Composite Structures 82 (2008) 209–216
www.elsevier.com/locate/compstructAnalytical solutions using a higher order refined computational
model with 12 degrees of freedom for the free vibration analysis
of antisymmetric angle-ply plates
K. Swaminathan *, S.S. Patil
Department of Civil Engineering, National Institute of Technology Karnataka, Srinivasnagar, Karnataka 575 025, India
Available online 9 January 2007Abstract
Analytical formulations and solutions to the natural frequency analysis of simply supported antisymmetric angle-ply composite and
sandwich plates hitherto not reported in the literature based on a higher order refined computational model with 12 degrees of freedom
already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which
account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displace-
ments with respect to the thickness coordinate thus modelling the warping of transverse cross sections more accurately and eliminating
the need for shear correction coefficients. In addition, another higher order computational model with five degrees of freedom already
available in the literature is also considered for comparison. The equations of motion are obtained using Hamilton’s principle. Solutions
are obtained in closed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying slenderness ratios, number
of layers, degrees of anisotropy, edge ratios and thickness of core to thickness of face sheet ratios are considered for analysis. Numerical
results with real properties using above two computational models are presented and compared for the free vibration analysis of mul-
tilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.
 2007 Elsevier Ltd. All rights reserved.
Keywords: Free vibration; Higher order theory; Shear deformation; Angle-ply plates; Analytical solutions1. Introduction
Laminated composite and sandwich plates and shells are
finding extensive usage in the aeronautical and aerospace
industries as well as in other fields of modern technology.
It has been observed that the strength and deformation
characteristics of such structural elements depend upon
the fibre orientation, stacking sequence and the fibre con-
tent in addition to the strength and rigidities of the fibre
and matrix material. Though symmetric laminates are
simple to analyse and design, some specific application of
composite and sandwich laminates requires the use of
unsymmetric laminates to fulfil certain design requirements.
Antisymmetric cross-ply and angle-ply laminates are the
special form of unsymmetric laminates and the associated* Corresponding author. Tel.: +91 824 2474340; fax: +91 824 2474033.
E-mail address: swami7192@yahoo.co.in (K. Swaminathan).
0263-8223/$ - see front matter  2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruct.2007.01.001theory offers some simplification in the analysis. The Classi-
cal Laminate Plate Theory [1] which ignores the effect of
transverse shear deformation becomes inadequate for the
analysis of multilayer composites. The First Order Shear
Deformation Theories (FSDTs) based on Reissner [2] and
Mindlin [3] assume linear in-plane stresses and displace-
ments respectively through the laminate thickness. Since
FSDTs account for layerwise constant states of transverse
shear stress, shear correction coefficients are needed to rec-
tify the unrealistic variation of the shear strain/stress
through the thickness. In order to overcome the limitations
of FSDTs, higher order shear deformation theories
(HSDTs) that involve higher order terms in the Taylor’s
expansions of the displacement in the thickness coordinate
were developed. Hildebrand et al. [4] were the first to intro-
duce this approach to derive improved theories of plates
and shells. Using the higher order theory of Reddy [5] free
vibration analysis of isotropic, orthotropic and laminated
210 K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–216plates was carried out by Reddy and Phan [6]. A selective
review of the various analytical and numerical methods
used for the stress analysis of laminated composite and
sandwich plates was presented by Kant and Swaminathan
[7]. Using the higher order refined theories already reported
in the literature by Kant [8], Pandya and Kant [9–13] and
Kant and Manjunatha [14], analytical formulations, solu-
tions and comparison of numerical results for the buckling,
free vibration and stress analyses of cross-ply composite
and sandwich plates were presented by Kant and Swamina-
than [15–18] and the finite element formulations and solu-
tions for the free vibration analysis of multilayer plates
were presented by Mallikarjuna [19], Mallikarjuna and
Kant [20], Kant and Mallikarjuna [21,22]. Recently the the-
oretical formulations and solutions for the static analysis of
antisymmetric angle-ply laminated composite and sandwich
plates using various higher order refined computational
models were presented by Swaminathan and Ragounadin
[23], Swaminathan et al. [24] and Swaminathan and Patil
[25]. In this paper, analytical formulations developed and
solutions obtained for the first time using a higher order
refined computational model with 12 degrees of freedom
is presented for the free vibration analysis of antisymmetric
angle-ply laminated composite and sandwich plates. In
addition, another higher order model with five degrees of
freedom already reported in the literature is also considered
for the analysis. Results generated using both the models
are presented for the antisymmetric angle-ply composite
and sandwich plates with real properties.
2. Theoretical formulation
2.1. Displacement model
In order to approximate the three-dimensional elasticity
problem to a two-dimensional plate problem, the displace-
ment components uðx; y; z; tÞ; vðx; y; z; tÞ and wðx; y; z; tÞ at
any point in the plate space are expanded in Taylor’s series
in terms of the thickness coordinate. The elasticity solution
indicates that the transverse shear stresses vary parabolically
through the plate thickness. This requires the use of a dis-
placement field in which the in-plane displacements are
expanded as cubic functions of the thickness coordinate. In
addition, the transverse normal strain may vary nonlinearly
through the plate thickness. The displacement field which
satisfies the above criteria may be assumed in the form [14]:
uðx; y; z; tÞ ¼ uoðx; y; tÞ þ zhxðx; y; tÞ
þ z2uoðx; y; tÞ þ z3h

xðx; y; tÞ
vðx; y; z; tÞ ¼ voðx; y; tÞ þ zhyðx; y; tÞ
þ z2v
ð1Þ
oðx; y; tÞ þ z3h

yðx; y; tÞ
wðx; y; z; tÞ ¼ woðx; y; tÞ þ zhzðx; y; tÞ
þ z2woðx; y; tÞ þ z3h

z ðx; y; tÞ
The parameters uo; vo are the in-plane displacements and wo
is the transverse displacement of a point ðx; yÞ on the mid-dle plane. The functions hx; hy are rotations of the normal
to the middle plane about y and x axes respectively. The
parameters u; vo o;w

o; h

x ; h
 
y ; hz and hz are the higher order
terms in the Taylor’s series expansion and they represent
higher order transverse cross sectional deformation modes.
Though the above theory was already reported earlier in
the literature and numerical results were presented using fi-
nite element formulations, analytical formulations and
solutions are obtained for the first time in this investigation
and hence the results obtained using the above theory are
referred to as present in all the tables. In addition to the
above, the following higher order shear deformation theory
[HSDT] with five degrees of freedom already reported in
the literature for the analysis of laminated composite and
sandwich plates are also considered for the evaluation pur-
pose. Results using these theories are generated indepen-
dently and presented here with a view to have all the
results on a common platform.
Reddy [5] 
uðx; y; z; tÞ ¼ uoðx; y; tÞ þ z hxðx; y; tÞ 
4z2 hxðx; y; tÞ þ
owo
3 h ox
vðx; y; z; tÞ ¼ voðx; y; tÞ þ
ð2Þ
z hyðx; y; tÞ
4z  2 ð owh x y tÞ þ oy ; ;
3 h oy
wðx; y; z; tÞ ¼ woðx; y; tÞ
In this paper the analytical formulations and solution meth-
od followed using the higher order refined theory given by
Eq. (1) is presented in detail. The geometry of a two-dimen-
sional laminated composite and sandwich plates with posi-
tive set of coordinate axes and the physical middle plane
displacement terms are shown in Figs. 1 and 2 respectively.
By substitution of the displacement relations given by Eq.
(1) into the strain–displacement equations of the classical
theory of elasticity, the following relations are obtained.
e ¼ e þ zj þ z2e þ z3jx xo x xo x
ey ¼ eyo þ zj þ z2e 3 y yo þ z jy
ez ¼ ezo þ zjz þ z2ezo
c ¼ e þ zj þ z2e þ z3jxy xyo xy xyo xy
cyz ¼ /y þ zjyz þ z2/ 3y þ z jyz
cxz ¼ /x þ zjxz þ z2/x þ z3jxz ð3Þ
where  	
ð Þ ¼ ouo ove o ouo ovoxo; eyo; exyo  ; ; þox oy oy ox   	
ð    ouo ovo oue o ovoxo; eyo; exyoÞ ¼ ; ; þox oy oy ox
ðe  zo; ezoÞ ¼ ðhz; 3hzÞ 	
ð  Þ ¼ ohx ohj j j j y ohx ohyx; y ; z ; xy ; ; 2w; þox oy o oy ox
K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–216 211
TYPICAL
3, z LAMINA
2
1
α
α
(1,2,3) - LAMINA REFERENCE AXES 
y
    	
ðj j j Þ ¼ oh
oh
x y oh ohx y
x ; y ; xy  ; ; þox oy oy o	x
ð oh ohj  z  zxz; jyzÞ ¼ 2uo þ 	; 2vox o þ oy
oh ohðjxz; j z zyzÞ ¼ ;ox oy
ð ow ow

/ / ox; x ;/y ;/

yÞ ¼ hx þ ; 3h
 þ ox 	;ox ox
h þ owoy ; 3h þ
owo ð4Þ
oy y oyθx
z, 3 νo
LAMINATE MID-
PLANE
wo
L = NL 
b
z L+1 u o θ z yL x
2
L = 2 L = 1 
a
(x,y,z) - LAMINATE REFERENCE AXES 
Fig. 1. Laminate geometry with positive set of lamina/laminate reference
axes, displacement components and fibre orientation.2.2. Constitutive equations
Each lamina in the laminate is assumed to be in a three-
dimensional stress state so that the constitutive relation for
a typical lamina L with reference to the fibre–matrix coor-
d8inate9axes2(1–2–3) can be written as>> 3 8 9> r1 >>>
L
6C
L L
11 C12 C13 0 0 0 >> e1>< r2 => 666
>>
C12 C22 C23 0 0 0 77 > e2 >
> r3 > 6
7 > >
6C13 C23 C33 0 0 0 7
< e =3
> ¼ 7>>s12>>> 646 0 0 0 C44 0 0 77
7 >>>c >12>>
ð5Þ
>:s23;> 0 0 0 0 C55 0 5 >>:c23;>>
s13 0 0 0 0 0 C66 c13
where ðr1; r2; r3; s12; s23; s13Þ are the stresses and
ðe1; e2; e3; c12; c23; c13Þ are the linear strain components re-
ferred to the lamina coordinates (1–2–3) and the Cij’s are
the elastic constants or the elements of stiffness matrix
[25] of the Lth lamina with reference to the fibre axes (1–
2–3). In the laminate coordinate ðx; y; zÞ the stress strain
relations fo2r the Lth lamina can be written as8 9 3Q L>> r >>L 66 11
Q12 Q13 Q14 0 0 8 9L
> x > 6 7 >> ex>> r >
Q >>
> 6 22
Q23 Q24 0 0 7
< = 6 7 > e >y y6 Q Q > >33 34 0 0 777 >r < e =>z z
>>>
¼ 6 Q
s 44
0 0 7
> xy>> 6 7 >cxy>>: >syz >;>
666 symmetric4 Q Q 77
7 >>>> >cyz
55 565 : ;
>>>
sxz c
Q xz66
ð6Þ
where ðrx; ry ; rz; sxy ; syz; sxzÞ are the stresses and
ðex; ey ; ez; cxy ; cyz; cxzÞ are the strains with respect to the lam-
inate axes. Qij’s are the transformed elastic constants or the
stiffness matrix [25] with respect to the laminate axes x; y; z:
2.3. Hamilton’s principle
Hamilton’s principle [26] can be written in analytical
foZrm as follows:t2
d ½K  ðU þ V Þdt ¼ 0 ð7Þ
t1where U is the total strain energy due to deformations, V is
the potential of the external loads, K is the kinetic energy
and U þ V ¼ P is the total potential energy and d denotes
the variational symbol. Substituting the appropriate energy
expression in the above equation, the final expression can
thus beZwr"it Z
ttenZash
2
0 ¼  ðrxdex þ rydey þ rzdez þ sxydcxy þ syzdcZ  yz0 h2 A
þs þ þxzdc
Z xz
ÞdAdz pz dw dA dt
t Z Ah Z2
þ d q½ðu_ Þ2 þ ðv_Þ2 þ ðw_ Þ2dAdzdt ð8Þ
2 0 h2 A
where q is the mass density of the material of the laminate
and pþz is the transverse load applied at the top surface of
the plate and wþ ¼ wo þ ðh=2Þhz þ ðh2=4Þwo þ ðh
3=8Þhz is
the transverse displacement of any point on the top surface
of the plate and the superposed dot denotes differentiation
with respect to time. Using Eqs. (1), (3) and (4) in Eq. (8)
and integrating the resulting expression by parts, and col-
lecting the coefficients of du0; dv0; dw0; dhx; dhy ; dh z; du0;
dv; dw0 0; dh
 
x ; dhy ; dh

z the following equations of equilib-
rium are obtained:
212 K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–216
TYPICAL LAMINA
3, z IN FACE SHEET 
2
1
α
α
( 1,2,3 ) - LAMINA REFERENCE AXES 
y
θx
z, 3 νo
LAMINATE MID-
PLANE
FACE
w o SHEET
L = NL 
b
z L+1 u θz L CORE
o y x
L = 2 L = 1 
a
( x,y,z ) - LAMINATE REFERENCE AXES 
Fig. 2. Geometry of a sandwich plate with positive set of
lamina/laminate reference axes, displacement components and fibre
orientation.oNx þ oNdu xy: ¼ I €  €o x y 1€uo þ I2hx þ I3€uo þ I4ho o x
oN
dv yo : þ
oNxy ¼ I €v þ I €h þ I €v þ I €h
oy ox 1 o 2 y 3 o 4 y
oQx oQdw þ y: þ pþ €  €o x y z ¼ I1w€o þ I2hz þ I3w€o þ I4ho o z
oMx þ oMdh xy: Q ¼ I u€ þ I €h þ I €u þ I €hx x y x 2 o 3 x 4 o 5o o x
oM oM
dh yy : þ xy Qy ¼ I2€v €o þ I3hy þ I4€v þ I €hoy ox o 5 y
oSx þ oSy  þ h


dh : N pþ

z z z ¼ I €  €x y 2w€o þ I3hz þ I4w€o þ I5ho o 2 z
N  oN o
du x þ xyo :  2Sx ¼ I3€u €  €x o þ I4hx þ I5€u þ I6ho oy o x
oN  y oN

dv þ xyo :  2Sy ¼ I3€vo þ I4h€ þ I €vy 5 o þ I €6hoy ox y
 oQ
 oQ 2 
 
dw x yo : þ   þ
h
2M pþz z ¼ I3w€o þ I €4hz þ I w€5 o þ I6h€ox oy 4 z
 
dh
oM oMx þ xyx :  3Qx ¼ I4u€o þ I €5hx þ I6€uo þ I €x y 7ho o xoM 
dh y
oMxy
y : þ  3Qy ¼ I4€v þ I € o 5hy þ I6€vo þ I €y x 7ho o y
S oSo 3
dh x þ y h

 
:  3N  þ pþ ¼ I w€ þ I €h þ I w€ þ I €z x y z z 4 o 5 z 6 o 7ho o 8 z
ð9Þ
and boundary conditions are the form:
On the edge x = constant
u ¼ u or N ¼ N u ¼ uo o x x o o or N  ¼ N x x
vo ¼ vo or Nxy ¼ N v   xy o ¼ vo or Nxy ¼ Nxy
wo ¼ wo or Qx ¼ Q w   x o ¼ wo or Qx ¼ Qx
hx ¼ hx or Mx ¼ Mx h ¼ h or M ¼ Mx x x x
hy ¼ hy or Mxy ¼ M h ¼ h or M ¼ Mxy y y xy xy
h    z ¼ hz or Sx ¼ Sx hz ¼ hz or Sx ¼ Sx
ð10Þ
On the edge y = constant
uo ¼ u or N ¼ N u ¼ u o xy xy o o or Nxy ¼ N xy
vo ¼ vo or N ¼ N vy y o ¼ vo or N  ¼ N y y
w o ¼ wo or Qy ¼ Qy wo ¼ w or Qo y ¼ Qy
h ¼ h or M ¼ M h ¼ h x x xy xy x x or Mxy ¼ Mxy
hy ¼ hy or My ¼ My hy ¼ hy or My ¼ My
hz ¼ hz or S y ¼ Sy hz ¼ hz or Sy ¼ Sy
ð11Þ
2where the st3ress resultants8are d9efined by
6 M M

x x > rx6 >6 M M 777 XNL Z z > >y Lþ1 < =4 y ry  
 ¼ z z3 dz ð12ÞMz 0 5 L¼1 z >>L rz >>
"Mxy M #
 : ;
xy
Q Q XNL Z
sxy
z  Lþ1
x x s¼ xz
 
1 z2 dz ð13Þ
2Qy Qy 3 L¼1 zL 8syz> 96 Nx Nx rx >
466 Ny Ny 77
7 XNL Z z <> >Lþ1 =
5 ¼ >>
ry
>>
 
1 z2 dz ð14Þ
Nz Nz L¼ r1 zL z
" N  : ;xy N#xy  sxy
S S XNL Z zx Lþ1x sxz  
3
S S
¼ z z dz ð15Þ
y y L¼ zL s1 yzand the inertias are given byZ h
2
I1; I2; I 23; I4; I5; I6; I7;¼ qð1; z; z ; z3; z4; z5; z6; Þdz ð16Þ
h2
The resultants in Eqs. (12)–(15) can be related to the total
strains in Eq. (3) by the following equations:
K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–216 2138 9 >8> 9>>> Nx > >
ouo
ox >>>ov
>> Ny> >
>> >>> o >oy > 8> 9ouo >ou > > oy >
>> N
 >
x >
>  >>>
>>> o >> >x > ovo >o ox >
N ov

>> y< >>
> o
oy
= >><> >
>> >> >>> ou
>
o >
oy >
N >z hz >= >< ovo =>
>> N z >> ¼ ½A
ox
>>M >> >>>>
hz >>þ ½A0>ohx >> >>
ohx
> x > > ox > >
>
oy >
> >> > > > > ohy >>M > > ohy > > ox >> y > > oy oh>>M
 >> >> ohx x >>> >:>
x
oy >>
> > > x > >
>
:My ; >
o
ohy ;
ohy >
M >: oy >z 8  ;
ox
w
> o 9> ð17Þ
>>
ouo
> ox >> ovo >> >8> oy>  >>> >>>
ouo
oy >>9
> ouo ov >oox >8 ov > > ox >><>
9> >N => >
> o >> >> ou >>o
xy
 >>
oy
< > >h >zN => >><
oy >>
ovo >=
> xy:>Mxy >
¼ ½ 0> h >þ ½ > ox> B; > z
B oh
>>
x >
ohx > > oy >
M > ox >>> >>> oh >xy > yoh >> y >> >> ox >
>>>
oy
ohx
> >>>> >
oh >>x
oy >
> ox>: oh

y > >: oh ;>y>; oxoy
8 wo>> 9>> >8> h 98 9 >
h
>>
x > > y >>>ow ow> o
o
>> >> > ox >> >> oy >><Qx = < > >  >h > >Q x = < hy =>
>>
x
>>
 
¼ ½ owD> o> ox >>þ ½
ow
D0 o
: Sx ; > oy> >u >> >>> v >S >>
o >> >>
>
o >>
x > ohz: ox ;> >
ohz
: >oy >oh oh >z ;
8 zox 9 8 oy> > > 9> ð18Þh hy
8> 9> >>
>> x > > >> ow
ow
o > > o >
>Q > > ox >
> >> oy >< >y > >  >h hx > > y >
>Q
 = < = < =
> y >

> ¼ ½E0>
ow
> o >>þ ½E>>
owo
: S ; > ox > > oyy > u  > o >> >> vo >
>>>Sy >> ohz: ox >>; >>
ohz
: >oy >oh  >z oh ;z
ox oy
where the matrices ½A; ½A0; ½B; ½B0; ½D; ½D0; ½E; ½E0 are the
matrices of plate stiffnesses whose elements are already re-
ported in article [25].
3. Analytical solutions
Here the exact solutions of Eqs. (9)–(18) for antisym-
metric angle-ply plates are considered. Assuming that theplate is simply supported with SS-2 boundary conditions
[27] in such a manner that tangential displacement is
admissible, but the normal displacement is not, the follow-
ing boundary conditions are appropriate:
At edges x ¼ 0 and x ¼ a;
uo ¼ 0; wo ¼ 0; hy ¼ 0; hz ¼ 0; Mx ¼ 0; Nxy ¼ 0;
u ¼ 0; wo o ¼ 0; h
   
y ¼ 0; hz ¼ 0; Mx ¼ 0; Nxy ¼ 0
ð19Þ
At edges y ¼ 0 and y ¼ b;
vo ¼ 0; wo ¼ 0; hx ¼ 0; hz ¼ 0; My ¼ 0; Nxy ¼ 0;
vo ¼ 0; wo ¼ 0; h

x ¼ 0; h
 
z ¼ 0; My ¼ 0; N xy ¼ 0
ð20Þ
Following Navier’s approach [27–29], the solution to the
displacement variables satisfying the above boundary con-
ditions can be expressed in the following forms:X1 X1
u ¼ u sin ax cos byeixto omn
Xm¼11 Xn¼11
u ¼ u ixto o sin ax cos byemn
Xm¼1 n¼11 X1
v ¼ v ixto omn cos ax sin bye
Xm¼1 n¼11 X1
v ¼ vo o cos ax sin byeixtmn
mX¼1 n¼11 X1
w ¼ w ixto omn sin ax sin bye
Xm¼1 n¼11 X1
w ¼ wo o sin ax sin byeixtmn
Xm¼1 n¼11 X1 ð21Þ
hx ¼ hxmn cos ax sin byeixt
Xm¼11 Xn¼11
h ¼ h cos ax sin byeixtx xmn
Xm¼1 n¼11 X1
h ixty ¼ hy sin ax cos byemn
Xm¼1 Xn¼11 1
h ¼ hy y sin ax cos byeixtmn
Xm¼11 Xn¼11
hz ¼ hzmn sin ax sin byeixt
mX¼1 n¼11 X1
h ¼ h sin ax sin byeixtz zmn
m¼1 n¼1
pþz ¼ 0
where a ¼ mpa ; b ¼ npb and x is the natural frequency of the
system. Substituting Eqs. (19)–(21) in to Eq. (9) and col-
lecting the coefficients one obtains
214 K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–2168>> 9> uo> >
>
> v> o
>
>>>>>wo> >>>> hx >><> hy =>>
ð½X 1212  k½M 1212Þ>>
hz
 >> ¼ f0g ð22Þ>> uo >
>>
>> v >o >>w >>> o >
>>>
h >x >
 >: h >y >
h
;
z 121
where k ¼ x2 for any fixed values of m and n. The elements
of coefficient matrix [X] and mass matrix [M] are already
reported in Refs. [25,17] respectively.
4. Numerical results and discussion
In this section, various numerical examples solved are
described and discussed for establishing the accuracy of
the theory for the free vibration analysis of antisymmetric
angle-ply laminated composite and sandwich plates. For
all the problems a simply supported plate with SS-2 bound-
ary conditions is considered for the analysis. Results are
obtained in closed-form using Navier’ssolution technique
by solving the eigenvalue equation. The non-dimensional-
ized natural frequencies computed for two, four and eight
layer antisymmetric angle-ply square laminate with layers
of equal thickness are given in Tables 1 and 2.
The orthotropic material properties of individual layers
in all the above laminates considered are E1=E2 ¼ open,
E2 ¼ E3, G12 ¼ G13 ¼ 0:6E2, G23 ¼ 0:5E2, t12 ¼ t13 ¼ t23 ¼
0:25:
The variation of natural frequencies with respect to
side-to-thickness ratio a/h is presented in Table 1. The nat-
ural frequencies obtained using the present theory are
compared with Reddy’s theory. In the case of thick plates
(a/h ratios 2, 4, 5 and 10) there is a considerable differenceTable 1 pffiffiffiffiffiffiffiffiffiffi
Non-dimensionalized fundamental frequencies x ¼ ðxb2=hÞ q=E2 for a simp
Lamination and number of layers Source a/h
2 4 5
(45/45)1 Present 5.3325 8.8426 10.0
HSDT [5]a 6.2837 9.7593 10.8
(45/45)2 Present 5.5674 10.0731 11.9
HSDT [5]b 6.1067 10.6507 12.5
(45/45)4 Present 5.9234 10.7473 12.7
HSDT [5]a 6.2836 10.9905 12.9
E1=E2 ¼ 40, E2 ¼ E3, G12 ¼ G13 ¼ 0:6E2, G23 ¼ 0:5E2, t12 ¼ t13 ¼ t23 ¼ 0:25.
a Results using this theory are computed independently and are found to be
b Results using this theory are computed independently for the first time.exists between the results computed using the present and
the Reddy’s theory. The variation of natural frequencies
with respect to side-to-thickness ratio a/h for different
E1/E2 ratio is presented in Table 2. For a four layered
thick plate with a/h ratio equal to 2 and E1/E2 ratio equal
to 3 and 10, the percentage difference in values predicted
by present theory are 0.13% and 3.51% lower as compared
to Reddy’s theory. At higher range of E1/E2 ratio equal to
20–40, the percentage difference in values between both the
theories is very much higher and Reddy’s theory very
much over predicts the natural frequency values. For a
four layered thick plate with a/h ratio equal to 2 and
E1/E2 ratio equal to 20, 30 and 40, the percentage differ-
ence in values predicted by present theory are 6.08%,
7.99% and 9.70% lower as compared to Reddy’s theory.
The difference between the models tends to reduce for thin
and relatively thin plates. Irrespective of the number of
layers the percentage difference in values between the
two theories increases with the increase in the degree of
anisotropy. As the number of layer increases, the percent-
age difference in values between the two theories decreases
significantly.
The variation of fundamental frequency with respect to
the various parameter like the side-to-thickness ratio (a/h),
thickness of the core to thickness of the flange (tc/tf) and
the aspect ratio (a/b) of a five layer sandwich plate with
antisymmetric angle-ply face sheets are given in Tables 3
and 4. The following material properties are used for face
sheets and the core [23]:
• Face sheets (Graphite-epoxy T300/934)
E1 ¼ 19 106 psi ð131 GPaÞ
E2 ¼ 1:5 106 psi ð10:34 GPaÞ
E2 ¼ E3; G 612 ¼ 1 10 psi ð6:895 GPaÞ
G13 ¼ 0:90 106 psi ð6:205 GPaÞ
G 623 ¼ 1 10 psi ð6:895 GPaÞ
t12 ¼ 0:22; t13 ¼ 0:22; t23 ¼ 0:49
q ¼ 0:057 lb=in:3 ð1627 kg=m3Þly supported antisymmetric angle-ply square laminated plate
10 12.5 20 25 50 100
350 12.9115 13.4690 14.1705 14.3500 14.6012 14.6668
401 13.2630 13.7040 14.2463 14.3827 14.5723 14.6214
465 17.8773 19.4064 21.6229 22.2554 23.1949 23.4499
331 18.3221 19.7621 21.8063 22.3798 23.2236 23.4507
523 19.1258 20.7784 23.1829 23.8713 24.8959 25.1741
719 19.2659 20.8884 23.2388 23.9091 24.9046 25.1744
the same as reported in the Ref. [6].
K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–216 215
Table 2 pffiffiffiffiffiffiffiffiffiffi
Non-dimensionalized fundamental frequencies x ¼ ðxb2=hÞ q=E2 for a simply supported antisymmetric angle-ply square laminated plate
Lamination and number of layers E1/E2 Source a/h
2 4 10 20 50 100
(45/45)1 3 Present 4.5312 6.1223 7.1056 7.3001 7.3583 7.3666
HSDT [5]b 4.5052 6.0861 7.0739 7.2704 7.3292 7.3373
10 Present 4.9742 7.2647 8.9893 9.3753 9.4943 9.5123
HSDT [5]b 5.1718 7.3469 8.9660 9.3265 9.4377 9.4538
20 Present 5.1817 8.0490 10.6412 11.2975 11.5074 11.5385
HSDT [5]b 5.7094 8.4151 10.7151 11.2772 11.4553 11.4819
30 Present 5.2771 8.5212 11.8926 12.8422 13.1566 13.2035
HSDT [5]b 6.0681 9.1752 12.0971 12.8659 13.1154 13.1521
40 Present 5.3325 8.8426 12.9115 14.1705 14.6012 14.6668
HSDT [5]a 6.2837 9.7593 13.2630 14.2463 14.5723 14.6214
(45/45)2 3 Present 4.6498 6.4597 7.6339 7.8724 7.9442 7.9545
HSDT [5]b 4.6546 6.4554 7.6267 7.8649 7.9366 7.9472
10 Present 5.2061 8.3447 11.4116 12.2294 12.4952 12.5351
HSDT [5]b 5.3887 8.5119 11.4674 12.2380 12.4866 12.5238
20 Present 5.4140 9.3306 14.4735 16.2570 16.8949 16.9927
HSDT [5]b 5.7431 9.6855 14.6609 16.3146 16.8964 16.9848
30 Present 5.5079 9.7966 16.4543 19.2323 20.3134 20.4839
HSDT [5]b 5.9481 10.2785 16.7750 19.3499 20.3277 20.4807
40 Present 5.5674 10.0731 17.8773 21.6229 23.1949 23.4499
HSDT [5]b 6.1067 10.6507 18.3221 21.8063 23.2236 23.4507
a Results using this theory are computed independently and are found to be the same as reported in the Ref. [6].
b Results using this theory are computed independently for the first time.
Table 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Non-dimensionalized fundamental frequencies x ¼ ðxb2=hÞ ðq=E2Þf for a simply supported antisymmetric angle-ply (45/45/core/45 /45) square
sandwich plate
tc/tf Source a/h
2 4 10 20 50 100
4 Present 2.6404 4.5712 9.8197 15.0371 19.1695 20.0845
HSDT [5]a 3.0986 5.7985 12.0510 16.8312 19.6858 20.2163
10 Present 1.2805 2.1911 5.0653 9.2740 16.2062 19.3098
HSDT [5]a 1.6929 3.2171 7.4895 12.6964 18.4604 20.1355
20 Present 0.7538 1.3487 3.2154 6.1552 12.4654 16.7293
HSDT [5]a 0.9806 1.8783 4.5392 8.4083 14.9592 18.0073
50 Present 0.6079 1.1836 2.8972 5.5259 10.8499 14.1053
HSDT [5]a 0.6473 1.2696 3.1080 5.8904 11.2731 14.3233
a Results using this theory are computed independently for the first time.
Table 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Non-dimensionalized fundamental frequencies x ¼ ðxb2=hÞ ðq=E2Þf for
a simply supported antisymmetric angle-ply (45/45/core/45 /45)
sandwich plate with a=h ¼10
a/b Source tc/tf
4 10 20 50
1.0 Present 9.8197 5.0653 3.2154 2.8972
HSDT [5]a 12.0510 7.4895 4.5392 3.1080
1.5 Present 5.7975 2.9101 1.8354 1.6498
HSDT [5]a 7.2503 4.3308 2.5939 1.7706
2.0 Present 4.1579 2.0562 1.2900 1.1557
HSDT [5]a 5.2441 3.0627 1.8216 1.2405
2.5 Present 3.2833 1.6054 1.0020 0.8939
HSDT [5]a 4.1585 2.3878 1.4122 0.9595
3.0 Present 2.7355 1.3268 0.8241 0.7315
HSDT [5]a 3.4698 1.9660 1.1577 0.7849
a Results using this theory are computed independently for the first time.• Core properties (isotropic)
E1 ¼ E2 ¼ E3 ¼ 2G ¼ 1000 psi ð6:90 103 GPaÞ
G12 ¼ G13 ¼ G23 ¼ 500 psi ð3:45 103 GPaÞ
t12 ¼ t13 ¼ t23 ¼ 0
q ¼ 0:3403 102 lb=in:3 ð97 kg=m3Þ
The results clearly show that in the case of thick plates
for all the parameters considered, there is a considerable
difference exists between the results computed using the
present theory and Reddy’s theory. In the case of a
square plate with tc/tf ratio equal to 4 and a/h ratio
equal to 10, the percentage difference in values predicted
by Reddy’s theory is 22.72% higher compared to present
theory. For a rectangular plate with a/b ratio equal to 2
216 K. Swaminathan, S.S. Patil / Composite Structures 82 (2008) 209–216and tc/tf ratio equal to 10, Reddy’s theory overestimates
the natural frequency by 48.95%. The Reddy’s theory
very much overestimates the natural frequency values
both for square and rectangular plates.
5. Conclusion
Analytical formulations and solutions to the natural fre-
quency analysis of simply supported antisymmetric angle-
ply composite and sandwich plates hitherto not reported
in the literature based on a higher order refined theory
which takes in to account the effects of both transverse
shear and transverse normal deformations are presented.
The accuracy of the present computational model with 12
degrees of freedom in comparison to other higher order
model with five degrees of freedom has been established.
It has been concluded that for all the parameters considered
Reddy’s theory very much over predicts the natural fre-
quency values both for the composite and sandwich plates.
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