Number 2
Volume 18 February 2012
Journal of Engineering
FREE VIBRATION ANALYSIS OF COMPOSITE LAMINATED
PLATES USING HOST 12
Dr. Adnan N. Jameel
Mechanical Eng. Dep.
University of Baghdad
Salam Ahmed Abed
Mechanical Eng. Dep.
University of Baghdad
ABSTRACTE
This paper presents an application of a Higher Order Shear Deformation Theory (HOST 12) to problem
of free vibration of simply supported symmetric and antisymmetric angle-ply composite laminated plates.
The theoretical model HOST12 presented incorporates laminate deformations which account for the effects
of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane
displacements with respect to the thickness coordinate – thus modeling the warping of transverse crosssections more accurately and eliminating the need for shear correction coefficients. Solutions are obtained in
closed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying number of
layers, degrees of anisotropy and slenderness ratios are considered for analysis. The results compared with
those from exact analysis and various theories from references.
‫الخالصة‬
‫) لمسألة األهتزاز الحر للصفائح‬Higher Order Shear Deformation Theory (HOST 12)( ‫هذا البحث يقدم التطبيق لنظرية‬
‫م توفهلتئ الطبقفئ التفس تا فر‬.‫ تف‬HOST 21 ‫مف‬.‫ لن الة رلف المق‬.‫الغيفر متمئثلف ذاا الليفئا الغيفر متدئمف‬
‫الطبقيف المركبف المتمئثلف‬
‫ هكذا توكل توفهله‬- ‫ إجتئد التهزلع الالخطس لزاحئ الم تهي بئلة ب لل مك‬/ ‫ إجتئد طبيدس م تدرض‬،‫تأثيرا توهله القص الم تدرض‬
‫ق أكثر تزلل الحئج لمدئمال تصحي القصن الحلهل للمدئدال تضمةت الحل المضفبهط للصفائح الطبقيف‬.‫المقئطع الدرضي الم تدرض ب‬
‫د الطبقئ درجف التتر بفس ت فب ال فمك الفر الدفرضن‬.‫ لن أُخذ بئلعتبئر تأثير تةهلع مهاصائ الصائح كد‬Navier Solution ‫المركب ذ‬
. ‫د‬.‫ت رلئ متةهع من مصئدر متد‬
‫الةتئح قهرتت مع حلهل مضبهط‬
Keywords: Free vibration; Higher order theory; Shear deformation; Angle-ply plates;
Analytical solutions.
267
Dr. Adnan N. Jameel
Salam Ahmed Abed
FREE VIBRATION ANALYSIS OF COMPOSITE
LAMINATED PLATES USING HOST 12
INTRODUCTION
order shear deformation theories (HSDTs)
Laminated composite plates and shells are
that involve higher order terms in the Taylor’s
finding extensive usage in the aeronautical
expansions of the displacement in the
and aerospace industries as well as in other
thickness
fields of modern technology. It has been
(Hildebrand et al., 1949) were the first to
observed that the strength and deformation
introduce this approach to derive improved
characteristics of such structural elements
theories of plates and shells. Using the higher
depend upon the fiber orientation, stacking
order theory of (Reddy, 1984) free vibration
sequence and the fiber content in addition to
analysis
the strength and rigidities of the fiber and
laminated plates was carried out by (Reddy
matrix material. Though symmetric and
and Phan, 1985). A generalized Levy-type
antisymmetric laminates are simple to analyze
solution in conjunction with the closed form
and design, some specific application of
solution was developed for the bending,
composite laminates requires the use of
buckling and vibration of antisymmetric
symmetric and antisymmetric laminates to
angle-ply laminated plates by A. (Khdeir A.,
fulfill certain design requirements. Symmetric
1989). The exact solutions were obtained for
and antisymmetric angle-ply laminates are the
the classical Kirchhoff theory and the
special form of symmetric and antisymmetric
numerical results were compared with their
laminates and the associated theory offers
counterparts using the first order transverse
some simplification in the analysis. The
shear deformation theory. The comparisons
Classical Laminate Plate Theory (Reissner E.
showed that the results obtained within the
and Stavsky Y., 1961) which ignores the
classical
effect
deformation
significantly inaccurate. A selective review of
becomes inadequate for the analysis of
the various analytical and numerical methods
multilayer composites. The First Order Shear
used for the stress analysis of laminated
Deformation Theories (FSDTs) based on
composite and sandwich plates was presented
(Reissner E., 1945) and (Mindlin RD., 1951)
by (Kant and Swaminathan, 2001). Using the
assume
of
transverse
linear
shear
of
were
isotropic,
laminated
developed.
orthotropic
theory
could
and
be
stresses
and
higher order refined theories already reported
through
the
in the literature by (Kant, 1982), (Pandya and
laminate thickness. Since FSDTs account for
Kant, 1988) and (Kant and Manjunatha,
layerwise constant states of transverse shear
1988), analytical formulations, solutions and
stress, shear correction coefficients are needed
comparison of numerical results for the
to rectify the unrealistic variation of the shear
buckling, free vibration and stress analyses of
strain/stress through the thickness. In order to
cross-ply composite and sandwich plates were
overcome the limitations of FSDTs, higher
presented by (Kant and Swaminathan, 2002).
displacements
in-plane
coordinate
respectively
268
Number 2
Volume 18 February 2012
5. The in-plane stresses  X ,  Y and  XY
Recently the theoretical formulations and
solutions
for
the
static
analysis
Journal of Engineering
of
are cubic functions of z.
antisymmetric angle-ply laminated composite
6. The normal to the mid-surface before
and sandwich plates using various higher
deformation
order refined computational models were
necessarily remaining normal to the
presented by (Swaminathan and Ragounadin,
mid-surface after deformation.
2004), (Swaminathan et al., 2006) and
are
straight,
but
not
7. The transverse normal strain  Z is not
(Swaminathan and Patil, 2008).
zero.
The parameters uo, vo are the in-plane
THEORETICAL FORMULATION
displacements and wo is the transverse
Higher Order Shear Deformation Theory
(HSDT 12)
displacement of a point (x, y) on the middle
For the first time, derived the equation of
normal to the middle plane about y and x axes
motion
based
on
higher-order
plane. The functions x, y are rotations of the
shear
respectively.
deformation theory (HOST 12) in the present
u o* , v o* , wo* ,
in the Taylor’s series expansion and they
The assumptions of a higher order plate
represent
theory can also be used within equivalent
higher-order
transverse
cross
sectional deformation modes. This is done by
single layer formulation from (Swaminathan
taking into account the parabolic variation of
and Patil, 2008):
transverse shear stresses through the thickness
u  x , y , z , t   u o  x , y , t   z x  x , y , t 
 z 2 u o x, y, t   z 3 x x, y, t 
v  x , y , z , t   v o  x , y , t   z y  x , y , t 
 z 2 vo x, y, t   z 3 y  x, y, t 
of the plate (Swaminathan and Patil, 2008).
The strain components will be derived,
(1)
based on the displacement, as:
w x, y, z , t   wo  x, y, t   z z  x, y, t 
ε *xo 
κ *x 
ε x  ε xo  κ x 
* 
 *
ε   ε   κ 
 y   yo   y  2 ε yo  3 κ y 
      z *   z  *   z  
ε z  ε zo  κ z 
ε zo 
0 
γ xy  ε xyo  κ xy 
ε * 
κ * 
     
 xy 
 xyo 
 z w  x, y , t   z   x, y , t 

o
parameters
 x* , y* , z* and  z are the higher-order terms
study.
2
The
3

z
1. The plate may be moderately thick.
2. The in-plane displacement u (x, y, z, t)
*
*
γ yz  φ y  κ yz  2 φ y  3 κ yz 
      z   z  *   z  * 
φx 
κ xz 
γ xz  φx  κ xz 
and v (x, y, z, t) are cubic functions of z.
3. The transverse displacement w (x, y, z,
t)of any point (x, y) cubic functions of z.
where:
4. The transverse shear stress  XZ ,  YZ are
parabolic in z.
269
(2)
Dr. Adnan N. Jameel
Salam Ahmed Abed
 uo

 x

 vo

 y



 uyo  vxo  



 uo*

*
x

 xo
 

 *  
*
 vo

 yo    y

 *  

 xyo   uo*  vo*  

x 
 y


 zo    z 

 *    *

 zo  3 z 

  x

x

 x   


y
 

y


 y  

 * 

*
 z   2 w

o
 xy  

    x  y 
 x 

 y

*
  x

*

x


 x 



*
 *    y



 y   y

 * 

*

 xy    x*  y 
 x 

 y

 z  
*
 xz  
2u o  x  
    *  z  
2v  y 
 yz  
 o


  z* 
*




 xz   x 

 *    *






 yz   yz 




  wo  
x
x 
 x  

 *  3 *  wo*  
x  
 x   x
   
wo  
 y   y  y  
 *  

 y   * wo*  
3  y
 y
 

FREE VIBRATION ANALYSIS OF COMPOSITE
LAMINATED PLATES USING HOST 12
  xo 


  yo  
 
 xyo 
 x 
 
 y
 z 
 
 xy 
 yz 
 
 xz  k
Q 11
Q 12
Q 13

Q 22
Q 23


Q 33



Symmetric


Q 14
Q 24
0
0
Q 34
Q 44
0
0
Q 55
0

0
0

0
Q 56 

Q 66  k
    x   xo*   x* 
 xo     *   * 
  yo   y   yo   y 
   *   *  0 
  zo   z  2  zo  3  
   z    z  *   z  * 
 xyo   xy   xyo   xy 
  y     *   * 
   yz   y   yz 
  x   xz   x*   xz* 


where [ Q ] from equ.
(4)
Q  T  QT ,
T
[Q] given by :
(3)
Q11=E1(1-v23v32)/Δ
Q12=E1(v12-v31v23)/Δ
Q13=E1(v31-v21v32)/Δ
Q23=E2(v32-v12v31)/Δ
Q33=E3(1-v31v23)/Δ
(5a)
Q44=G12
Q55=G23
Q66=G13
where:
Δ= (1-v12v21 -v23v32 -v31v13 -2v12v23v31)
(5b)
And the transformation matrix [T] is given by
the transformation equations:
Substituting eq. (2) in the stress- strain
relation of the lamina, the constitutive
relations for any layer in the (x, y) can be
expressed in the form:
Q 11= Q11c4
+ 2(Q12 + 2Q33)s2c2+ Q22s4
270
(5c)
Number 2
Volume 18 February 2012
 x 
 N x* 
h
*
 N  2  
 y
 y 2
 *      z dz
 N z   h 2  z 
 N xy* 
 xy 
 k
Q 12= (Q11 + Q22 - 4Q33)s2c2
+ Q12(s4 + c4)
Q 13= Q13c2 + Q23s2
Q 14= (Q11 - Q12- 2Q44) sc3
+(Q12 - Q22 +2Q44)cs3
Q 22= Q11s4+2(Q12 + 2Q33)s2c2
+ Q22c4
Q 23= Q13s2 + Q23c2
N

k 1
Q 11





Q 13 Q 14 

Q 23 Q 24 
symmetric Q 33 Q 34 

Q 44  k
Q 12
Q 22
  
 x 
 Z k  xo 

Zk 
   yo  2
 y  3
    z dz    * z dz
Z k 1   z 
Z k 1   zo 

  xyo 
 xy 

Q 24= (Q11 - Q12 - 2Q44)cs3
+(Q12 - Q22 + 2Q44)sc3
Q 33= Q33
Q 34= (Q31- Q32)cs
Q 44= (Q11 - 2Q12 + Q22 - 2Q44)c2 s2
+ Q44 (c4 + s4)
Q 55= Q55s2 + Q66c2
Journal of Engineering
(6b)

  x* 
  xo* 

Zk  * 
Zk  * 
  yo  4
 y  5 
   *  z dz    z dz 
Z k 1   zo 
Z k 1  0 

*
 xyo 
 xy* 

 

(5d)
Q 56= (Q66 - Q55)cs
Q 66= Q55s2+Q66c2
Mx 
 M  h 2  x 
x
 *     y  z dz k
 M z   h 2  
 xy  k
M xy 
All other elements of [Qij] and [ Q ij] are zero.
The entire collection of forces and
moments resultants for N-layered laminated
are defined as:
(6a)
 x 
 Nx 
h 
 N y  2  y 
      dz
 N z   h 2  z 
 N xy 
 xy 
 k
Q 11
Q 12
Q 13 Q 14 


N
Q 23 Q 24 
Q 22


k 1 
symmetric Q 33 Q 34 


Q 44  k

  
 x 
 Z k  xo 

Zk 
   yo 
 y 
    dz    * z dz
Z k 1   z 
Z k 1   zo 
  xyo 
 xy 


  x* 
  xo* 

Zk  * 
Zk  * 
  yo  2
 y  3 
   *  z dz    z dz 
Z k 1   zo 
Z k 1  0 

*
 xyo 
 xy* 

 

N

k 1
Q 11





Q 13 Q 14 

Q 23 Q 24 
symmetric Q 33 Q 34 

Q 44  k
Q 12
Q 22
  
 x 
 Z k  xo 

Zk 
   yo 
 y  2
    z dz    * z dz
Z k 1   z 
Z k 1   zo 
  xyo 
 xy 


  x* 
  xo* 

Zk  * 
Zk  * 
  yo  3
 y  4 
   *  z dz    z dz 
Z k 1   zo 
Z k 1  0 

*
 xyo 
 xy* 

 

271
(6c)
Dr. Adnan N. Jameel
Salam Ahmed Abed
FREE VIBRATION ANALYSIS OF COMPOSITE
LAMINATED PLATES USING HOST 12
 M x 
 x 
 M   h 2  
 y
 y 3

     z dz
 0   h 2  z 
 xy  k
M xy 
N

k 1
Q 11





Q y  2  yz  2
       z dz
Q x   h 2  xz  k
h
N Q
Q 56 
55
 

k 1 Q
 56 Q 66  k
Q 12
Q 13 Q 14 

Q 23 Q 24 
Q 22
symmetric Q 33 Q 34 

Q 44  k
  
 x 
 Z k  xo 

Zk 
   yo  3
 y  4
    z dz    * z dz
Z k 1   z 
Z k 1   zo 
  xyo 
 xy 

  y  2
 yz  3
z
dz

  
Z   z dz
Z K 1  x 
K 1  xz 
ZK
(6d)
*
ZK  *
 yz  5 
 y  4
   *  z dz    *  z dz 
Z K 1 
Z K 1  xy 
 x 
  
h
S y  2  yz 
ZK
      zdz
S x  h 2  xz  k

  x* 
  xo* 

Zk  * 
Zk  * 
  yo  5
 y  6 
   *  z dz    z dz 
Z k 1   zo 
Z k 1  0 

*
 xyo 
 xy* 

 

N Q
55
 
k 1 Q
 56
h
Q 56 

Q 66  k
ZK
 Z K  y 
 yz 
dz

  
   zdz
Z K 1  x 
Z K 1  xz 
Q 56 

Q 66  k
ZK
 Z K  y 
 yz  2
    zdz     z dz
Z K 1  x 
Z K 1  xz 
ZK
 y*  3
   *  z dz
Z K 1 
 x 
Z K  * 

 yz 
   *  z 4 dz 
Z K 1  xy 

 
h
S y  2  yz  3
       z dz
S x   h 2  xz  k
Q y  2  yz 
      dz
Q x   h 2  xz  k
N Q
55
 
k 1 Q
 56
(6f)
ZK
(6e)
*
*
ZK
 y  2
 yz  3 
   *  z dz    *  z dz 
Z K 1 
Z K 1  xy 

 x 
 
ZK
Q 55
 
k 1 Q
 56
N
Q 56 

Q 66  k
ZK
 Z K  y  3
 yz  4
    z dz     z dz
Z K 1  x 
Z K 1  xz 
ZK
 *yz  6 
 y*  5
   *  z dz    *  z dz 
Z K 1 
Z K 1  xy 

 x 
 
ZK
Laminate Constitutive Equations
272
(6g)
(6h)
Number 2
Volume 18 February 2012
 N 
  x    A11 A12 A13 A14   B11 B12
  N y   
A22 A23 A24  
B22
  N z   
 


A33 A34  
 N  
  xy   SYM
A44  SYM
 N x*   
 D11 D12
 *   
N



D22
 y  


*
 Nz  

 *   


SYM
N
 xy   


 M x  
 M  
 y  
 M z*  
M xy  
   
 M x  
 M y  
SYM
  
 0  
M xy  
 
Q y    A55 A56   B55
Q   
 
 x    A56 A66   B56
Q y  
 D55
Q   
D
 x 
 56

 

S y 
Sx  
 * 
 S y   SYM
 S x*  
B13 B14   D11
B23 B24  

B33 B34  
B44  SYM
D13 D14   E11
D23 D24  

D33 D34  
D44  SYM
 F11



SYM
B56   D55
B66   D56
D56   E55
D66   E56
 F55
F
 56
D56 
D66 
E56 
E 66 
F56 
F66 
Journal of Engineering
D12 D13 D14 
D22 D23 D24 

D33 D34 
D44 
E12 E13 E14 
E22 E23 E24 

E33 E34 
E44 
F12 F13 F14 
F22 F23 F24 

F33 F34 
F44 
 E55
E
 56
 F55
F
 56
G55
G
 56
 H 55
H
 56
 E11



SYM
 F11



SYM
 G11



SYM
 H11



SYM
  
E12 E13 E14    x 

E22 E23 E24    y 
   
E33 E34    zo 

E44    xyo 
F12 F13 F14     x  
   
F22 F23 F24    y  
  *
F33 F34    z  
F44    xy  


G12 G13 G14     xo* 

G22 G23 G24     *yo 
 

G33 G34     * 

zo 
G44    * 
  xyo 
H12 H13 H14    * 
 
H 22 H 23 H 24     x*  

   y  
H 33 H 34      
H 44     0*  
  xy  


(7)
E56     y 
 
E 66    x 
F56    yz 
 
F66    xz 
 *
G56     y 
 *
G66    x 
*
H 56    yz 
H 66    xz* 
where the overall laminate stiffnesses Aij,
Bij, Dij, Eij, Fij, Gij and Hij are:
A
ij
h

, Bij , Dij , E ij , Fij , Gij , H ij 
 Q 1, z, z
2
k 
ij
h
2
, z , z , z , z dz
3
4
5
6
Differential Equations of Equilibrium of
Laminated Plates
(8)
The equilibrium differential equations in
2
i , j =1, 2, 3, 4, 5, 6, 7)
If Aij, Bij, etc, are written in terms of the ply
terms of the moments and forces resultants for
a plate are (Swaminathan and Patil, 2008):
k 
stiffness Qij and the ply coordinates zk and
zk-1, the following is obtained:
 A , B , D , E , F , G , H 
 Q   z  z 
ij
ij
ij
ij
ij
ij
ij
N
1
n
k
k 1
ij
n
k 1
n
k
(9)
(n =1, 2, 3, 4, 5, 6, 7)
273
Dr. Adnan N. Jameel
Salam Ahmed Abed
FREE VIBRATION ANALYSIS OF COMPOSITE
LAMINATED PLATES USING HOST 12
N x , x  N xy, y  I 1u0  I 2x
 I u*  I *
3 0
If the material for all the layer is identical,
that is if the density   k  is the same for all k,
4 x
N y , y  N xy, x  I 1v0  I 2y
 I v*  I *
3 0
4
then
I2=I4=I6=0
y
Qx , x  Q y , y  Pz  I 1 w0  I 2z
 I w *  I *
3
0
Exact Solution for Simply Supported
4 z
M x , x  M xy, y  Q x  I 2 u0
 I 3x  I 4 u0*  I 5x*
M y , y  M xy, x  Q x  I 2 v0
Rectangular Plates
The exact analytical solution of the
differential eq. (1) (HOST 12) for a general
 I 3y  I 4 v0*  I 5y*
h
S x , x  S y , y  N z  ( Pz )  I 2 w0
2
 I 3z  I 4 w0*  I 5z*
N x*, x  N xy* , y  2S x  I 3 u0
 I 4x  I 5 u0*  I 6x*
laminate plate under arbitrary boundary
conditions are impossible task. However,
closed-form solution for ‘simply-supported’
(10)
rectangular plates is to be considered.
The following simply supported boundary
N y*, y  N xy* , x  2S y  I 3 v0
 I 4y  I 5 v0*  I 6y*
Q x*, x  Q xy* , y  2M 2* 
conditions are assumed (see fig. 1).
B. C. of cross-ply laminated plate associated
h2
( Pz )  I 3 w0
4
 I 4z  I 5 w0*  I 6z*
SS-1 :
M x*, x  M xy* , y  3Q x*  I 4 u0
 I 5x  I 6 u0*  I 7x*
M *  M *  3Q *  I v  I 
y, y
xy, x
y
4 0
5
(12)
(13a)
y
 I v  I 7y*
*
6 0
S x*, x  S y*, y  3N z* 
h3
( Pz )  I 4 w0  I 5z
8
 I 6 w0*  I 7z*
B. C. of angle-ply laminated plate associated
SS-2 :
The following plate inertia can be
introduced:
I , I
1
2
, I3 , I4 , I5 , I6 , I7  
  ρ   1, z, z
N z k 1
k
k 1 z k
2
, z 3 , z 4 , z 5 , z 6 dz
(13b)
(11)
274
Number 2
Volume 18 February 2012
Journal of Engineering

u
u0 
m , n 1
u 0* 
v0 
cos x sin ye imnt ,
0 mn

u
*
0 mn
cos x sin ye imnt ,
0 mn
sin x cos ye
*
0 mn
sin x cos ye imnt
m , n 1
(14b)

v
m , n 1
i mnt
,
Fig. 1 Geometry and the co-ordinate system
of a rectangular plate of thickness h
Equation of Motion in Terms of
v0* 
Displacements HOST12
For antisymmetric angle-ply laminates:
u0 
For the first time, the equations of
motion to the HOST12 eq. (1) can be
expressed
u ,
0
0
in
terms
of
displacements
, w0 ,  x ,  y ,  z , u , , w ,  ,  , 
*
0
*
0
u 0* 
*
0
*
x
*
y
*
z

v0 
by
Substituting eq. (6) into eq. (10) as in (Salam
v0* 
Ahmed A., 2008).
Free Vibration Solution by HOST12

v
m , n 1

u
m , n 1
0 mn
sin x cos ye imnt ,
*
0 mn
sin x cos ye imnt ,

u
m , n 1
(14 c)

 v0 mn cosx sin ye imnt ,
m , n 1

v
m , n 1
*
0 mn
cos x sin ye imnt ,
By substituting eq. (14) into the
The following form of solution satisfies
equations of motion and expressing the a
the differential the equations of motion and
result in matrix form the following is
the boundary condition eq. (13), when the
obtained:
K   w M   0
2
mn
applied load q (x, y, t) on the right hand side
(15)
of the equations of motion is set to zero.
w0 
w0* 
 w0 mn sin x sin ye imnt ,
m , n 1

w
m , n 1
*
0 mn


x 
m , n 1
 x* 
m , n 1

m , n 1
 y* 
*
x mn
cos x sin ye imnt ,
*
y mn


m , n 1
RESULT AND DISCUSSION
cos x sin ye


sin x sin ye imnt ,
x mn

y 
The elements of the matrix [K] (Stiffness
Matrix) [M] (Mass Matrix) are given in
(Salam Ahmed A., 2008).

*
y mn
i mnt
sin x cos ye
,
In the following, it is assumed that the
material is fiber-reinforced and remains in the
(14a)
i mnt
elastic range. The boundary conditions are
,
SSSS, and the analytical procedure (HOST
sin x cos ye imnt ,
12) is used in this work.
The material properties are :-

 z mn sin x sin ye i t ,
z 
mn
E2 =6.92 x109 N/m2, E1= 40E2,
m , n 1
 z* 


m , n 1
*
z mn
G12 =G13 =0.5E2, G23 =0.6E2, v12=0.25
sin x sin ye imnt
For antisymmetric cross-ply laminates:
275
Dr. Adnan N. Jameel
Salam Ahmed Abed
FREE VIBRATION ANALYSIS OF COMPOSITE
LAMINATED PLATES USING HOST 12
Dimensions of plate:
b=1 m
, h=0.02 m
Fundamental frequency
a=1 m ,
5
Table 1 Effect of degree of orthotropy of
individual layers on the fundamental
frequency of simply supported symmetric
1
square laminates: a/h=5,   10 x  h 2 / E2  2
No.
of
laye
rs
E1/E2
Source
4.5
4
3.5
3
3-Layers
2.5
2
5-Layers
7-Layers
1.5
1
0.5
0
0
3
10
20
30
40
2.64
74
2.62
86
2.52
71
4.54
%
2.65
87
2.64
16
2.50
09
5.93
%
2.66
40
2.64
60
2.45
08
8.00
%
3.28
41
3.26
79
3.21
97
1.96
%
3.40
89
3.38
02
3.29
13
3.44
%
3.44
32
3.42
02
3.27
29
4.94
%
3.82
41
3.70
11
3.68
34
4.99
%
3.97
92
3.94
39
3.89
05
2.22
%
4.05
47
4.03
10
3.92
04
3.31
%
4.10
89
3.94
56
3.94
42
4.00
%
4.31
40
4.28
09
4.24
76
1.53
%
4.42
10
4.40
08
4.31
20
2.46
%
4.30
06
4.11
50
4.11
98
4.20
%
4.53
74
4.51
06
4.49
01
1.04
%
4.66
79
4.65
33
4.57
80
1.92
%
10
20
30
40
50
E1/E2
Fig. 2 Effect of degree of orthotropy of
Exact
GTTR
3
Present
HOST
12%
error*
Exact
GTTR
5
Present
HOST
12%
error
Exact
GTTR
7
Present
HOST
12%
error
individual layers on the fundamental
frequency of simply supported symmetric
square laminates using (HOST 12):
a/h=5,   10 x  h 2 / E2  2
1
Table 2 shows analytical solutions of the
variation of natural frequencies with respect
to side-to-thickness ratio a/h for different
E1/E2 ratio for two and four layered a simply
supported
antisymmetric
angle-ply
(45/-
45/…) square laminated plate E1/E2 = open,
E2 = E3, G12 = G13 = 0.6E2, G23 = 0.5E2,
υ12 = υ13 = υ23 = 0.25.
*
Values in parenthesis the give percentage
error for natural frequency with respect to
exact solution mentioned above (Bose P. &
Reddy J. N., 1998). GTTR (General Third
Order Theory of Reddy) (Bose P. & Reddy J.
N., 1998).
276
Number 2
Volume 18 February 2012
*Values in parenthesis the give percentage
Table 2 Analytical effect of degree of
orthotropy and (a/h) ratio of individual layers
on the fundamental
frequency   ( a 2 / h) x / E2 
error for natural frequency with respect to
exact
(Swaminathan
a/h
Source
GTTR
3
HOST1
2
Error %
GTTR
10
HOST1
2
Error %
2
GTTR
20
HOST1
2
Error %
GTTR
40
HOST1
2
Error %
GTTR
3
HOST1
2
Error %
GTTR
10
HOST1
2
Error %
4
GTTR
20
HOST1
2
Error %
GTTR
40
HOST1
2
Error %
2
4.531
2
4.615
9
1.87
%
4.974
2
5.031
2
1.15
%
5.181
7
5.272
5
1.75
%
5.332
5
5.381
7
0.92
%
4.649
8
4.762
8
2.43
%
5.206
1
5.330
7
2.40
%
5.414
0
5.530
8
2.16
%
5.567
4
5.614
7
0.85
%
mentioned
above
and
Patil,
2008).
GTTR
(General Third Order Theory of Reddy)
4
10
100
6.1223
7.1056
7.3666
(Swaminathan and Patil, 2008).
18
6.2572
2.20 %
7.2647
7.4041
1.91 %
7.2879
2.56 %
8.9893
9.1163
7.5013
Fundamental frequency
E1
/E2
solution
2
1.82 %
9.5123
9.6453
16
14
12
10
8
2-Layers
6
4-Layers
4
2
0
1.41 %
1.39 %
10.641
2
10.772
4
11.538
5
11.673
8
1.23 %
1.17 %
12.911
5
13.076
2
14.666
8
14.741
4
0.57 %
1.27 %
0.50 %
6.4597
7.6339
7.9545
6.6133
7.8131
8.1412
2.37 %
2.34 %
2.35 %
11.411
6
11.577
7
12.535
1
12.635
3
1.45 %
0.79 %
14.473
5
14.612
8
16.992
7
17.103
4
It is also demonstrated that increasing the
1.73 %
0.96 %
0.65 %
fundamental frequency with increases the
10.073
1
10.153
5
17.877
3
17.926
8
23.449
9
23.591
2
degrees of orthotropy (E1/E2) for laminate
0.80 %
0.30 %
0.60 %
8.0490
8.1604
1.38 %
0
20
40
60
80
100
120
a/h
Fig. 3 Analytical effect of degree of (a/h)
ratio of individual layers on the fundamental
frequency using (HOST 12): E1 /E2 =20,
  ( a 2 / h) x / E2  2
1
8.8426
8.8933
8.3447
8.4752
1.56 %
9.3306
9.4927
Fundamental frequency
No.
of
laye
rs
1
Journal of Engineering
20
18
16
14
12
10
8
6
4
2-ayers
4-Layers
2
0
0
10
20
30
40
50
E1/E2
Fig. 4 Analytical effect of degree of
orthotropy of individual layers on the
fundamental frequency using (HOST 12):
a/h=10,   ( a 2 / h) x / E2 
1
2
plate due to the increase plate stiffness. The
number of layers has different effects in
laminated plates. As the span-to-thickness
277
Dr. Adnan N. Jameel
Salam Ahmed Abed
ratio
(a/h)
FREE VIBRATION ANALYSIS OF COMPOSITE
LAMINATED PLATES USING HOST 12
increases,
the
fundamental
values both for the composite and sandwich
frequency decreases, due to the decrease in
plates.
the stiffness of the plate, but the factor of
nondimensional is gives opposite relation.
REFERENCES

Bose P. & Reddy J. N., 1998,“
Analysis of Composite Plates Using
Various Plate Theories. Part 1:
Formulation
and
Analytical
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and Mechanics, Vol. 6, No. 6, 583612,.

Bose P.& Reddy J. N., 1998, “
Analysis of Composite Plates Using
Various Plate Theories. Part 2: Finite
Element Model and Numerical
Results”, J. Structural Engineering and
Mechanics, Vol. 6, No. 7, p.p. 727746,.

Hildebrand FB, Reissner E, Thomas GB.,
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
Kant T., 1982, "Numerical analysis of
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
Kant T., Manjunatha BS., 1988, "An
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
Kant T., Swaminathan K., 2001,
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
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CONCLUSIONS
1.
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 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.
2.
The effect of degree of (a/h) ratio
becomes more pronounced as the number
of layers increases. Increasing the ratio
(a/h) from (2 to 20) the natural frequency
very increases and from (20 to 100)
remains stable roughly for (E1/E2= 20).
3.
The effect of degree of orthotropy
(E1/E2) becomes more pronounced as the
number of layers increases (for the same
laminate thickness). Increasing the ratio
(E1/E2) from (10 to 40) increases the
natural frequency for (a/h = 5 ) and
(a/h=10).
It has been concluded that for all the
parameters considered Reddy’s theory very
much over predicts the natural frequency
278
Number 2




Volume 18 February 2012
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
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
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
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
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