briefly recalled in Sec. 3. The proposed
Rayleigh-Ritz approximation method is widely described in Sec.
4. The basis functions chosen, and an example of the surface integrals, performed analytically, that appear in the Rayleigh-Ritz formulation are highlighted in Sec. 4.1, whereas all the surface
integrals are illustrated in the Appendix. Finally, an assessment of
the proposed models of simply-supported laminated plates is carried out in Sec. 5, where the effects of the various parameters (material, number of layers, fiber orientation) upon the frequencies
and mode shapes are discussed.
2
Preliminaries
The salient features of Plate geometry are shown in Fig. 1. A
laminated plate composed of Nl layers is considered. The integer
k, used as superscript or subscript, denotes the layer number
which starts from the plate bottom. The layer geometry is denoted
by the same symbols as those used for the whole multilayered
plate and vice versa. With x and y the plate middle surface Xk
coordinates are indicated. Ck is the layer boundary on Xk . z and zk
are the plate and layer thickness coordinates; h and hk denote the
plate and layer thicknesses, respectively. fk ¼ 2zk =hk is the nondimensioned local plate-coordinate; Ak denotes the k-layer thickness
domain. Symbols that are not affected by the k subscript/superscripts refer to the whole plate. The notation for the displacement
vector is:
uðx; y; zÞ ¼
uy
ux
uz
T
(1)
Superscript T represents the transposition operator. The stress, r,
and the strain, e, are grouped as it follows:
Multilayered plate
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rp ¼ rxx
rn ¼ rxz
T
ryy rxy ;
T
ryz rzz ;
ep ¼ exx
en ¼ exz
T
eyy exy
T
eyz ezz
(2)
The subscripts n and p denote transverse (out-of-plane, normal)
and in-plane values, respectively. In case of small displacements
with respect to in plan dimension, the strain-displacement relations are:
ep ¼ Dp u
en ¼ Dn u ¼ ðDnX þ Dnz Þu
(3)
where Dp , DnX and Dnz are differential matrix operators:
2@
3
2
3
2@
3
@
0 0 @x
@x 0 0
@z 0 0
6 0 @ 07
@ 5; D ¼ 4 0 @ 0 5 (4)
Dp ¼ 4
5; DnX ¼ 4 0 0 @y
nz
@y
@z
@ @
@
0 0 @z
0
0
0
0
@y @x
erence surface X. Linear and higher order distributions in the
z-direction are introduced by the r-polynomials (see Fig. 2). The
assumed models can be written with the same notations that will
be adopted for the layer-wise model Eq. (9) is therefore rewritten
as
u ¼ Ft ut þ Fb ub þ Fr ur ¼ Fs us s ¼ t; b; r r ¼ 1; 2; 3; ; N 1
(10)
Subscript b denotes values related to the plate reference surface X
(ub ¼ u0 ) while subscript t refers to the highest term (ut ¼ uN ).
The Fs functions assume the following explicit form:
Fb ¼ 1;
Ft ¼ zN ;
Fr ¼ zr ;
r ¼ 2; 3; ; N 1
(11)
Classical models violate interlaminar equilibrium of the transverse
stresses. Further, they do not describe zigzag form of the displacement field in plate thickness direction.
In the case of orthotropic materials, Hooke’s law holds:
r ¼ Ce
(5)
According to Eq. (2), the previous equation becomes:
rp ¼ C~pp ep þ C~pn en
rn ¼ C~np ep þ C~nn en
where matrices C~pp , C~nn , C~pn and C~np are:
2
3
2
C~11 C~12 C~16
C~56
6
7
6
C~pp ¼ 4 C~12 C~11 C~26 5; C~nn ¼ 4 C~45
0
C~16 C~26 C~66
2
3
0 0 C~13
6
7
C~pn ¼ C~Tpp ¼ 4 0 0 C~23 5
(6)
C~45
C~45
0
3
0
7
0 5;
C~33
(7)
0 0 C~36
Refined Plate Theories
In the framework of Carrera unified formulation (CUF), the displacement field is assumed as:
u ¼ Fs us ;
u ¼ u0 þ ð1Þk fk uZ þ zr ur ;
s ¼ 1; 2; ::::; N
(8)
r ¼ 1; 2; ; N
(12)
Subscript Z refers to the introduced zigzag term. Note that the
unknown function u0 , uZ , ur , are k-independent. The geometrical
meaning of the zig-zag function is explained in Fig. 3. In the Murakami’s function MðzÞ ¼ ð1Þk fk the exponent k changes the sign
of the zig-zag term in each layer. Such an artifice permits one to
reproduce the discontinuity of the first derivate of the displacement variables in the z-direction which physically comes from the
intrinsic transverse anisotropy of multilayer structures. With unified notations Eq. (12) becomes,
u ¼ Ft ut þ Fb ub þ Fr ur ¼ Fs us
For the sake of conciseness, the dependence of the coefficients C~ij
versus Young’s moduli, Poisson’s ratio, the shear moduli, and the
fiber angle is not reported. It can be found in Tsai [48], Reddy
[49] or Jones [50].
3
3.2 Zig-Zag Models. The expansion given in Eq. (9) does
not permit the description of the zig-zag effects. Such a limitation
could somehow be overcome by referring to Murakami’s idea.
Murakami [51] proposed adding a zig-zag function to Eq. (9),
s ¼ t; b; r
r ¼ 1; 2; 3; ; N
(13)
Subscript t refers to the introduced zigzag term
(ut ¼ uN ; Fs ¼ ð1Þk fk ). It should be noticed that Fs assumes the
values 61 in correspondence to the bottom and the top interface
of the k-layer (see Fig. 3). A comprehensive documentation can
be found in Refs. [30,52,53].
3.3 Layer-Wise. By assuming the expansion in Eq. (9) in
each layer, layer-wise description is obtained. Nevertheless,
Taylor-type expansion of Eq. (9) is not convenient for a layerwise description. In fact, the fulfillment of continuity requirements
where FT are functions of the coordinates z in the thickness layerplate direction. uT is the displacement vector and N stands for the
number of terms of the expansion. According to Einstein’s notation, the repeated subscript s indicates summation. The maximum
expansion order, N, is supposed to be 4. A thorough description of
the refined model used can be found in Ref. [34].
3.1 Classical Equivalent Single Layer Models. Firstly, classical models are considered. As usual, the displacement variables
are expressed in Taylor series in terms of unknown variables
which are defined on the plate reference surface X.
u ¼ u0 þ zr ur ;
r ¼ 1; 2; ; N
(9)
N is a free parameter of the model. Different values for different
modelings and different displacement components are assumed.
The repeated r indexes are summed over their ranges. Subscript 0
denotes displacement values with correspondence to the plate refJournal of Vibration and Acoustics
Fig. 2
Linear and cubic case of ESLM
DECEMBER 2011, Vol. 133 / 061017-3
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The top and bottom values have been used as unknown variables.
The interlaminar compatibility of displacement at each interface
is easily linked:
ukt ¼ ubkþ1 ;
4
k ¼ 1;
Nl 1
(18)
Rayleigh-Ritz Formulation
The Rayleigh-Ritz method is a useful method for the approximate solutions to boundary value problems. This approach is
equally applicable to bending, buckling and vibration problems.
The stiffness and mass matrices in Rayleigh-Ritz approximation
are obtained via the principle of virtual displacements in dynamic
case:
Nl ð
X
Fig. 3
Linear and cubic case of ZZM
k¼1
¼
C0z
for the displacements at the interfaces, i.e., the
-requirement
could be easily introduced by using the interface variable as
unknown functions:
uk ¼ Ft ukt þ Fb ukb þ Fr ukr ¼ Fs uks
s ¼ t; b; r
r ¼ 1; 2; 3; ; N
(14)
The subscripts t and b denote values related to the layer’s top and
bottom surfaces, respectively. They consist of the linear part of
the expansion. The thickness functions Fs ðfk Þ have now been
defined at the k-layer level,
Ft ¼
P0 þ P1
P0 P1
; Fb ¼
; Fr ¼ Pr Pr2 ; r ¼ 2; 3; ; N
2
2
(15)
in which Pj ¼ Pj ðfk Þ is the Legendre polynomial of the j-order
defined in the fk -domain: 1 fk 1. Linear and cubic displacement field are shown in Fig. 4. The related polynomials are:
P0 ¼ 1;
5f3k
P1 ¼ fk ;
3f
P3 ¼
k;
2
2
3fk 1
2
35f4k 15f2k 3
P4 ¼
þ
8
8
4
ð Xk
Nl ð
X
k¼1
ð
Xk
(19)
qk duk u€k dV
Ak
where qk denotes mass density while double dots signifies accelerations. The subscript T signifies an array transposition and d virtual variations. The subscript H underlines that stresses are
computed via Hooke’s law. The variation of the internal work has
been split into in-plane and out-plane parts and involves stress
from Hooke’s law and strain from geometrical relations (subscript
G).
In Rayleigh-Ritz method the displacement vector, uks , that
appears in Eq. (8), is expressed in series expansions:
uks ¼ f ðtÞgi Uksi
where i ¼ 1; :::; N s ¼ t; b; r r ¼ 2; 3; :::::; N
(20)
N indicates the order of expansion in the approximation. Consequently, the displacement field, in compact way, assume the following form:
P2 ¼
The chosen functions have the following properties:
1 : Ft ¼ 1; Fb ¼ 0; Fr ¼ 0
fk ¼
1 : Ft ¼ 0; Fb ¼ 1; Fr ¼ 0
Ak
T
T
dekpG rkpH þ deknG rknH dXk dz
(16)
uk ¼ f ðtÞFs gi Uksi
(21)
8 k 9
< Uxsi =
k
Uksi ¼ Uysi
: k ;
Uzsi
(22)
where:
(17)
the matrix gi is a diagonal matrix
functions:
2
gxi 0
gi ¼ 4 0 gyi
0
0
formed by approximation
3
0
0 5
gzi
end in complete form it can be written, in two ways:
2
0 gx2 0
0 … gxN
0
gx1 0
0 gy2 0 … 0 gyN
g ¼ 4 0 gy1 0
0 gz2 … 0
0
0
0 gz1 0
(23)
3
0
0 5
gzN
(24)
or, in form of blocks
g ¼ ½ g1
Fig. 4 Linear and cubic case of LWM
061017-4 / Vol. 133, DECEMBER 2011
g2
…
gN (25)
therefore, Eq. (20) can be written as:
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uks ¼ f ðtÞgUks
The function f ðtÞ represent the temporal evolution of the solution
variables and is required for dynamic analysis. The functions gxi ,
gyi , gzi are chosen appropriately on the type of problem. The
results depend strongly on the functions that will be chosen. Convergence to the exact solution is guaranteed if the basis functions
are admissible functions, i.e., they satisfy the following three
points:
•
•
•
ekn ¼ Fs DnX gi Uksi þ Fs;z gi Uksi
(26)
By substituting the previous expression in Eq. (19) and using Eq.
(6), the internal work becomes:
dLkint ¼
be continuous as required in the variational statement (i.e.,
should be such that it has a nonzero contribution to the virtual
work statement)
satisfy the homogeneous form of the specified geometric
boundary condition
the set is linearly independent and complete
Xk
si
p
pn
k
ðA
Xk
Recalling general expression of the virtual work:
Pk ¼ Lkint LkFin
(28)
the total potential energy functional, the Eq. (27) corresponds to a
minimization of the functional
dPk ¼ 0
dLkint ¼ dUksi Kks sij Uksj
(27)
where Lint is the internal work and LFin is the work done from the
inertial force. Being:
(29)
The minimization is respect to the undetermined coefficients of
linear combination that derived to the approximation solution in
k
k
k
, Uysi
, Uzsi
and the
Eq. (21). In particular, Pk is a function of Uxsi
condition given in Eq. (29) can be written in the following form:
(33)
Ak
T
dLkint dLkFin ¼ 0
sj
kT T
k
~
þ
dUsi DnX gi Cnp
Fs Fs dz Dp gj Uksj dX
Xk
Ak
ð
ð
T
dUksi DTnX gi C~knn
Fs Fs dz DnX gj Uksj dX
þ
k
Xk
ðA
ð
T
þ
dUksi DTnX gi C~knn
Fs Fs;z dz gj Uksj dX
Xk
Ak
ð
ð
kT
k
dUsi gi C~np
Fs;z Fs dz Dp gj Uksj dX
þ
k
Xk
ðA
ð
T
k
k
~
dUsi gi Cnn
Fs;z Fs dz DnX gj Uksj dX
þ
Xk
Ak
ð
ð
T
dUksi gi C~knn
Fs;z Fs;z dz gj Uksj dX
þ
gxi ðx; yÞ ¼ genericfunction
gxi ðx; yÞ ¼ /xi ð xÞwxi ð yÞ
gxi ðx; yÞ ¼ gxmn ðx; yÞ ¼ trigonometricfunction
) /xm ð xÞ
np ¼ cos mp
a x ; wxn ð yÞ ¼ sin b y
In the first case, gxi ðx; yÞ is a general function of x and y, in the
second case is the product of two generic functions of x and y and
in the third case is the product of simple harmonic functions. The
subscript m and n identify the number of half-waves along x and y
respectively.
Applying Rayleigh-Ritz method, it is useful rewrite the PVD
as:
ð
T
dUksi DTp gi C~kpp
Fs Fs dz Dp gj Uksj dX
Xk
Ak
ð
ð
T
dUksi DTp gi C~kpn
Fs Fs dz DnX gj Uksj dX
þ
k
Xk
ðA
ð
kT T ~k
dU D gi C
Fs Fs;z dz gj Uk dX
þ
ð
ð
In this case, it is not implemented a process of discretization of
the domain, like in FEM, but it is looking for a kind of global solution on the entire domain. The functions gxi , gyi , gzi can be represented through three ways, for the sake of simplicity only gxi is
taken into account:
•
•
•
(32)
(34)
By comparison with Eq. (33):
Kkssij ¼
ð
ð
DTp gi C~kpp
Fs Fs dz Dp gj þ C~kpn
Fs Fs dz DnX gj
k
Xk
Ak
ð A
ð
k
T
k
~
~
þ Cpp
Fs Fs;z dz gj þ DnX gi Cnp
Fs Fs dz Dp gj
k
Ak
ðA
ð
þ C~knn
Fs Fs dz DnX gj þ C~knn
Fs Fs;z dz gj
k
Ak
A ð
ð
Fs;z Fs dz Dp gj þ C~knn
Fs;z Fs dz DnX gj
þ gi C~knp
k
Ak
ð A
þ C~knn
Fs;z Fs;z dz gj dX
ð
Ak
(35)
k
@P
¼ 0 with
k
@Uxsi
i ¼ 1; :::; N ;
s ¼ t; b; r
r ¼ 2; 3; :::::; N 1
@Pk
¼ 0 with
k
@Uysi
i ¼ 1; :::; N ;
s ¼ t; b; r
r ¼ 2; 3; :::::; N 1
@Pk
¼ 0 with
k
@Uzsi
i ¼ 1; :::; N ;
s ¼ t; b; r
r ¼ 2; 3; :::::; N 1
(30)
Stiffness and mass matrices can be determined in more direct way
using Eq. (19). The strain vectors can be written by coupling Eqs.
(3) and (21).
ekp ¼ Fs Dp gi Uksi
Journal of Vibration and Acoustics
(31)
The stiffness matrix determined represents fundamental nucleus
related to principle of virtual displacements application in static
case. For the sake of accuracy, the nine terms K kssij are:
ð
ð
ð
Fs Fs dz gxi;x gxj;x dX þ C~k16 Fs Fs dz gxi;y gxj;x dX
Ak
X
Ak
X
ð
ð
k
~
þ C16 Fs Fs dz gxi;x gxj;y dX
k
ðA
ðX
k
~
þ C66 Fs Fs dz gxi;y gxj;y dX
k
X
ðA
ð
k
~
þ C55 Fs;z Fs;z dz gxi gxj dX
ssij
Kxx
¼ C~k11
ð
Ak
X
DECEMBER 2011, Vol. 133 / 061017-5
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ð
ð
ð
Fs Fs dz gxi;x gyj;y dX þ C~k26 Fs Fs dz gxi;x gyj;y dX
Ak
X
Ak
X
ð
ð
k
þ C~16 Fs Fs dz gxi;x gyj;x dX
k
ðA
ðX
k
þ C~66 Fs Fs dz gxi;x gyj;x dX
k
X
ðA
ð
k
þ C~45 Fs;z Fs;z dz gxi gyj dX
ssij
Kxy
¼ C~k12
ð
Ak
ð
Ak
ssij
Kyx
¼ C~k12
ð
Fs Fs dz gyi;x gxj;x dX þ C~k16
X
ð
ð
k
~
þ C26 Fs Fs dz gyi;x gxj;y dX
k
ðA
ðX
þ C~k66 Fs Fs dz gyi;x gxj;y dX
k
X
ðA
ð
k
þ C~45 Fs;z Fs;z dz gyi gxj dX
ð
Fs Fs dz gyi;x gyj;y dX þ C~k26
Ak
X
ð
ð
k
~
þ C26 Fs Fs dz gyi;x gyj;x dX
k
ðA
ðX
þ C~k66 Fs Fs dz gyi;x gyj;x dX
k
X
ðA
ð
þ C~k44 Fs;z Fs;z dz gyi gyj dX
Ak
ssij
Kyz
¼ C~k23
gyi;x gxj;x dX
Ak
X
Fs Fs;z dz gyi;x gzj dX þ C~k36
Ak
X
ð
ð
k
~
þ C45 Fs;z Fs dz gyi gzj;x dX
k
ðA
ðX
k
~
þ C44 Fs;z Fs dz gyi gzj;y dX
Ak
Xk
Xk
ssij
Kzx
¼ C~k55
Fs Fs;z dz gzi;x gxj dX þ C~k45
Ak
X
ð
ð
k
~
þ C13 Fs;z Fs dz gzi gxj;x dX
k
ðA
ðX
k
~
þ C36 Fs;z Fs dz gzi gxj;y dX
Ak
Fs Fs dz
Ak
gyi;x gyj;y dX
Mkssij ¼ qk I
ð
ð
Fs Fs dz
Ak
X
k
gi gj dX
(41)
where I is the unit array. The three terms M kssij are:
kssij
¼ qk I
Mxx
ð
Ak
ð
Fs Fs;z dz
kssij
Myy
k
¼q I
ð
Fs Fs dz
k
ðA
kssij
¼ qk I
Mzz
ðX
k
ðX
k
X
k
Fs Fs dz
Ak
gyi;x gzj dX
X
ð
ð
Fs Fs dz
Ak
gxi gxj dX
gyi gyj dX
gzi gzj dX
(42)
The discrete form of the governing equations is finally obtained in
terms of fundamental nuclei:
ð
Ak
dUksi :
ð
Fs Fs;z dz
T
Kkssij Usj þ Mkssij U€sjk ¼ 0
(43)
gzi;x gxj dX
X
The free-vibration response lead to the following eigenvalue
problem
jjKkssij x2ij Mkssij jj ¼ 0
(44)
X
ð
Ak
(40)
Ak
X
ð
ð
ð
Fs Fs;z dz gzi;x gyj dX þ C~k44 Fs Fs;z dz gzi;x gyj dX
Ak
X
Ak
X
ð
ð
k
~
þ C23 Fs;z Fs dz gzi gyj;y dX
k
ðA
ðX
k
~
þ C36 Fs;z Fs dz gzi gyj;x dX
ssij
¼ C~k45
Kzy
(39)
Eq. (40) can be rewritten as:
ð
X
ð
T
comparing the two relations:
ð
ð gi qk
Fs Fs dz gj dX
Mkssij ¼
ð
(38)
Ak
Since
T
ð
ð
T
€ kT dXk
dUksi gi qk
Fs Fs dz gj U
sj
T
X
ð
ð
(37)
Ak
dLkFin ¼ dUksi Mkssij U€sjk
X
ð
ð
using Eq. (21):
ð
Fs Fs dz
Ak
Ak
ssij
Kyy
¼ C~k22
Xk
dLkFin ¼
ð
X
From the work done by the inertial force:
ð ð
qk duk u€k dV
dLkFin ¼
X
ð
ð
Ak
X
ð
ð
ð
Fs Fs;z dz gxi;x gzj dX þ C~k36 Fs Fs;z dz gxi;x gzj dX
Ak
X
Ak
X
ð
ð
k
þ C~55 Fs;z Fs dz gxi gzj;x dX
k
ðA
ðX
k
þ C~45 Fs;z Fs dz gxi gzj;y dX
ssij
Kxz
¼ C~k13
ð
ð
ð
Fs Fs dz gzi;x gzj;x dX þ C~k45 Fs Fs dz gzi;x gzj;x dX
Ak
X
Ak
X
ð
ð
k
Fs;z Fs dz gzi;x gzj;y dX
þ C~45
k
ðA
ðX
k
Fs Fs dz gzi;x gzj;y dX
þ C~44
k
X
ðA
ð
k
Fs;z Fs;z dz gzi gzj dX
(36)
þ C~33
Kzzssij ¼ C~k55
X
061017-6 / Vol. 133, DECEMBER 2011
where kij ¼ x2ij are the eigenvalues of the problem, the double bar
denote the determinant. This procedure has been used for the different case theories and results discussed later.
4.1 Basis Functions. A crucial point in Rayleigh-Ritz
method is the chosen of the basis functions on which build, with
linear combinations of unknown coefficients, approximate solutions. In the present work, they have been only considered trigonometric functions,
Transactions of the ASME
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mpx npy
sin
cos
a
b
m¼1 n¼1
M
N
XX
mpx
npy
gymn ðx; yÞ ¼
cos
sin
a
b
m¼1 n¼1
M X
N
mpx npy
X
gzmn ðx; yÞ ¼
sin
sin
a
b
m¼1 n¼1
gxmn ðx; yÞ ¼
M X
N
X
Table
X ¼ xa 2
m,n DOF
The comprehensive set of surface integrals that appear in fundamental nucleus are accounted for in Appendix. With the symbols
y
x
and J1nq
, the value of integrals of trigonometric functions are
J1mp
defined, also these quantities are listed in Appendix. This procedure has been coded for the different case theories and results discuss in Sec. 5.
Results and Discussion
4
6
8
10
12
a
16
36
64
100
144
X5
b
X5
a
X6
b
X6 a
51.657 — 57.543 —
50.407 — 54.436 —
49.984 49.99 53.863 53.87
49.749 49.75 53.591 53.59
49.595 49.60 53.425 53.43
X7 b
X7 a
X8 b
X8 a
75.723 — 75.767 —
71.325 — 73.473 —
70.586 70.59 72.849 72.85
70.240 70.24 72.536 72.54
70.033 70.04 72.346 72.35
Leissa and Narita [29].
Present analysis.
b
out. A good agreement between present analysis and references
has been found. The lamination scheme used is:
•
LS : ½30 = 30 =30 and the material is graphite-epoxy:
E1 ¼ 138 GPa, E2 ¼ 8:96 GPa, G12 ¼ 7:1 GPa, 12 ¼ 0:30
In Fig. 5, the influence of lamination angle upon the first height
frequencies is shown. The graphic shows a symmetric trend due to
similar frequency values obtained with lamination of 30 60 ,
15 75 and 0 90 . In following analysis different lamination
schemes (LS), material and refined theories have been considered:
•
•
•
•
LS1 : ½0 = 0 =0 LS2 : ½15 = 15 =15 LS3 : ½30 = 30 =30 LS4 : ½45 = 45 =45 The material proprieties of the used lamina in the analysis is
E-glass-epoxy (E/E):
•
Free-vibration analyses of anisotropic, simply supported composite laminated plates are considered. Different materials, lamination schemes and theories are taken into account. Acronyms
have been introduced to denote the different analysis in tables and
diagrams. Three characters have been used to build up these acronyms. The first character can be L or E which states Layer-wise or
Equivalent single layer analysis. The second one is D which states
classical analysis on the basis of Displacement formulation. The
third character can assume the numbers 1, 2, 3 or 4 which state
the order N of the displacement field. For instance, LD3 means
Layer-wise Displacement analysis with cubic displacement field
in each layer. Thin and thick as well as square plate geometries
have been analyzed. Regular symmetric angle-ply laminate
½h = h =h and cross-ply laminate are considered.
parameter
X
(45)
Using these kind of functions, the surface integral present in the
fundamental nucleus can be solved in analytical way. For the sake
of brevity, it is only shown the first surface integral present in the
first component of fundamental nucleus, for instance:
ð
xx
¼ gxi;x gxj;x dX
I K1mnpq
X
ð ð
mp2 p a b mpx npy ppx qpy
¼ 2
sin
sin
sin
dx dy
sin
a
a
b
a
b
0 0
ð
ð
npy qpy
b
mp2 p a mpx ppx
¼ 2
sin
dx sin
sin
dy
sin
a
a
a
b
b
0
0
mp2 p x y
¼ 2 J1mp
J1nq
a
(46)
5
2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Convergence
of
the
frequency
q
a
12qð1m12 m21 Þ
for
square
plate
¼
1
b
E1 h2
E1 ¼ 60:7 GPa, E2 ¼ 24:8 GPa, G12 ¼ 12:0 GPa, 12 ¼ 0:23
First and second natural frequencies with different thicknessratio and theories are evaluated. From the results reported in
Tables 3–6, it is remarked that frequencies predicted by the linear
models LD1, ED1 and EDZ1 are higher than those obtained with
the other models. This confirms the pathology that is peculiar to
linear models retaining the full 3D constitutive law and that has
been referred to in Ref. [54] as Poisson locking. Figs. 6–9 highlight as the fundamental frequency parameter is more influenced
from thickness parameter than lamination angle. For thick plate
a=h ¼ 4, refined theories such as FSDT and ED4 show a better
behavior with respect to CLPT, the ineffectiveness of CLPT has
had due to limits of validity to thin plates. For moderately thick
5.1 Regular Symmetric Angle-Ply. In Tables 1 and 2 convergence of first eight frequency parameters with CLPT is carried
Table
X ¼ xa 2
1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Convergence
of
the
frequency
q
a
12qð1m12 m21 Þ
for
square
plate
¼
1
b
E1 h2
parameter
X
m, n DOF
4
6
8
10
12
a
16
36
64
100
144
X1
b
X1
a
X2
b
X2 a
12.329 — 22.747 —
12.159 — 22.333 —
12.068 12.07 22.148 22.15
12.009 12.01 22.039 22.04
11.967 11.97 21.965 21.97
X3 b
X3 a
X4 b
X4 a
36.631 — 37.267 —
36.231 — 36.565 —
36.054 36.05 36.288 36.29
35.949 35.95 36.134 36.14
35.878 35.88 36.034 36.04
Leissa and Narita [29].
Present analysis.
b
Journal of Vibration and Acoustics
Fig. 5 Effect
of the laminationﬃ angle on frequency parameter
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
X ¼ xa 2 ð12qð1 m12 m 21 Þ=E1 h2 Þ, ða=b Þ ¼ 1
DECEMBER 2011, Vol. 133 / 061017-7
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Table p
3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
First and
second circular frequency parameters
ﬃ
X ¼ x ða 4 q=E1 h2 Þ related to fundamental and higher order
modes, m, n 5 10, square plate ða=b Þ ¼ 1, lamination scheme:
LS1
a/h
X
CLPTa
FSDT
ED1
ED2
ED3
ED4
EDZ1
EDZ2
EDZ3
LD1
LD2
LD3
LD4
4
10
20
100
X1
X2
X1
X2
X1
X2
X1
X2
4.223
3.730
3.810
3.730
3.665
3.664
3.769
3.698
3.665
3.707
3.665
3.664
3.664
8.668
7.271
7.435
7.274
7.111
7.107
7.326
7.195
7.111
7.214
7.110
7.107
7.107
4.398
4.289
4.400
4.289
4.271
4.271
4.389
4.280
4.271
4.293
4.271
4.271
4.271
9.524
9.117
9.390
9.118
9.058
9.058
9.350
9.088
9.058
9.119
9.059
9.058
9.058
4.425
4.396
4.514
4.396
4.392
4.392
4.511
4.394
4.392
4.407
4.392
4.392
4.392
9.669
9.554
9.860
9.554
9.536
9.536
9.848
9.545
9.536
9.580
9.537
9.536
9.536
4.434
4.433
4.553
4.433
4.433
4.433
4.553
4.433
4.433
4.446
4.433
4.433
4.433
9.716
9.712
10.030
9.712
9.711
9.711
10.030
9.711
9.711
9.747
9.711
9.711
9.711
a
Qatu [27] using CLPT with algebric polynomial and 49 DOF, for thin
plates obtained X1 ¼ 4.434 and X2 ¼ 9.717.
Table p
6ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
First and
second circular frequency parameters
ﬃ
X ¼ x ða 4 q=E1 h2 Þ related to fundamental and higher order
modes, m, n 5 10, square plate ða=b Þ ¼ 1, lamination scheme:
LS4
a/h
X
CLPTa
FSDT
ED1
ED2
ED3
ED4
EDZ1
EDZ2
EDZ3
LD1
LD2
LD3
LD4
4
10
20
100
X1
X2
X1
X2
X1
X2
X1
X2
4.485
3.936
4.010
3.937
3.862
3.861
3.962
3.898
3.861
3.901
3.861
3.859
3.859
7.902
7.819
7.902
7.821
7.603
7.597
7.812
7.706
7.601
7.723
7.600
7.595
7.595
4.674
4.550
4.657
4.550
4.530
4.530
4.643
4.540
4.530
4.552
4.530
4.529
4.529
10.567
10.028
10.284
10.029
9.945
9.945
10.225
9.984
9.945
10.014
9.946
9.944
9.944
4.702
4.670
4.784
4.670
4.664
4.664
4.780
4.667
4.664
4.680
4.664
4.664
4.664
10.728
10.575
10.866
10.575
10.550
10.550
10.848
10.562
10.550
10.595
10.550
10.549
10.549
4.712
4.710
4.827
4.711
4.710
4.710
4.827
4.710
4.710
4.724
4.710
4.710
4.710
10.781
10.775
11.080
10.775
10.774
10.774
11.079
10.774
10.774
10.810
10.774
10.774
10.774
a
Qatu [27] using CLPT with algebraic polynomial and 49 DOF, for thin
plates obtained X1 ¼ 4.696 and X2 ¼ 10.76.
Table p4ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
First and
second circular frequency parameters
ﬃ
X ¼ x ða 4 q=E1 h2 Þ related to fundamental and higher order
modes, m, n 5 10, square plate ða=b Þ ¼ 1, lamination scheme: LS2
a/h
X
CLPTa
FSDT
ED1
ED2
ED3
ED4
EDZ1
EDZ2
EDZ3
LD1
LD2
LD3
LD4
4
10
20
100
X1
X2
X1
X2
X1
X2
X1
X2
4.287
3.779
3.858
3.780
3.712
3.712
3.815
3.746
3.712
3.755
3.712
3.711
3.711
8.702
7.384
7.542
7.386
7.212
7.208
7.426
7.300
7.212
7.319
7.210
7.207
7.207
4.466
4.352
4.462
4.353
4.334
4.334
4.451
4.343
4.334
4.356
4.334
4.334
4.334
9.753
9.314
9.583
9.315
9.250
9.250
9.539
9.282
9.250
9.313
9.250
9.249
9.249
4.494
4.463
4.581
4.464
4.458
4.458
4.577
4.461
4.458
4.474
4.458
4.458
4.458
9.901
9.777
10.079
9.777
9.758
9.758
10.066
9.767
9.758
9.802
9.758
9.758
9.758
4.502
4.501
4.621
4.501
4.501
4.501
4.621
4.501
4.501
4.515
4.501
4.501
4.501
9.949
9.945
10.259
9.945
9.944
9.944
10.259
9.944
9.944
9.980
9.944
9.944
9.944
a
Qatu [27] using CLPT with algebric polynomial and 49 DOF, for thin
plates obtained X1 ¼ 4.499 and X2 ¼ 9.942.
Fig. 6 Fundamental circular frequency parameter versus lamination angle, thickness ratio a/h 5 4
Table p5ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
First and
second circular frequency parameters
ﬃ
X ¼ x ða 4 q=E1 h2 Þ related to fundamental and higher order
modes, m, n 5 10, square plate ða=b Þ ¼ 1, lamination scheme: LS3
a/h
X
CLPTa
FSDT
ED1
ED2
ED3
ED4
EDZ1
EDZ2
EDZ3
LD1
LD2
LD3
LD4
4
10
X1
X2
4.419
3.882
3.958
3.883
3.811
3.810
3.912
3.846
3.810
3.855
3.810
3.801
3.801
8.225
7.646
7.793
7.649
7.448
7.443
7.659
7.546
7.447
7.563
7.446
7.441
7.441
X1
20
X2
X1
100
X2
X1
X2
4.604 10.259 4.632 10.415 4.641 10.467
4.483 9.756 4.600 10.272 4.640 10.461
4.591 10.015 4.715 10.567 4.758 10.769
4.483 9.757 4.600 10.273 4.640 10.461
4.464 9.680 4.595 10.250 4.640 10.460
4.464 9.680 4.595 10.249 4.640 10.460
4.578 9.963 4.712 10.551 4.758 10.768
4.473 9.717 4.598 10.261 4.640 10.461
4.464 9.679 4.595 10.249 4.640 10.460
4.486 9.747 4.611 10.295 4.654 10.496
4.464 9.680 4.595 10.250 4.640 10.460
4.463 9.679 4.595 10.249 4.640 10.460
4.463 9.679 4.595 10.249 4.640 10.460
a
Qatu [27] using CLPT with algebric polynomial and 49 DOF, for thin
plates obtained X1 ¼ 4.631 and X2 ¼ 10.45.
061017-8 / Vol. 133, DECEMBER 2011
Fig. 7 Fundamental circular frequency parameter versus lamination angle, thickness ratio a/h 5 10
Transactions of the ASME
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Table 7 Nondimensionalized
fundamental frequency paramepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ter X ¼ x ða 4 q=E2 h2 Þ of simply supported antisymmetric
cross-ply square plates (G12 5 G13 5 0.5 E2, G23 5 0.2 E2,
m 12 5 0.25), and the shear corrector coefficient of K 5 1.0
E1/E2
Fig. 10 Fundamental circular frequency parameter versus
thickness ratio, lamination scheme:LS4
Journal of Vibration and Acoustics
40
Theory (0 /90 ) (0 /90 )4 (0 /90 ) (0 /90 )4 (0 /90 ) (0 /90 )4
2
FSDT
ED1
ED2
ED3
ED4
LD1
LD2
LD3
LD4
4.235
4.267
4.232
4.139
4.135
4.239
4.147
4.076
4.075
4.643
4.657
4.638
4.346
4.336
4.012
3.992
3.992
3.992
4.530
4.546
4.502
4.402
4.384
4.502
4.401
4.323
4.322
4.946
4.949
4.936
4.598
4.590
4.218
4.195
4.192
4.192
4.696
4.705
4.622
4.514
4.484
4.608
4.557
4.427
4.425
5.049
5.050
5.039
4.708
4.700
4.316
4.293
4.287
4.287
4
FSDT
ED1
ED2
ED3
ED4
LD1
LD2
LD3
LD4
6.126
6.208
6.119
6.032
6.029
6.138
6.027
5.982
5.982
7.276
7.326
7.270
6.944
6.939
6.572
6.550
6.550
6.550
6.948
6.998
6.926
6.818
6.794
6.936
6.861
6.734
6.734
8.547
8.565
8.528
7.985
7.978
7.423
7.389
7.388
7.388
7.504
7.539
7.416
7.291
7.224
7.401
7.343
7.154
7.153
9.101
9.110
9.075
8.432
8.425
7.783
7.743
7.741
7.741
10
CLPTa
FSDT
ED1
ED2
ED3
ED4
LD1
LD2
LD3
LD4
7.832
7.513
7.649
7.509
7.479
7.478
7.546
7.474
7.462
7.462
10.268
9.573
9.682
9.571
9.440
9.438
9.278
9.267
9.267
9.267
9.566
9.002
9.102
8.997
8.953
8.945
9.025
8.969
8.923
8.923
14.816
12.927
12.987
12.914
12.571
12.568
12.181
12.156
12.156
12.156
11.011
10.174
10.255
10.143
10.086
10.052
10.156
10.110
10.019
10.019
18.265
15.029
15.068
15.003
14.461
14.456
13.872
13.833
13.833
13.833
a
25
a/h
Fig. 8 Fundamental circular frequency parameter versus lamination angle, thickness ratio a/h 5 20
Fig. 9 Fundamental circular frequency parameter versus lamination angle, thickness ratio a/h 5 100
10
Reddy [49].
plate enhance the behavior of CLPT. Taking into account thin
plates all the theories are agree with the exception of ED1 model
affected by Poisson locking. In Fig. 10, it is possible to observe
the monotonous increasing of the circular frequency parameters as
the thickness ratio a/h increase.
5.2 Cross-Ply Laminates. Free-vibration analysis of both
symmetric and antisymmetric cross-ply laminates is addressed.
In Tables 7 and 8, antisymmetric laminate are considered. Fundamental frequency parameter for different thickness and modulus ratio is evaluated. Lamination schemes used are ð0 =90 Þ and
ð0 =90 Þ4 , and the material proprieties are G12 ¼ G13 ¼ 0:5E2 ,
G23 ¼ 0:2E2 , 12 ¼ 0:25. Increasing the number of layers, the
fundamental frequency parameter increases, as well. More
refined theories such as LD3, LD4 show a better behavior both
for thick plate and for an higher number of layers. In Figs. 11
and 12 increasing the thickness ratio CLPT theory shows a good
agreement with LD3, LD4. For thick plate the error committed
by CLPT is bigger with lamination scheme ð0 =90 Þ4 than
ð0 =90 Þ. The error percentage, generated using FDST by comparison with LD4 model and an anisotropic ratio of E1 =E2 ¼ 10,
is strongly dependent form lamination scheme. More specifically
the error ¼ 3:93% with lamination scheme ð0 =90 Þ and
error ¼ 16:31% with ð0 =90 Þ4 . Figures 13 and 14 highlight the
rise of fundamental frequency, always using the same model, as
increase the anisotropic ratio. For symmetric cross-ply laminate,
lamination scheme ð0 =90 =0 Þ have been considered. As illustrated in Table 9 for thick plate EDZ3, LD2 and LD4 give the
best results, indeed refer to ZZT has been shown in Ref. [53]
that it can be more convenient to enhance a plate theory by
DECEMBER 2011, Vol. 133 / 061017-9
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Table 8 Nondimensionalized
fundamental frequency paramepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ter X ¼ x ða 4 q=E2 h2 Þ of simply supported antisymmetric
cross-ply square plates (G12 5 G13 5 0.5 E2, G23 5 0.2 E2,
m 12 5 0.25), and the shear corrector coefficient of K 5 1.0
E1/E2
10
20
CLPTa
FSDT
ED1
ED2
ED3
ED4
LD1
LD2
LD3
7.906
7.819
7.970
7.818
7.809
7.809
7.860
7.807
7.804
10.337
10.139
10.268
10.139
10.099
10.098
10.049
10.044
10.044
9.663
9.505
9.621
9.504
9.491
9.489
9.538
9.495
9.482
14.910
14.347
14.429
14.342
14.220
14.218
14.073
14.063
14.063
11.125
10.886
10.984
10.877
10.859
10.848
10.902
10.866
10.837
18.381
17.355
17.417
17.345
17.124
17.122
16.867
16.849
16.848
LD4
CLPTa
FSDT
ED1
ED2
ED3
ED4
LD1
LD2
LD3
LD4
7.804
7.931
7.927
8.084
7.927
7.927
7.927
7.972
7.927
7.926
7.926
10.044
10.354
10.346
10.482
10.346
10.344
10.344
10.344
10.342
10.342
10.342
9.841
9.695
9.689
9.811
9.689
9.688
9.688
9.724
9.688
9.688
9.688
14.063
14.941
14.917
15.009
14.917
14.911
14.911
14.906
14.904
14.904
14.904
10.837
11.162
11.152
11.257
11.152
11.151
11.151
11.183
11.152
11.150
11.150
16.848
18.419
18.374
18.448
18.374
18.363
18.363
18.351
18.349
18.349
18.349
7.933
7.932
8.089
7.932
7.932
7.932
7.977
7.931
7.932
7.932
10.355
10.355
10.491
10.355
10.355
10.355
10.360
10.352
10.353
10.350
9.726
9.696
9.819
9.697
9.697
9.697
9.732
9.696
9.696
9.697
14.942
14.942
15.034
14.942
14.942
14.942
14.943
14.941
14.945
14.939
11.230
11.164
11.269
11.164
11.164
11.164
11.195
11.164
11.164
11.164
18.421
18.420
18.495
18.420
18.420
18.420
18.422
18.421
18.423
18.422
a
40
Theory (0 /90 ) (0 /90 )4 (0 /90 ) (0 /90 )4 (0 /90 ) (0 /90 )4
1000 CLPT
FSDT
ED1
ED2
ED3
ED4
LD1
LD2
LD3
LD4
25
a/h
100
Fig. 12 Lamination scheme (0 /90 )4 and anisotropic ratio
E1/E2 5 10
Reddy [49].
introducing Murakami zig-zag function (MZZF) than refining it
by adding two or three higher order terms. For thin plate instead
all used theory are agree enough. Mode shapes in Figs. 15–20,
in order to shown the variable kinematic approach, are plotted.
Comparison between ESLM and LWM have been accounted
for.
Fig. 11 Lamination scheme (0 /90 ) and anisotropic ratio
E1/E2 5 10
061017-10 / Vol. 133, DECEMBER 2011
Fig. 13 Lamination scheme (0 /90 ) and LD4 model
Fig. 14 Lamination scheme (0 /90 )4 and LD4 model
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Table 9 Nondimensionalized
fundamental
circular frequency
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
parameter Xmn ¼ xmn ða 4 q=E2 h2 Þ of symmetric cross-ply
square plates with hk 5 h/3, E1/E2 5 25, G12 5 G13 5 0.5 E2,
G23 5 0.2 E2, m 12 5 0.25
(0 /90 /0 )
a/h
Theory
a
a
m
n CLPT
FSDT
ED2
ED4
EDZ3
LD2
LD4
10
1
1
1
2
2
1
2
1
2
3
1
2
4
3
15.104
22.421
38.738
55.751
59.001
62.526
67.980
12.527
19.203
31.921
32.931
36.362
47.854
44.720
12.523
19.210
31.968
32.911
36.339
47.967
44.724
11.756
18.490
30.938
29.262
32.914
46.219
41.510
11.461
18.267
30.786
28.219
31.956
46.132
40.732
11.464
18.285
30.843
28.246
31.988
46.236
40.789
11.457
18.212
30.564
28.182
31.892
45.653
40.537
100
1
1
1
2
2
1
2
1
2
3
1
2
4
3
15.227
22.873
40.283
56.874
60.891
66.708
71.484
15.192
22.827
40.174
56.319
60.322
66.421
70.882
15.191
22.827
40.176
56.318
60.321
66.426
70.882
15.174
22.810
40.152
56.058
60.059
66.374
70.625
15.165
22.803
40.145
55.934
59.934
66.368
70.504
15.165
22.803
40.146
55.935
59.934
66.371
70.505
15.165
22.802
40.140
55.934
59.933
66.350
70.501
a
Fig. 17 z/h versus displacement component uy, ESL models,
m5n51
Reddy [49].
Fig. 15 z/h versus displacement component ux, ESL models,
m5n51
Fig. 16 z/h versus displacement component ux, LW models,
m5n51
Journal of Vibration and Acoustics
Fig. 18 z/h versus displacement component uy, LW models,
m5n51
Fig. 19 z/h versus displacement component uz, ESL models,
m5n51
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xx
I K2mnpq
¼
Fig. 20 z/h versus displacement component uz, LW models,
m5n51
6
Conclusions
In this paper Rayleigh-Ritz method is derived in the framework of
Carrera’s unified formulation. Classical formulations based on displacement assumptions have been considered. A variable Kinematics
approach with hierarchical capabilities has been considered to establish the accuracy of a large variety of classical and advanced plate
theories to evaluate circular frequency parameters. LW, ZZ and
ESLM have been implemented. Linear up to fourth order displacement fields in the layer-plate thickness direction have been considered. The conducted investigation restricted to vibration of simply
supported plate have mainly lead to the following conclusions:
•
•
•
•
Layer-wise furnishes a better description of free-vibration of
laminate thick plates, while the accuracy of ESLM analyses
is very much subordinate to laminate lay-outs and to the mechanical properties of the lamina.
As already highlighted in Ref. [53], for bending analysis, also
in free-vibration analysis the inclusion of MZZF is very
effective, especially if the order of the expansion along the
thickness is low.
The accuracy of the different modelings is very much subordinate to the order N of the used displacement expansion. Very
accurate layer-wise results have been obtained for N 2.
The convergence is reasonably fast, by virtue of the fulfillment of both mechanical and geometrical boundary conditions of the basis function chosen.
It is concluded that CUF has shown its own strength in building
classical and advanced plate theories to perform accurate freevibration analysis of laminated composite plates. Future works
will propose the same Rayleigh-Ritz approximate solution method
in order to address bending and buckling analyses.
Appendix
All the integrals that appear in the Rayleigh-Ritz fundamental
nucleus formulations are listed below:
• Calculus of the integrals for Kxx
Ð
xx
I K1mnpq
¼ X gxi;x gxj;x d X
mp2 p Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼ 2 0 0 sin
a
a
b
a
b
2 Ð
Ð
mp p a
mpx
ppx
npy
qpy
b
¼ 2 0 sin
sin
dx 0 sin
sin
dy
a
a
a
b
b
2
mp p x y
¼ 2 J1mp
J1nq
a
(A1)
061017-12 / Vol. 133, DECEMBER 2011
Ð
X gxi;x
2
gxj;x d X
np p Ð a Ð b mpx npy ppx qpy
cos
sin
sin
dx dy
¼
cos
ba 0 0
a
b
a
b
np2 p Ð a mpx ppx Ð b npy qpy
¼
sin
dx 0 cos
sin
dy
cos
ba 0
a
a
b
b
np2 p x y
¼
J J
ba 3mp 3nq
(A2)
Ð
Kxx
I 3mnpq ¼ X gxi;x gxj;y d X
mp2 q Ð a Ð b mpx npy ppx qpy
sin
cos
cos
dxdy
¼
ba 0 0 a
b
a
b
mp2 q Ð a mpx ppx Ð b npy qpy
¼
cos
dx 0 sin
cos
dy
sin
ba 0
a
a
b
b
mp2 q x y
¼
J J
ba 4mp 4nq
(A3)
Ð
xx
I K4mnpq
¼ X gxi;x gxj;y d X
np2 q Ð a Ð b mpx npy ppx qpy
cos
cos
cos
dx dy
¼ 2 0 0 cos
b
a
b
a
b
2 Ð
Ðb
np q a
mpx
ppx
npy
qpy
¼ 2 0 cos
cos
dx 0 cos
cos
dy
b
a
a
b
b
2
np p x y
¼ 2 J2mp
J2nq
b
(A4)
Ð
xx
¼ X gxi gxj d X
I K5mnpq
Ð a Ð b mpx npy ppx qpy
sin
cos
sin
dx dy
¼ 0 0 cos
Ð a mpxa ppxb Ð b anpy bqpy
cos
dx 0 sin
sin
dy
¼ 0 cos
a
a
b
b
y
x
¼ J2mp
J1nq
(A5)
Calculus of the integrals for Kxy
Ð
Kxy
I 1mnpq
¼ X gxi;x gyj;y d X
mp2 q Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dxdy
¼
sin
ab 0 0
a
b
a
b
2 Ð
Ðb
mp q a
mpx
ppx
npy
qpy
¼
sin
dx 0 sin
sin
dy
sin
0
ab
a
a
b
b
2
mp q x y
¼
J J
ab 1mp 1nq
Ð
(A6)
Kxy
I 2mnpq
¼ X gxi;x gyj;y d X
np2 q Ð a Ð b mpx npy ppx qpy
cos
sin
sin
dx dy
¼ 2 0 0 cos
b
a
b
a
b
2 Ð
Ð
np q a
mpx
ppx
npy
qpy
b
¼ 2 0 cos
sin
dx 0 cos
sin
dy
b
a
a
b
b
2
np q x y
¼ 2 J3mp
J3nq
b
Ð
(A7)
Kxy
I 3mnpq
¼ X gxi;x gyj;x d X
2 Ð Ð
mp p a b
mpx
npy
ppx
qpy
sin
cos
cos
dx dy
¼ 2 0 0 sin
a
a
b
a
b
mp2 p Ð a mpx ppx Ð b npy qpy
¼ 2 0 sin
cos
dx 0 sin
cos
dy
a
a
a
b
b
2
mp p x y
¼ 2 J4mp
J4nq
a
(A8)
Ð
Kxy
I 4mnpq
¼ X gxi;x gxj;y d X
np2 q Ð a Ð b mpx npy ppx qpy
cos
cos
cos
dx dy
¼ 2 0 0 cos
b
a
b
a
b
np2 q Ð a mpx ppx Ð b npy qpy
¼ 2 0 cos
cos
dx 0 cos
cos
dy
b
a
a
b
b
2
np p x y
¼ 2 J2mp
J2nq
b
(A9)
•
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K
xy
I 5mnpq
¼
Ð
dX
X gxi gyj ÐaÐb
mpx npy ppx qpy
sin
sin
cos
dx dy
¼ 0 0 cos
¼
Ða
0
K
yx
I 4mnpq
¼
X gyi;x gxj;y d X
mp2 q Ð a Ð b mpx npy ppx qpy
cos
cos
cos
dx dy
¼
cos
ba 0 0
a
b
a
b
mp2 q Ð a mpx ppx Ð b npy qpy
¼
cos
dx 0 cos
cos
dy
cos
ba 0
a
a
b
b
2
mp q x y
¼
J J
ba 2mp 2nq
(A18)
b Ð
mpxa ppx
anpy bqpy
b
sin
dx 0 sin
cos
dy
a
a
b
b
y
cos
x
¼ J3mp
J4nq
(A10)
•
Calculus of the integrals for Kxz
Ð
xz
I K1mnpq
¼ X gxi;x gzj d X
mp Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼
sin
a 0 0
a
b
a
b
mp Ð a mpx ppx Ð b npy qpy
sin
dx 0 sin
sin
dy
sin
¼
a 0
a
a
b
b
mp x y
¼ J1mp J1nq
a
(A11)
Ð
Kxz
I 2mnpq ¼ X gxi;x gzj d X
mpx
npy ppx qpy
np Ð a Ð b
cos
sin
sin
dx dy
¼
cos
b 0 0 a
b
anpy bqpy
np Ð a
mpx ppx Ð b
sin
dx 0 cos
sin
dy
cos
¼
b 0
a
a
b
b
np x y
¼ J3mp J3nq
b
(A12)
Ð
Kxz
I 3mnpq ¼ X gxi gzj;x d X
mpx npy
ppx qpy
pp Ð a Ð b
sin
cos
sin
dx dy
¼
0 0 cos
a
mpxa
ppxb Ð
anpy bqpy
pp Ð a
b
cos
dx 0 sin
sin
dy
cos
¼
a 0
a
a
b
b
pp x y
¼ J2mp J1nq
a
(A13)
Ð
xz
¼ X gxi gzj;y d X
I K4mnpq
mpx npy ppx
qpy
qp Ð a Ð b
sin
sin
cos
dxdy
¼
0 0 cos
b
mpxa ppxb Ð
anpy
bqpy
qp Ð a
b
sin
dx 0 sin
cos
dy
cos
¼
b 0
a
a
b
b
qp x y
¼ J3mp J3nq
b
(A14)
• Calculus of the integrals for Kyx
Kyx
I 1mnpq
K
yx
I 2mnpq
K
yx
I 3mnpq
¼
K
yx
I 5mnpq
¼
gyi;x gxj;x d X
np2 p Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼
sin
ab 0 0
a
b
a
b
2 Ð
Ð
np p a
mpx
ppx
npy
qpy
b
¼
sin
dx 0 sin
sin
dy
0 sin
ab
a
a
b
b
2
np p x y
¼
J J
ab 1mp 1nq
(A15)
Ð
¼ X gyi;x gxj;x d X
mp2 p Ð a Ð b mpx npy ppx qpy
cos
sin
sin
dx dy
¼ 2 0 0 cos
a
a
b
a
b
2 Ð
Ð
mp p a
mpx
ppx
npy
qpy
b
¼ 2 0 cos
sin
dx 0 cos
sin
dy
a
a
a
b
b
2
mp p x y
¼ 2 J3mp
J3nq
a
(A16)
Ð
¼ X gyi;x gxj;y d X
np2 q Ð a Ð b mpx npy ppx qpy
sin
cos
cos
dx dy
¼ 2 0 0 sin
b
a
b
a
b
np2 q Ð a mpx ppx Ð b npy qpy
¼ 2 0 sin
cos
dx 0 sin
cos
dy
b
a
a
b
b
2
np q x y
¼ 2 J4mp
J4nq
b
(A17)
(A19)
•
Calculus of the integrals for Kyy
K
Ð
yy
I 1mnpq
¼
K
X gyi;x gyj;y d X
np2 q Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼ 2 0 0 sin
b
a
b
a
b
np2 q Ð a mpx ppx Ð b npy qpy
¼ 2 0 sin
sin
dx 0 sin
sin
dy
b
a
a
b
b
2
np q x y
¼ 2 J1mp
J1nq
b
(A20)
yy
¼
I 2mnpq
Ð
X gyi;x gyj;y d X
mp2 q Ð a Ð b mpx npy ppx qpy
cos
sin
sin
dx dy
¼
cos
ab 0 0
a
b
a
b
2 Ð
Ð
mp q a
mpx
ppx
npy
qpy
b
¼
sin
dx 0 cos
sin
dy
0 cos
ab
a
a
b
b
2
mp q x y
¼
J3mp J3nq
ab
(A21)
K
X
Journal of Vibration and Acoustics
Ð
X gyi gxj d X
Ð a Ð b mpx npy ppx qpy
cos
cos
sin
dx dy
¼ 0 0 sin
b Ð
anpy bqpy
Ð a mpxa ppx
b
cos
dx 0 cos
sin
dy
¼ 0 sin
a
a
b
b
y
x
¼ J4mp
J3nq
yy
I 3mnpq
¼
Ð
Ð
Ð
gyj;x d X
np p Ð a Ð b mpx npy ppx qpy
sin
cos
cos
dx dy
¼
sin
ab 0 0
a
b
a
b
2 Ð
Ðb
np p a
mpx
ppx
npy
qpy
¼
cos
dx 0 sin
cos
dy
0 sin
ab
a
a
b
b
2
np p x y
¼
J J
ab 4mp 4nq
(A22)
K
yy
¼
I 4mnpq
X gyi;x
2
Ð
X gyi;x gyj;x d X
mp2 p Ð a Ð b mpx npy ppx qpy
cos
cos
cos
dx dy
¼ 2 0 0 cos
a
a
b
a
b
mp2 p Ð a mpx ppx Ð b npy qpy
¼ 2 0 cos
cos
dx 0 cos
cos
dy
a
a
a
b
b
mp2 p x y
¼ 2 J2mp J2nq
a
(A23)
K
yy
¼
I 5mnpq
Ð
X gyi gyj d X
Ð a Ð b mpx npy ppx qpy
cos
sin
cos
dx dy
¼ 0 0 sin
anpy bqpy
Ð a mpxa ppxb Ð b
sin
dx 0 cos
cos
dy
¼ 0 sin
a
a
b
b
y
x
¼ J1mp J2nq
(A24)
DECEMBER 2011, Vol. 133 / 061017-13
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•
Calculus of the integrals for Kyz
Ð
Kyz
I 1mnpq ¼ X gyi;x gzj d X
np Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼
sin
b 0 0
a
b
a
b
np Ð a mpx ppx Ð b npy qpy
sin
dx 0 sin
sin
dy
sin
¼
b 0
a
a
b
b
np x y
¼ J1mp J1nq
b
(A25)
Ð
Kyz
I 2mnpq ¼ X gyi;x gzj d X
mpx npy ppx qpy
mp Ð a Ð b
cos
sin
sin
dx dy
¼
cos
a 0 0 a
b
anpy bqpy
mp Ð a
mpx ppx Ð b
sin
dx 0 cos
sin
dy
cos
¼
a 0
a
a
b
b
mp x y
J J
¼
a 3mp 3nq
(A26)
Ð
Kyz
I 3mnpq ¼ X gyi gzj;x d X
npy
ppx qpy
pp Ð a Ð b mpx
cos
cos
sin
dx dy
¼
0 0 sin
a
a
ppxb Ð
anpy bqpy
pp Ð a mpx
b
cos
dx 0 cos
sin
dy
sin
¼
a 0
a
a
b
b
pp x y
¼ J4mp J3nq
a
(A27)
Ð
Kyz
I 4mnpq
¼ X gyi gzj;y d X
npy ppx
qpy
qp Ð a Ð b mpx
cos
sin
cos
dx dy
¼
sin
b 0 0
a
b
anpy
bqpy
qp Ð a mpx ppx Ð b
sin
dx 0 cos
cos
dy
sin
¼
b 0
a
a
b
b
qp x y
¼ J1mp J2nq
b
(A28)
•
Calculus of the integrals for Kzx
Ð
zz
I K1mnpq
¼ X gzi;x gxj d X
mpx npy ppx qpy
mp Ð a Ð b
sin
cos
sin
dx dy
¼
0 0 cos
a
mpxa ppxb Ð
anpy bqpy
mp Ð a
b
cos
dx 0 sin
sin
dy
cos
¼
a 0
a
a
b
b
mp x y
J J
¼
a 2mp 1nq
(A29)
Ð
Kzx
I 2mnpq ¼ X gzi;x gxj d X
npy
ppx qpy
np Ð a Ð b mpx
cos
cos
sin
dx dy
¼
sin
0
0
b
a
ppxb Ð
anpy bqpy
np Ð a mpx
b
cos
dx 0 cos
sin
dy
sin
¼
b 0
a
a
b
b
np x y
¼ J4mp J3nq
b
(A30)
Ð
Kzx
I 3mnpq ¼ X gzi gxj;x d X
pp Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼
sin
a 0 0
a
b
a
b
pp Ð a mpx ppx Ð b npy qpy
sin
dx 0 sin
sin
dy
sin
¼
a 0
a
a
b
b
pp x y
¼ J1mp J1nq
a
(A31)
Ð
Kzx
I 4mnpq ¼ X gzi gxj;y d X
ppx
qpy
qp Ð a Ð b mpx npy
sin
cos
cos
dx dy
¼
0 0 sin
b
a
ppxb Ð
anpy
bqpy
qp Ð a mpx
b
cos
dx 0 sin
cos
dy
sin
¼
b 0
a
a
b
b
qp x y
¼ J4mp J4nq
b
(A32)
061017-14 / Vol. 133, DECEMBER 2011
•
Calculus of the integrals for Kzy
Ð
¼ X gzi;x gyj d X
mpx npy ppx qpy
mp Ð a Ð b
sin
sin
cos
dx dy
¼
cos
a 0 0 a
b
a
bqpy
mp Ð a
mpx ppx Ð b npy
sin
dx 0 sin
cos
dy
cos
¼
a 0
a
a
b
b
mp x y
J J
¼
a 3mp 4nq
(A33)
Ð
Kzy
¼ X gzi;x gyj d X
I 2mnpq
np Ð a Ð b
mpx
npy
ppx
qpy
cos
sin
cos
dx dy
¼
sin
b Ð0 0 a b Ð
a b np a
mpx
ppx
npy
qpy
b
sin
dx 0 cos
cos
dy
sin
¼
b 0
a
a
b
b
np x y
¼ J1mp J2nq
b
(A34)
Ð
Kzy
I 3mnpq ¼ X gzi gyj;y d X
qp Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼
sin
b 0 0
a
b
a
b
qp Ð a mpx ppx Ð b npy qpy
sin
dx 0 sin
sin
dy
sin
¼
b 0
a
a
b
b
qp x y
¼ J1mp J1nq
b
(A35)
Ð
Kzy
¼ X gzi gyj;x d X
I 4mnpq
pp Ð a Ð b
mpx
npy
ppx
ppy
sin
cos
cos
dx dy
¼
sin
0
0
a
a
b
a
a
pp Ð a mpx ppx Ð b npy ppy
cos
dx 0 sin
cos
dy
sin
¼
a 0
a
a
b
a
pp x y
¼ J4mp J4nq
a
(A36)
Kzy
I 1mnpq
•
Calculus of the integrals for Kzz
zz
I K1mnpq
¼
zz
I K2mnpq
zz
I K3mnpq
zz
I K4mnpq
Ð
X gzi;x gzj;x d X
mpx npy ppx qpy
mp2 p Ð a Ð b
sin
cos
sin
dx dy
¼ 2 0 0 cos
a
a
b
a
b
2 Ð
Ð
mp p a
mpx
ppx
npy
qpy
b
¼ 2 0 cos
cos
dx 0 sin
sin
dy
a
a
a
b
b
2
mp m x y
¼ 2 J2mp
J1nq
a
(A37)
Ð
¼ X gzi;x gzj;x d X
np2 p Ð a Ð b mpx npy ppx qpy
cos
cos
sin
dx dy
¼
sin
ab 0 0
a
b
a
b
2 Ð
Ð
np p a
mpx
ppx
npy
qpy
b
¼
cos
dx 0 cos
sin
dy
0 sin
ab
a
a
b
b
2
np p x y
¼
J J
ab 4mp 3nq
(A38)
Ð
¼ X gzi;x gzj;y d X
mp2 q Ð a Ð b mpx npy ppx qpy
sin
sin
cos
dx dy
¼
cos
ab 0 0
a
b
a
b
2 Ð
Ðb
mp q a
mpx
ppx
npy
qpy
¼
sin
dx 0 sin
cos
dy
0 cos
ab
a
a
b
b
mp2 q x y
¼
J J
ab 3mp 4nq
(A39)
Ð
¼ X gzi;x gzj;y d X
np2 q Ð a Ð b mpx npy ppx qpy
cos
sin
cos
dx dy
¼ 2 0 0 sin
b
a
b
a
b
2 Ð
Ð
np q a
mpx
ppx
npy
qpy
b
¼ 2 0 sin
sin
dx 0 cos
cos
dy
b
a
a
b
b
2
np q x y
¼ 2 J1mp
J2nq
b
(A40)
Transactions of the ASME
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zz
I K5mnpq
¼
Ð
X gzi gzj d X
Ð a Ð b mpx npy ppx qpy
sin
sin
sin
dx dy
¼ 0 0 sin
b Ð
anpy bqpy
Ð a mpxa ppx
b
sin
dx 0 sin
sin
dy
¼ 0 sin
a
a
b
b
y
x
¼ J1mp
J1nq
(A41)
The possible combination of trigonometric integrals are reprex
x
x
x
sented by J1mn
, J2mn
, J3mn
, J4mn
, if the integration is evaluated
y
y
y
y
, J2mn
, J3mn
, J4mn
, along y. Defining am ¼ ðmp=aÞ,
along x or J1mn
bn ¼ ðnp=bÞ and ap ¼ ðpp=aÞ, bq ¼ ðqp=bÞ, the previous integrals assume the following values, when the integration is evaluated along x direction:
8
<0
¼ sinðam xÞ sinðan xÞdx ¼
: a
0
82
ða
<0
x
¼ cosðam xÞ cosðan xÞdx ¼ a
J2mn
:
0
2
ð
x
J1mn
ða
m 6¼ n
m¼n
m 6¼ n
m¼n
a
x
¼
J3mn
cosðam xÞ sinðan xÞdx
80
m¼n
>
<0
¼
2an
>
m 6¼ n
:
pðn2 m2 Þ
ða
x
¼ sinðam xÞ cosðan xÞdx
J4mn
80
m¼n
>
<0
¼
2am
>
m 6¼ n
:
pðm2 n2 Þ
and n þ m even
(A42)
and n þ m odd
and n þ m even
and n þ m odd
instead, when the integration is evaluated along y direction:
8
>
ðb
< 0 m 6¼ n
y
J1mn ¼ sinðbm yÞ sinðbn yÞdx ¼ b
>
0
: m¼n
2
8
>
ðb
< 0 m 6¼ n
y
¼ cosðbm yÞ cosðbn yÞdx ¼ b
J2mn
>
0
: m¼n
2
ðb
y
J3mn ¼ cosðbm yÞ sinðbn yÞdx
80
(A43)
m ¼ n and n þ m even
>
<0
¼
2bn
>
m 6¼ n and n þ m odd
:
pðn2 m2 Þ
ðb
y
¼ sinðam xÞ cosðan xÞdx
J4mn
80
m ¼ n and n þ m even
>
<0
¼
2bm
>
m 6¼ n and n þ m odd
:
pðm2 n2 Þ
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