Accepted Manuscript
Bending, buckling, and free vibration analysis of MSGT microcomposite Reddy
plate reinforced by FG-SWCNTs with temperature- dependent material properties under hydro-thermo-mechanical loadings using DQM
M. Mohammadimehr, M. Salemi, B. Rousta Navi
PII:
DOI:
Reference:
S0263-8223(15)01058-2
http://dx.doi.org/10.1016/j.compstruct.2015.11.055
COST 7010
To appear in:
Composite Structures
Please cite this article as: Mohammadimehr, M., Salemi, M., Rousta Navi, B., Bending, buckling, and free vibration
analysis of MSGT microcomposite Reddy plate reinforced by FG-SWCNTs with temperature- dependent material
properties under hydro-thermo-mechanical loadings using DQM, Composite Structures (2015), doi: http://
dx.doi.org/10.1016/j.compstruct.2015.11.055
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Bending, buckling, and free vibration analysis of MSGT microcomposite Reddy
plate reinforced by FG-SWCNTs with temperature- dependent material properties
under hydro-thermo-mechanical loadings using DQM
M. Mohammadimehra*, M. Salemia, B. Rousta Navia
a
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan,
Kashan, Iran
Abstract
In this paper, using third-order shear deformation theory (TSDT) and modified strain gradient theory (MSGT),
bending, buckling and free vibration behaviors of microcomposite plate reinforced by functionally graded singlewalled carbon nanotube (FG-SWCNT) under hydro-thermal environments are investigated. The generalized rule of
mixture is employed to predict mechanical, moisture and thermal properties of micro composite plate. The
governing equations of motion are obtained using energy method and Hamilton's principle, and solved by
differential quadrature method (DQM). There is a good agreement between the obtained results and the other results.
The influences of the material length scale, elastic foundation parameters and temperature and moisture changes for
various boundary conditions on the natural frequency, critical buckling load and deflection of the micro composite
plate reinforced by FG-SWCNT are presented. The obtained results show that critical buckling load and natural
frequency for MSGT are more than that of for classic theory (CT) and modified coupled stress theory (MCST), and
vice versa for the deflection. The material length scale parameters lead to increase the stiffness of system. Also the
effect of moisture on microcomposite plate reinforced by SWCNT is similar to thermal effect; also, with increasing
of moisture change reduces the natural frequency and critical buckling load and increases the deflection of micro
composite plate. Considering the environmental conditions and temperature results are closer to reality.
Keywords: Bending, buckling and free vibration analysis; Micro composite Reddy plate; Hygro-thermal
environments; Generalized rule of mixtures; Temperature- dependent material properties; MSGT.
*
Corresponding author: E-mail: mmohammadimehr@kashanu.ac.ir
Tel:+ 98 31 55912423 Fax: +98 31 55912424
1
1-Introduction
The polymeric composite material is employed in various industrial aspects. They have some
disadvantages which limited them in particular applications. For enhancement their material
properties, various reinforcement can be added to them. This reinforced composite material can
be used in microactuators, microtransducers, microsensor and so on, also because of the high
strength and low weight can be widely used in different industries, including aerospace and
shipping.
Recently many researches have been done in functionally graded carbon nanotube reinforced
composite (FG-CNTRC) materials field that Liew et al. [1] have a review of these papers. Also,
in this section, some of these important works are presented. Abdollahzadeh Shahrbabaki and
Alibeigloo [2] investigated three-dimensional free vibration of carbon nanotube (CNT)
reinforced composite rectangular plates with various boundary conditions by developing a set of
orthogonal admissible functions using Ritz method. They achieved non-dimensional frequencies
vary from minimum values for the plate with CFFF to the maximum values for the plate with
CCCC boundary conditions. Using the first-order shear deformation theory (FSDT) and elementfree improved moving least-squares-Ritz (IMLS-Ritz) method, Zhang et al. [3] carried out free
vibration analysis of functionally graded carbon nanotube (FG-CNT) reinforced composite
triangular plates. Ghorbanpour Arani et al. [4] analyzed the surface stress effect on the biaxial
critical buckling load of nonlocal polymeric nanocomposite rectangular plate reinforced by
CNTs. They showed the biaxial critical buckling load decreases with increasing the CNTs
volume fraction in the inclusion (agglomeration effect).
Yin et al. [5] performed the dynamic analysis of micro non-classical Kirchohof plate based on
modified couple stress theory (MCST). They found that as the thickness to be comparable with
2
material length scale parameter, the MCST natural frequency is dependent on the size effect.
Hasani Baferani et al. [6] studied vibration of rectangular functionally graded Reddy plate based
on elastic foundation. They observed that the frequency more increases as the power law index
of functionally graded material increases. Wang and co-workers [7] presented vibration and
static analysis of rectangular Kirchhoff plate based on the strain gradient theory (SGT). They
found that the stiffness, the critical buckling load and natural frequency affected significantly
using the size-dependent effect. Jomehzadeh et al. [8] analyzed vibration micro plate based on
MCST. They obtained by applying the material length scale parameter increases the stiffness and
natural frequency of the micro plate. Ramezani [9] developed buckling and vibration of micro
plate based on SGT and FSDT. He also showed that the size dependent effect at the micro scale
is significant. Ke et al. [10] examined the vibration analysis of micro plate based on MCST and
FSDT. They concluded that as the thickness is close to the material length scale parameter, the
size effect on the natural frequency of micro plate is important. Thai and Choi [11] extended the
size dependent effect on the bending, buckling and vibration of Kirchhoff and Mindlin
functionally graded plate based on MCST. Their numerical results displayed that considering the
small-scale parameters leads to reduce bending and increases critical buckling load and natural
frequency. Zhang and Liew [12] studied geometrically nonlinear large deformation analysis of
FG-CNTRC quadrilateral plates, and using IMLS approximation for the field variables, they
improved the obtained results. Sahmani and Ansari [13] investigated vibration of functionally
graded micro plate based on SGT. They showed that as the thickness reaches to the material
length scale parameter, the natural frequency is significantly increased. Thai and Kim [14]
studied bending and free vibration analysis of functionally graded Reddy plate. They discovered
that when the thickness is smaller than the small scale parameter, size dependent effect on the
3
natural frequency is important but as the thickness increases, its effect will be negligible. Kim
and Reddy [15] presented exact solution for functionally graded Reddy plate according to
MCST. They illustrated that the plate becomes stiffener considering the small scale effects.
Furthermore the critical buckling load reduced by decreasing of power law index of functionally
graded plate. Zhang and his co-workers [16] used mesh-free kp-Ritz for analysis flexural
strength and free vibration of carbon nanotube reinforced composite cylindrical panels. Thai and
Choi [17] developed two variable plate theories for bending, buckling and vibration of
rectangular plate. Their results have a best agreement with results of FSDT and third order shear
deformation theory (TSDT).
Reddy and Berry [18] used Kirchhoff and FSDT to analyze the bending of circular plate based
on MCST. They presented the analytical solutions of bending, buckling, and free vibration for
the linear case and finite-element models for the nonlinear case to determine the effect of the
geometric nonlinearity, power law index, and microstructure-dependent constitutive relations on
linear and nonlinear response of axisymmetric analysis of circular plates. Mozafari and Ayob
[19] presented the exact solution for buckling of functionally graded plate using FSDT and
TSDT. They concluded that the critical buckling load decreases with increasing of aspect ratio
and power law index. Zhu et al. [20] developed a meshless local Petrov-Galerkin approach for
geometric nonlinear thermoelastic analysis of functionally graded plates in thermal
environments. Jarali et al. [21] studied the effect of electrical, thermal and moisture properties of
carbon nanotubes (CNTs) in composites considering the CNTs accumulation. They noticed that
CNT agglomeration has a strong influence on the effective hygro-thermo-electric properties of
the nanocomposites. Nonlinear bending of spherical shell under hydro-thermo-mechanical loads
is examined by Lal et al. [22]. They used higher-order shear deformation theory and micro-
4
mechanical model in their study and examined the hygrothermal effects are more detrimental as
the working temperature increases and reaches closer to the glass transition temperature. Lei et
al. [23] investigated the buckling analysis of functionally graded carbon nanotube reinforced
composite plate using Ritz method. They employed Mori-Tanaka- model to estimate material
properties of the composite plate. They showed that the effects of carbon nanotube volume
fraction, plate width-to-thickness ratio and temperature change have distinct effects on buckling
strength of CNT. The buckling, bending and free vibration analysis of functionally graded
carbon nanotubes reinforced composite beam using the finite element method (FEM) is carried
out by Mayandi and Jeyaraj [24]. They used the extended rule of mixture to define the
mechanical material properties, and then found the critical buckling temperature is not increasing
significantly with an increase in volume fraction of the CNT. Ghorbanpour Arani et al. [25]
investigated the buckling analysis of single-walled carbon nanotube (SWCNT) reinforced
composite plate using FEM. They reported that non uniform dispersion of CNTs in the polymer
matrix decreases the critical buckling load. Moreover, it increases the difference between the
critical buckling loads obtained by the analytical method and the FEM.
Rafiee et al. [26] studied the dynamic stability of FG-CNT reinforced piezoelectric composite
plate under electro-thermal loadings. They showed that the influence of the temperature rise on
the thin plates is more significant than the thicker one. Zhang et al. [27] carried out FG-CNTRC
moderately thick rectangular plates with edges elastically restrained against transverse
displacements and rotation of the plate cross section. They found when the elastic restraint
parameter Kφ is greater than 105, the natural frequency parameters vary slowly due to the fact
that the stiffness of the spring is great enough to serve as a rigid restraint. Shooshtari and Rafiee
[28] studied vibrations of composite plate reinforced by SWCNTs. It is assumed the properties of
5
SWCNTs to be related to temperature change and theirs obtained from molecular dynamics
simulation. They found that the nonlinear natural frequency to linear natural frequency ratio
increases with increasing of SWCNTs volume fraction. Mohammadimehr et al. [29] considered
the small-scale effect on torsional critical buckling load of double-walled carbon nanotubes
based on Winkler-Pasternak foundation using non-local elasticity theory. Their result disclosed
that the nonlocal critical buckling load increases with increasing of Pasternak shear constant.
Moreover nonlocal critical buckling load is lower than local critical buckling load. Liew et al.
[30] completed post-buckling analysis of cylindrical SWCNTs reinforced nanocomposite panel
subjected to axial compression. They investigated the effects of boundary conditions and
SWCNTs arrangement on the post buckling. Ghorbanpour Arani and his coworkers [31]
accomplished nonlinear vibrations of rectangular polymeric piezoelectric microcomposite plate
reinforced by double zigzag boron nitride nanotubes. They illustrated that with an increase in
elastic coefficients, aspect ratio and volume fraction of boron nitride nanotubes increase the
nonlinear natural frequency while with an increase in the non-local parameter, it reduces.
Bodaghi and Saidi [32] presented Levy solution for the buckling analysis of functional graded
thick rectangular plate. They evaluated effects of boundary conditions, power law index type of
loadings and thickness on the post buckling of the composite plate. Ansari et al. [33] established
vibration analysis of functionally graded Mindlin micro plate with nonlinear geometry. They
solved system of nonlinear equations using the quadratic differential generalized method and
examined the influences of power law index, length to thickness ratio, length scale and different
boundary conditions on natural frequency. Zhang et al. [34] used the IMLS method to present a
set of first known vibration frequencies and mode shapes for FG-CNTRC skew plates is
obtained. Sahmani and Ansari [35] studied free vibration of micro composite plate according to
6
micromechanical Mori-Tanaka approach. They demonstrated that with an increase in the
material length scale parameter, the natural frequency increases considerably. The effects of
material length scale parameter and hydro-thermal loadings on bending of nanoplate surrounded
in elastic medium are investigated by Alzahrani et al. [36]. Mohammadimehr and Salemi [37]
developed SGT for bending and buckling analysis of FG Mindlin nanoplate. They concluded that
considering SGT leads to increase stiffness of nanoplate. Alibeigloo [38] studied the bending
behavior of simply supported functionally graded carbon nanotube reinforced composite plate in
the piezoelectric layer using three dimensional elasticity theory. Bodaghi and Saidi [39] offered
obtained the exact solution for buckling of functionally graded rectangular plate under nonuniformly axial load. Zhang et al. [40] studied the mechanical and thermal buckling analyses of
different types of functionally grade plates by developed the local meshless method based on the
local Petrov–Galerkin weak-form formulation combined with shape functions having the
Kronecker delta function property, constructed by the Kriging interpolation. Jomehzadeh et al.
[41] analyzed the vibration analysis of microplate based on MCST. They concluded that with an
increase in the material length scale parameter increases the stiffness and the natural frequency.
The effect of the SWCNTs volume fraction on electro-magneto-thermal behavior of functionally
graded cylindrical nanocomposite plate is evaluated by Ghorbanpour Arani et al. [42]. Nonlinear
vibrations of rectangle graphene sheets in the thermal environment are established by Shen et al.
[43]. Their results exhibited that with the correct select of the small scale parameters and the
material properties, the nonlocal model presented exact prediction of nonlinear vibration
behavior in thermal environment. Mohammadimehr et al. [44] studied the buckling of doublewalled carbon nanotube surrounded by the elastic foundation under axial compression using
Timoshenko beam theory. Lei et al. [45] considered different distributions of SWCNTs through
7
the thickness of layers to study the free vibration analysis of laminated FG-CNT reinforced
composite rectangular plates using the element-free kp-Ritz method. Using this method, they
[46] studied dynamic stability of carbon nanotube-reinforced functionally graded cylindrical
panels. Rahmati and Mohammadimehr [47] analyzed axial vibration of non-uniform and
heterogeneous boron nitride nanorod embedded in an elastic foundation. They concluded that
with an increase in small-scale parameter and elastic foundation constants reduces the nondimensional frequency ratio.
Zhang et al. [48] presented the nonlinear bending behaviors of FG-CNT reinforced composite
thick plates. Using the element-free IMLS-Ritz method, they observed that the non-dimensional
central deflections of the FG-CNT reinforced composite plates rise as the load increases. Also
they developed this method to large deflection and buckling analysis of FG-CNT reinforced
composite skew plates for numerical solution of three-dimensional wave propagation equations
and for elasto-dynamic problems [49-52]. Mohammadimehr et al. [53] investigated vibration of
viscoelastic piezoelectric polymeric nanocomposit plate reinforced by FG-SWCNTs using
meshless method based on modified strain gradient theory (MSGT) and sinusoidal shear
deformation theory. They concluded that the natural frequency increases with increasing of
elastic foundation parameter, small scale parameters, and magnetic field.
In this article, bending and buckling, and free vibration analysis of micro composite Reddy plate
reinforced by FG-SWCNTs embedded in an elastic foundation and hydro-thermal environments
with temperature- dependent material properties are investigated. The generalized rule of
mixture is used to define hydro-thermo-mechanical material properties. Three material length
scale parameters are considered by MSGT. The governing equations of motion using Hamilton’s
principle are obtained and solved by differential quadrature method (DQM).
8
2- Geometry
According to Fig. 1, the micro composite rectangular Reddy plate is considered with length a ,
width b and thickness h . This micro composite plate rested on elastic foundation with Winkler
coefficient K w and Pasternak shear coefficient K G . Fig. 2 shows the distribution of FGSWCNTs in micro composite Reddy plate. Uniform distribution (UD), FG-O, FG-V and FG-X
are assumed to reinforce the micro composite plate. Volume fraction for these distributions is
defined as follows [54]:
V CNT
*
 V CNT

  1  2z  V *
 CNT
 
h 

2z  *
 
 z    2 1   V CNT
h 
 

2z
*
V r  2
V CNT
h



UD
CNTRC 
 FG V
CNTRC 
 FG  O
CNTRC 
 FG  X
CNTRC 
(1)
where:
*
V CNT

w CNT   
CNT
w CNT
/  m     CNT /  m w CNT
(2)
w CNT ,  m and  CNT are SWCNTs mass fraction, matrix density and SWCNTs density,
respectively. Mass and volume of SWCNTs is same for four SWCNT distributions.
9
3- Generalized rule of mixture
The generalized rule of mixture for estimation of hydro-thermo-mechanical properties of micro
composite plate reinforced by FG-SWCNTs are expressed as follows:
Mechanical properties:
E11  1V CNT E11CNT V m E m
2
(3-a)

V CNT V m

CNT
E 22
Em
(3-b)

V CNT V m

G12CNT G m
(3-c)
 V CNT CNT V m  m
(3-d)
E 22
3
G12
where, i (i  1,2,3) denotes force transformation between SWCNTs and polymeric matrix.
V m is matrix volume fraction.
Thermal properties:
11 V CNT 11CNT V m m
(4-a)
CNT
 22  1  12CNT V CNT  22
 1   m V m m  1211
(4-b)
and hydro properties:
11 V CNT 11CNT V m  m
(5-a)
22  1  12CNT V CNT 12CNT  1   m V m  m  12 11
(5-a)
10
4- Governing equations
Displacement fields of the micro composite plate according to Reddy plate theory (third-order
shear deformation theory (TSDT)) can be written as follows [54]:
w 

u ( x, y, z , t )  u0 ( x, y, t )  z x ( x, y, t )  c1 z 3   x  0 
x 


w 
v( x, y, z , t )  v0 ( x, y, t )  z y ( x, y, t )  c1 z 3   y  0 
y 

(6)
w( x, y, z, t )  w0 ( x, y, t )
where, u 0 , v 0 , and w 0 denote middle surface displacements.  x and  y are the rotation of
middle surface at z  0 . c1 
4
and c1  0 are for TSDT and FSDT, respectively.
3h 2
Strain- displacement relations according to TSDT can be expressed as:
x 
   2 w0 

1  w 
  0   z x  c1 z 3  x 

x 2  x 
x
x 2 
 x
u0
2
 y
  y  2 w0 
1  w 
y 
  0 z
 c1 z 3 


y 2  y 
y
y 2 
 y
2
v0
 y
 2 w0 
  x  y 
3   x
 xy 


 z

  c1 z  y  x  2 xy 
y
x
x y
x 
 y


u0
v0
w0 w0
w

 xz  1  3c1 z 2    x  0
x

w0 

2 
 ,  yz  1  3c1 z    y 

y 


Using Hook's law, stress- strain relations can be stated as follows:
11
(7)
0
0    xx  11 (z )T  11 (z )H 
 xx  Q11 Q12 0
  


0
0   yy   22 (z )T   22 (z )H 
 yy  Q12 Q 22 0


 yz
0 Q 44 0
0 
 yz    0


   0

 xz
0
0 Q 55 0 
 xz  


 xy
0
0
0 Q 66  
 xy   0

(8)
where, T and H denote temperature and moisture changes and Qij is defined as follows:
Q11 (z ) 
E 11 (z )
1  12 (z )21 (z )
Q 22 (z ) 
E 22 (z )
1  12 ( z )21 ( z )
Q12 (z ) 
21 (z )E 11 (z )
1  12 (z )21 (z )
Q 44 (z )  G 23 (z ) 
E 22 (z )
2 1  23 (z ) 
(9)
Q 55 (z )  Q 66 (z )  G12 (z )
The governing equations based on minimum potential energy principle are obtained as follows:
 U  V
t2
t1
 T  dt  0
(10)
where U , T and V are strain energy, kinematic energy and work done by external forces,
respectively. The variation of kinetic energy can be written as:
12
 T    u u  v v  w w    z  dAdz
V

  T   I 0  u0 u0  v0 v0  w0 w0   I1  x u0  u0 x   y v0   y v0  
A
 
 w0 
w 

I 2  x x   y y   c1 I 3 u0   x 
  u0   x  0  

x 
x 

 


 
 w0 
w0  
 w0 
v0   y 

   v0   y 
   c1 I 4  x   x 
y 
y  
x 
 


(11)


w 
 w0 
w0  

 x   x  0    y   y 
   y   y 
 
x 
y 
y  




w 
 w0  
w0 
 w0   
c12 I 6   x  0   x 





 y
 y
   dxdy
x 
x  
y 
y   

Work done by external forces can be stated as:
V    q (x , y )  k w w 0  k G 2w wdA
(12)
A
where q( x, y) is transverse load per length.
The variation of strain energy based on MSGT can be obtained as follows:
U  
A
h /2
 
x
 x   y  y   xy  xy   xz  xz   yz  yz  p x  x 
h / 2
p y  y  p z  z   xxx xxx   yyy  yyy   zzz  zzz  6 xyz  xyz 
3 xxy xxy  3 xxz xxz  3 yyx  yyx  3 yyz  yyz  3 zzx  zzx  3 zzy  zzy 
m xx  xx  m yy  yy  m zz  zz  2m xy  xy  2m xz  xz  2m yz  yz dzdA
13
(13)
where ijk(1) ,  ijs and  i are deviatoric stretch gradient tensor, symmetric rotation gradient tensor and
dilatation gradient vector according to TSDT and Eqs. (7)- (9) can be obtained as the following
form:

(1)
xxx
3
3
1   2u 0  2u 0
 2v 0
w 0
3 w0
3 w0
 2 2  2  2
 6c1z
 2c1z
 3c1z

5  x
y
y x
x
x 3
y 2x
2
 w 0    2w 0   w 0    2w 0 
 w 0    w 0 
2

2
 6c1z x 





2 
2 
 x   x 
 y   y x   x   y 
  2x  2x
 2y  
z c1z 3  2


2
 
 x 2 y 2

y

x

 


(1)
yyy

3
3
1   2v 0  2v 0
 2u 0
w 0
3 w0
3 w0
 2 2  2  2
 6c1z
 2c1z
 3c1z

5  y
x
y x
y
y 3
x 2y
2
2
 w 0    2w 0 
 w 0    w 0   w 0    w 0 
2

2

 6c1z  y 
 




2 
2 
 x   y x   y   x 
 y   y 
  2y  2 y
 2x  
z c1z 3  2


2
 
 y 2 x 2

y

x

 


  2w
1
 2w 0 x y   x y  
z(1)zz   6c1z 2  1  20 





5
y 2
x
y   x
y  
 x
(1)
(1)
(1)
(1)
(1)
xyz
  yzx
 zxy
 zyx
  yxz
 x(1)zy 
2


1
2   w 0

c
z
 x  y
1


x
3
  y x y
(1)
(1)
(1)
xxy
 xyx
  yxx

2

2  w0

c
z
 1
y x

8  2u 0
4  2v 0 1  2v 0 2


 c1z
15 y x 15 x 2 5 y 2 5

w 0 
 y 

y 

  2x
 2y
 2y

4
 8
2
x 2
 y x y
4   2w 0  w 0  8  w 0    2w 0  1  w 0    2w 0 


 


 

15  x 2   y  15  x   y x  5  y   y 2 
1 3   3w 0
 3w 0  1
c1z 

4

z c1z 3
3
2 
5

y

y

x
15




(1)
(1)
xxz
 xzx
 z(1)xx 
2
 2w 0   2 2 2   y
 
 2 2 1   w 0

4
  c1z   
4 x 
 c1z   
2
2 
15   y
x   5
15   y
x 
5
14

 


(1)
yyx

(1)
yxy

(1)
xyy
4  2u 0 1  2u 0 8  2v 0 2 
w 0 



 c1z  x 

2
2
15 y
5 x
15 y x 5 
x 
  2 y
 2x
 2x


4
 8
2
y 2
 y x x
4   2w 0   w 0  8  w 0    2w 0  1  w 0    2w 0 


 

 


15  y 2   x  15  y   y x  5  x   x 2 
1 3   3w 0
 3w 0  1
c1z 

4

z c1z 3
3
2 
5

x

x

y
15





 

(1)
(1)
(1)
 yyz
  yzy
 zyy

2
 
 2w 0   2 2 2   x
 2 2 1   w 0
c
z


4
  c1z   
4 y 
 1

2
2 
15   x
y   5
15   x
y 
5
(1)
(1)
zz(1)x  zxz
 xzz


1  2u 0 1  2u 0 2  2v 0 8


 c1z
5 x 2 15 y 2 15 y x 5
w 0 

 x 

x 

  2x
 2y
 2x

2

3

2
y x
x 2
 y
1  w 0    2w 0  2  w 0    2w 0  1  w 0    2w 0 
 
 





5  x   x 2  15  y   y x  15  x   y 2 
1 3   3w 0
 3w 0
c1z 

3
5
y 2x
 x
 1
c1z 3  z

 15



 

(1)
(1)
(1)
zzy
 zyz
  yzz


w 0 
 y 

y 

  2 y
 2 y
1 3   3w 0
 3w 0  1
 2x
3
c1z 


c
z

z

2

3


1
3
2
5
x 2y  15
y x
y 2
 y
 x
1  w 0    2w 0  2  w 0    2w 0  1  w 0    2w 0 
 
 





5  y   y 2  15  x   y x  15  y   x 2 

1  2v 0 1  2v 0 2  2u 0 8


 c1z
5 y 2 15 x 2 15 y x 5



 

(14-a)
 
1
 2w 0
 xxs   1  3c1z 2 
  1  3c1z 2  y 
2
x y
x 
1
 2w 0
 
 yys    1  3c1z 2 
  1  3c1z 2  x 
2
x y
y 
1
2
 y x  


y  
 x
 zzs   1  3c1z 2  
15
(14-b)
1
4

  2w 0  2w 0 
 

 1  3c1z 2  x  y
2
2 
x 
y
 x
 y
 xys   yxs   1  3c1z 2  

 
s
zx
1   2v
 2u 0
  20 
 z 1  c1z 2
4  x
x y
 
s
zy
1   2v 0  2u 0
 

 z 1  c1z 2
4  x y y 2
s
xz
s
yz







  2y
 2x


2
x y
 x

  6c1z


w 0  


 y

y  


  2y
 2x


2
 x y y

  6c1z

w 0  



 x

x  

3
 2u 0  2v 0
 3w 0 
3 w 0
x  2 
 c1z 
 2   z  c1z 3
3
x
y x
y x 
 x

  2x
 2y
 2 
y x
 x



  2x
 2y
 2 
y x
 x




2
2
 w 0    w 0   w 0    w 0 






2 
 x   x   y   y x 
3
 2u 0  2v 0
 3w 0
3  w 0
y 


c
z


1
3
y x y 2
x 2y
 y
2
 w 0    w 0


 x   y x

3
  z  c1z



(14-c)
  w 0    2w 0 


2 
  y   y 
 
  2w 0  2w 0 
 

 1  3c1z 2  x  y 
2
2 
y 
y 
 x
 x
 z  3c1z 2 


Also the following relations for higher-order stresses can be defined as:
M ijh  k s   ij z h dz i  x , y ; j  z & h  0,1, 2,3
z
Pi   p i z h dz i , j  x , y , z & h  0,1, 2
h
z
Y
h
ij
T
h
ijk
(15)
  m ijs z h dz i , j  x , y , z & h  0,1, 2
z
(1) h
  ijk
z dz i , j , k  x , y , z & h  0,1, 2
z
where k s is the shear correction factor.
16
Substituting Eqs. (9),(15) and (14) in Eq. (13), the variation of strain energy for MSGT Reddy
microcomposite plate reinforced by SWCNTs subjected to hydro-thermo-mechanical loadings
are given in detail in Appendix A.
Using Eq. (A.1) and separation of variables, the following governing equations of motion for
microcomposite plate reinforced by SWCNTs based on MSGT and TSDT are obtained as
follows:
For  u 0 :

0
2 0
2 0
0
0
M xx0 M xy  2 Px0  Py
2  2T xxx
1  2T xxx
2  T yyy







x
y
x 2 x y 5 x 2
5 y 2
5 x y
0
0
8  T xxy 4  T yyx 3  T yyx 3  2T zzx
1  2T zzx
2  T zzy






2
2
2
2
5 x y 5 y
5 x
5 x
5 y
5 x y
2
0
2
0
2
0
2
0
(16)
1  2Y xz0 1  Y yz
 2u 0
 2x
 3w 0



I

(
c
I

I
)

c
I
0
1 3
1
1 3
2 x y 2 y 2
t 2
t 2
x t 2
2
0
For v 0 :

M yy0
y

M xy0
x

 2 Py0
y 2
0
 2 Px0 2  T yyy 1  T yyy 2  2T xxx




x y 5 y 2
5 x 2
5 x y
2

0
2
0
0
8  T yyx 4  T xxy 3  T xxy 3  T zzy 1  T zzy 2  2T zzx






5 x y 5 x 2
5 y 2
5 y 2
5 x 2
5 x y
2
0
2
0
2
0
2
0
2
0
 y
1  Y xy 1  2Y xz0
 2v 0
 3w 0



I

(
c
I

I
)

c
I
0
1 3
1
1 3
2 x y 2 x 2
t 2
t 2
y t 2
2
0
2
For w 0 :
17
(17)
M
0
xx
2
2
 2 M xy3
 2w 0
 2 M xx3
 2 M xx3
0  w0
0  w0
 c1
 M yy
 c1
 2M xy
 2c1

x 2
x 2
y 2
x 2
x y
x y
0
  3Py3
M yz2
  3Px3
M xz0
M xz2 M yz
 3Px3 
 3c1

 3c1
 c1 
 2   c1 

3
 y 3
x
x
y
y
y x 
 x

1
3
3
 3Py3 
  2 Pz2  2 Pz2  6 T xxx
2  3T xxx
3  3T xxx

3
c


c

c

c



1
1
1
1
2
x 2y 
y 2  5
x
5
x 3
5 y 2x
 x
0
0
2
2
6 T yyy 2  T yyy 1  2T zzz
1  2T zzz
6  2T zzz
6  2T zzz
c1
 c1



c

c

1
1
5
y
5
y 3
5 x 2
5 y 2
5 x 2
5 y 2
1
0
 2T xyz
3
3
2
 2T xyz
1
3 3
2 3
0
6 T xxy 3  T xxy 12  T xxy 4  2T xxz
2
 12c1
 c1
 c1
 c1


y x
y x 5
y
5 y 3
5 y x 2 5 x 2
2 3
1
3 3
2 3
2
2
1  T xxy 24  2T xxz
6 T xxz
6 T yyx 3  T yyx 12  T yyx
 c1
 c1
 c1
 c1
 c1

5 y 2
5
x 2
5 y 2 5
x
5 x 3
5 x y 2
3
3
4  T yyz 1  T yyz 24  T yyz 6 T yyz 3  3T zzx
3  3T zzx


c

c


c

1
1
1
5 y 2
5 x 2
5
y 2
5 x 2 5 x 3
5 x y 2
2
0
2
3
2
2
2
3 3
3 3
1
1
24 T zzx
3  T zzy 3  T zzy
24 T zzy 1  2Y xx0 3  2Y xx2
c1


c

c

 c1

1
1
5
x
5 y 3
5 y x 2 5
y
2 y x 2 y x
2 0
2 2
2 0
2 0
2 2
2 2
1  Y yy 3  Y yy 1  Y xy 1  Y xy 3  Y xy 3  Y xy
 c1


 c1
 c1

2 y x 2 y x 2 y 2
2 x 2
2 y 2
2 y 2
Y yz
  3u 0
Y xz1
 2w 0
 3v 0
 3c1
 q  I0

I
c

3 1
2
2
y
x
t 2
 t x t y
1
3c1
  3
 3
c1  I 4  I 6c1   2 x  2 y
 t x t y




4

 4w 0 
2  w0

I
c

 6 1  2 2

t 2y 2 
 t x

(18)
For x :
18
1
1
M xy3
M xx
M xx3 M xy
 2 Px1
Px3

 c1

 c1
 M xz0  3c1M xz2 

c

1
x
x
y
y
x 2
x 2
 2 Py1
 c1
x y
2 T
c1
5
x
2
2
0
T xyz
y
3
xxx
2
Py3
x y

1
1
Pz0
P 2 2  2T xxx
1  2T xxx
6
1
 3c1 z 

 c1T xxx

2
2
x
x
5 x
5 y
5
3
0
1  2T xxx
2  T yyy 2  T yyy 2 T zzz
6 T zzz2
 c1

 c1

 c1

5 y 2
5 x y 5 x y 5 x
5 x
 6c1
2
2
T xyz
y
1
2
3
0
8  T xxy 8  T xxy 8 T xxz
24 T 2

 c1

 c1 xxz 
5 x y 5 x y 5 x
5
x
2
1
2
3
6 1
4  T yyx 3  T yyx 4  T yyx 3  T yyx 2 T yyz
c1T yyx 

 c1
 c1


5
5 y 2
5 x 2
5
y 2
5
x 2
5 x
2
1
2
1
2
3
2
3
0
1
3
3
6 T yyz 3  2T z1zx 1  2T zzx
3  2T zzx
1  2T zzx
24
1
c1



c

c
 c1T zzx

1
1
2
2
2
2
5
x
5 x
5 y
5 x
5 y
5
2
2 1
2 3
0
2
2  T zzy 2  T zzy 1 Y yy 3 Y yy 1 Y zz0 3 Y zz2
 c1

 c1

 c1

5 x y 5 x y 2 y
2
y
2 y
2 y
1 Y xy 3 Y xy 1  2Y xz1 1  2Y xz3 1  Y yz 1  Y yz
 c1

 c1

 c1

2 x
2
x
2 x y 2 x y 2 y 2
2 y 2
0
2
1
3cY
1 yz   I 1  c1I 3 
2
1
2
3
 2u 0
 2x
 3w 0
2

I

2
c
I

c
I

c
I

c
I
2
1 4
1 6
1 4
1 6
t 2
t 2
x t 2


For y :
19
(19)

1
M yy
 c1
y
M yy3

y
1
M xy
 c1
x
M xy3
M
x
0
yz
 3c1M
2
yz

 2 Py1
y 2
 c1
Py3
y 2

 2 Px1
Px3
P 0
P 2 2  T yyy 1  T yyy 6
1
 c1
 z  3c1 z 

 c1T yyy

2
2
x y
x y
y
y
5 y
5 x
5
2
1
2
1
1
3
0
2  T yyy 1  T yyy 2  2T xxx
2  2T xxx
2 T zzz
6 T zzz2
c1

c


c


c1

1
1
5
y 2
5 x 2
5 x y 5 x y 5 y
5 y
2
2
0
T xyz
x
3
 6c1
2
2
T xyz
3
8  T yyx 8  T yyx 8 T yyz 24 T yyz
 c1

 c1

5 x y 5 x y 5 y
5
y
2

x
1
2
3
0
2
0
6 1
4  T xxy 3  T xxy 4  T xxy 3  T xxy 2 T xxz
c1T xxy 

 c1
 c1


5
5 x 2
5 y 2
5
x 2
5
y 2
5 y
2
1
2
1
2
3
2
3
(20)
2
6 T xxz
3  T z zy 1  T zzy 3  T zzy 1  T zzy 24
1
c1


 c1
 c1
 c1T zzy

5
y
5 y 2
5 x 2
5 y 2
5 x 2
5
2
1
2
1
2
3
2
3
1
3
2  2T zzx
2  2T zzx
1 Y xx0 3 Y xx2 1 Y zz0 3 Y zz2
 c1

 c1

 c1

5 x y 5 x y 2 x
2
x
2 x
2 x
0
2
2 1
2 3
1 Y xy 3 Y xy 1  Y yz 1  Y yz 1  2Y xz1 1  2Y xz3
 c1

 c1

 c1

2 y
2
y
2 x y 2 x y 2 y 2
2 x 2
1
1 xz
3cY
 2 y
 2v 0
 3w 0
2
  I 1  c1I 3  2  I 2  2c1I 4  c1 I 6
 c1  I 4  c1I 6 
t
t 2
y t 2


Dimensionless parameters are defined as:
u0
a
z
 
h
q
q 
Em
v0
b
a
1 
h
u 

D
t
h
i,j
v 
Ii 
Em
m
Ii
 m h i 1
 
, D i , j , D i, j  
w0
x
y


h
a
b
b
l
2 
c1  c1h 2
li  i
h
h
2
2
12(1   )Na
N 
 x  x
 y  y
Emh3
w 
2
a
h
m
Em
Kw 
Kw h
Em
1
D , T D  i , j , HD  i , j 
j 1  i , j
Emh
where:
20
KG 
KG
Emh
(21)
D
1, i
,D

1, i
,D
h /2

1, i
 
Q11 (z ) * z i 1,11 (z ), 11 (z )dz
h / 2
D


2, i , D 4, i , D 4, i  
h /2

Q 22 (z ) * z i 1, 22 (z ),  22 (z )dz
h / 2


D 3, i , D 2, i , D
 

D 3, i , D 3, i


2,i
h /2
,

i
   Q12 (z ) * z 1,11 (z ), 11 (z ), 22 (z ),  22 ( z )dz
 h / 2

(22)
h /2
D   
Q 44 (z ) * z i dz
4, i
h / 2
h /2
D   
5, i
i
Q 55 (z ) * z i dz
 0...6 
h / 2
Using stresses and dimensionless parameters, the dimensionless governing equations of motion
for MSGT Reddy microcomposite plate reinforced by SWCNTs with considering hydro-thermomechanical loadings are expressed in detail in Appendix B.
Clamped and simply supported, free boundary conditions are considered for the micro composite
plate in detail in Appendix C.
5- Solving method
DQM is employed to solve the governing equations. In this method, derivative of any arbitrary
function in arbitrary point as f  x, y    xi , yi  can be rewritten in all interval as follows:
d rf
dx r
d sf
dy s
N
 x , y   x i , y j 
  A in r f nj
n 1
M
 x , y   x i , y j 
 r s f
x r y s
i  1, 2,..., N
s 
  A jm
f im
m 1
N
 x , y x i , y j 
j  1, 2,..., M
r  1, 2,..., N  1
(23)
M
s 
  A in r  A jm
f nm
n 1 m 1
s  1, 2,..., M  1
r
where f ij  f  x i , y j  and A   is weighted coefficient matrixes which are defined as ( in x
direction):
21
(1)
A ij
 N
  (x i  x m )
 mm 1i , j
, (i , j  1, 2,3,..., N ; i  j )
 N
  (x j  x m )
  m 1
 mj
N
1

, (i  j  1, 2,3,..., N )
 m 1 (x i  x m )
m i
A  r   A  r 1 A 1
(24)
2  r  N 1
A well-accepted set of the grid points is given by the Gauss–Lobatto–Chebyshev points for
interval [0, L] that these set of grid points in terms of natural coordinate directions xi and yi are
defined as:
xi 
1
(i  1) 
]
1  cos[
2
(N  1) 
yj 
1
( j  1) 
]
1  cos[
2
(M  1) 
(25)
In two dimensional problems, we have:
uv  f ij
v   j  1 N  i
(26)
Using above expression, the following relations are obtained as follows:
a f b   b 
T

 a  u 
(27)
where  symbol denotes Hadamard products. The following definitions in x and y directions
are used to simplification of relations:
x : A x  A 1
B x  A  2
C x  A  3
D x  A  4
y : A y  A 1
B y  A  2
C y  A  3
D y  A  4
22
(28)
Using above definitions, the governing equations  F  and boundary conditions T

rewritten as
differential quadrature form then according to boundary conditions and order of differential
equation for each of displacement variables, displacement vector divided to domain vector U d
and boundary displacement vector U b . In the next step, rows of  F  which belong to boundary
points are eliminated and matrix,  F  is obtained. Then column of boundary condition and the
governing equation matrices for boundary points and domain separated and matrix of the
governing equations of boundary points,  F b , domain  F d and matrices of boundary
conditions for boundary points T
b
and domain T
d
are obtained finally they are related
together with following relation:
F b U b  F d U d  0
(29)
T b U b  T d U d
(30)
U 
b
0
b T d U d
  T
1
(31)
Substituting Eq. (31) into Eq. (29), the following equation is derived as follows:
  F  T 
1
b
b
T d  F d U d
0 
 Z U d
0
(32)
For bending analysis, the following equation is used:
 K  U    q 
(33)
23
q 
where  K  and
U 
are stiffness and external distributed load matrixes.
denotes
displacement vector which are demonstrated as:
U   u
v
y
T
x
w
(34)
If Eq. (32) rewritten as Eq. (33), the deflection are determined as follows:
  K  T 
1
b
b
T d  K d U d

 [q ]d  U d    K b T
b T d  K d 
1
1
[q ]d
(35)
Also the critical buckling load is obtained as the following form:
 K1  U   N  K2  U 
(36)
where  K1  ,  K 2  , N , and U  are stiffness matrices, the critical buckling load and mode
shape. The above equation can be rewritten in the form of boundary points and domain, the
critical buckling load is obtained as:
 K  T  T   K  U   N  K  T  T   K  U 

 K  T  T   K    N  K  T  T   K  U 
1
1 b
1
b
1 d
d
2 b
d
b
1
1 b
2 d
d
d
1
b
d
1 d
2 b
b
d
2 d
d
0
(37)
For vibration analysis, at first the time expression replaced by following equations:
u0  , ,   U 0  ,  ei
v0  , ,   V0  ,  ei
w0  , ,   W0  ,  ei
x  , ,    x  ,  e
(38)
i
 y  , ,    y  ,  ei
24
where  , U 0  ,  , V0  ,  , W0  ,  ,  x  ,  and  y  ,  denote dimensionless
frequency and dimensionless vibration domains. The dimensionless natural frequency and
domain are obtained as follows:
 K  T  T   K  U     M  T  T   M  U 

 K  T  T   K      M  T  T   M  U 
1
1 b
1
2
b
1 d
d
2 b
d
1
1 b
b
b
1 d
d
1
2
d
2 d
d
2 b
25
b
d
2 d
d
0
(39)
6- Numerical results and discussion
The numerical results of vibration, buckling and bending of micro composite plate reinforced by
FG-SWCNTs under the distributed load q   0.1MPa and N x  f1 Ncr ; N y  f 2 Ncr ; N xy  0 are
presented.
The temperature- dependent mechanical properties of PmPV (Poly {(mphenylenevinylene)-co[(2,5-dioctoxy-p-phenylene)vinylene]}) as the matrix are considered as follows [55]:
vm = 0.34
(40)
αm = 45(1 + 0.0005 ΔT) × 10 −6/ K
βm = 2.67(1 + 0.0005 ΔT) × 10 −6/ K
Em = (3.51-0.0047T) GPa
where T = T0+ΔT and T0 = 300 K (room temperature) [55]. It is noted that the temperature of the
top and bottom surfaces is the same. On the other hand, the temperature throughout the thickness
is constant.
The temperature- dependent material properties of armchair SWCNTs (10, 10) as reinforcement
are listed in Table 1 [56].
i coefficients for different SWCNTs volume fractions are shown in Table 2 [57].
The obtained results for Mindlin plate are compared with results of Refs. [23] and [57]. These
results are listed in Tables 3, 4 and 5. A good agreement is observed between DQM results
(present work) and the obtained results by the other researchers.
26
6-1- Bending and buckling of the micro composite Reddy plate reinforced by SWCNTs
The dimensionless deflection ( VCNT  0.11 ) and the dimensionless critical buckling load
( VCNT  0.11 ) of the micro composite plate are calculated for various distribution types of
SWCNT and boundary conditions. Also, the effects of different parameters are examined for
them. Figs. 3(a) and 3 (b), Tables 6 and 7 depict the effects of various boundary conditions on
the dimensionless deflection and the critical buckling load of the micro composite plate
reinforced by SWCNTs for different boundary conditions. In this figure, letter C, S, and F denote
clamped, simply supported, and free boundary conditions in the edge of the micro composite
plate. As it is observed from Fig. 3 (a) that the dimensionless deflection for CCCC and SSSF
have the lowest and highest values. Because of the clamped boundary condition with respect to
simply supported and free boundary conditions leads to increase stiffer of the micro composite
plate. Thus the dimensionless critical buckling load increases and vice versa for dimensionless
deflection of micro composite Reddy plate.
Figs. 4(a) and 4(b) show the effects of material length scale parameters on the dimensionless
deflection and the critical buckling load of the micro composite Reddy plate reinforced by
SWCNTs based on MSGT and MCST and CT. It is observed from this figure that considering
material length scale parameters lead to increase stiffness of the micro composite Reddy plate,
therefore the dimensionless deflection reduces while dimensionless critical buckling load
increases.
Figs. 5(a) and 5(b) illustrate the effects of SWCNT volume fraction on the dimensionless
deflection and the critical buckling load of the micro composite Reddy plate reinforced by
SWCNTs for FSDT and TSDT. It can be seen that the dimensionless deflection for TSDT is
lower than that of for FSDT but the dimensionless critical buckling load for TSDT is higher than
27
that of for FSDT. Also for both plate theories, the dimensionless deflection decreases with
increasing of SWCNT volume fraction and vice versa for dimensionless critical buckling load. It
is noticeable that the increasing volume fraction of SWCNT leads to increase stiffness of micro
composite plate. In Fig. 5a, it can be seen that there are the difference between the FSDT results
and TSDT results. Due to the TSDT, consider higher order terms, thus TSDT results have more
accurate with respect to FSDT, then, in this research the TSDT result is suitable.
Figs. 6(a) and 6(b) display the effects of various distribution types of SWCNT on the
dimensionless deflection and the critical buckling load of the micro composite plate reinforced
by SWCNTs. It is obvious that the micro composite plate is stiffer as reinforced by FG-X
distribution type rather than other distribution types. Thus it means the highest dimensionless
critical buckling load and lowest dimensionless deflection is belong to FG-X distribution type.
Effects of elastic foundation parameters on the dimensionless deflection and the critical buckling
load of the micro composite plate are illustrated in Figs. 7(a) and 7(b), respectively. It reveals
that the elastic foundation parameter enhances the stiffness of the micro composite Reddy plate
hence dimensionless deflection decreases with presence of elastic foundation but the
dimensionless critical buckling load increases in its presence.
Effects of temperature and moisture changes on the dimensionless deflection and the critical
buckling load of the micro composite plate are shown in Figs. 8a, 8b and 8c, respectively. It can
be seen from these figures that the dimensionless deflection increases with an increase in
temperature and moisture changes and this is vice versa for the dimensionless critical buckling
load that means the micro composite plate becomes softer with increasing of moisture and
temperature changes. It is noted that Figs. 8a and 8c have been plotted with considering
temperature-dependent material properties of PmPV (Eq. (40)) and SWCNTs (Table 1).
28
Fig. 9 demonstrates the dimensionless critical buckling load with respect to different axial
loading. It can be observed that applying biaxial compression loading in x and y directions of
leads the micro composite plate buckles earlier. Also the elastic modulus of the micro composite
plate in x direction due to aligning with SWCNT orientation is greater than its elastic modulus in
y direction, applying compression loading in x direction rather than y direction increases the
critical buckling load.
6-2- Vibration of the micro composite plate reinforced by SWCNTs
The dimensionless natural frequency ( VCNT  0.11 ) of the micro composite plate are obtained and
the effects of different parameters such as SWCNT volume fraction, SWCNT distribution types,
boundary conditions and material length small scale parameters are investigated.
Fig. 10 illustrates the dimensionless natural frequencies of micro composite Reddy plate based
on FSDT and TSDT for various volume fraction and aspect ratio a / h . According to this figure,
the dimensionless natural frequency based on TSDT is larger than the dimensionless natural
frequency of FSDT. Also as the SWCNT volume fraction increases, the dimensionless natural
frequency increases.
The dimensionless natural frequency of the micro composite plate against a / h for various
distribution types of SWCNTs is shown in Fig. 11. It is predictable that the micro composite
Reddy plate reinforced by FG-X SWCNTs and FG-O SWCNTs have highest and lowest
dimensionless natural frequency. This is due to that in FG-X reinforcements, SWCNTs are
linearly distributed in all areas of the micro composite plate then the micro composite plate
easily tolerated different loadings.
29
Based on various theories such as MSGT, MCST and CT, dimensionless natural frequencies of
the micro composite plate are depicted for aspect ratio of a / b in Fig. 12. It is found that the
dimensionless natural frequency increases as the material length small scale parameters increases
particularly in higher aspect ratio of a / b .
The dimensionless natural frequency of micro composite plate against elastic foundation
parameters is shown in Fig. 13. As it is expressed, stability of micro composite plate improves
with considering elastic foundation therefore the dimensionless natural frequency increases with
increasing of Winkler coefficient, KW and Pasternak shear coefficient, K G .
The dimensionless natural frequency of micro composite Reddy plate for different moisture and
temperature changes is demonstrated in Fig. 14. It is shown that with an increase in the moisture
and temperature changes, the flexibility of the micro composite plate increases. Consequently the
dimensionless natural frequency decreases with increasing of moisture and temperature changes.
It should be stated that this figure have been plotted based on temperature-dependent material
properties of PmPV (Eq. (40)) and SWCNTs (Table 1).
First six vibration mode shapes of micro composite Reddy plate with clamped boundary
conditions in its four edges are illustrated in Fig. 15. It is shown that the CCCC boundary
conditions are satisfied as well for first six vibration mode shapes.
7- Conclusions
In this paper, bending, buckling, and free vibration analysis of micro composite Mindlin and
Reddy plate reinforced by FG-SWCNTs with temperature- dependent material properties
embedded in an elastic foundation for various boundary conditions such as simply supported,
clamped, and free boundary conditions were investigated. The generalized rule of mixture is
30
employed to predict mechanical, moisture and thermal properties of micro composite Reddy
plate material. The governing equations of motion for micro composite Mindlin and Reddy plate
reinforced by FG-SWCNTs based on Hamilton’s principle are obtained and DQM is used to
solve these equations. The results of research can be listed as follows:
1- The dimensionless natural frequency and critical buckling load of the micro composite
plate decreases with an increase in temperature change while the dimensionless
deflection increases.
2- Humidity effect is similar temperature change. With increasing both them, the natural
frequency and the critical buckling load decrease and vice versa for dimensionless
deflection. It means that the micro composite plate becomes softer with increasing of
moisture and temperature changes.
3- With increasing of Winkler and Pasternak, the dimensionless natural frequency and
critical buckling load of the micro composite Reddy plate increases while the
dimensionless deflection decreases. Also, the influence of Pasternak shear coefficient is
more than Winkler coefficient on the dimensionless natural frequency.
4- The dimensionless critical buckling load and the natural frequency of micro composite
plate for SGT is higher than those for MCST and CT and this behavior is inverse for the
dimensionless deflection. It is due to the material length scale parameter increases the
stiffness of microstructures.
5- Reinforcements enhance the stiffness of micro composite plate; hence the dimensionless
natural frequency and the critical buckling load increase while the dimensionless
deflection decreases. It is noticeable that the increasing volume fraction of SWCNT leads
to increase stiffness of micro composite plate.
31
6- It is obvious that the micro composite plate is stiffer as reinforced by FG-X distribution
type rather than other distribution types. Thus it means the highest dimensionless critical
buckling load and natural frequency and lowest dimensionless deflection is belong to FGX distribution type.
7- The dimensionless natural frequency and the critical buckling load of micro composite
plate Reddy is higher than those for micro composite Mindlin plate.
8- The clamped boundary condition with respect to simply supported and free boundary
conditions leads to increase stiffer of the micro composite plate. Thus the dimensionless
critical buckling load increases and vice versa for dimensionless deflection of micro
composite Reddy plate.
Acknowledgments
The authors would like to thank the referees for their valuable comments. They are also grateful
to the Iranian Nanotechnology Development Committee for their financial support and the
University of Kashan for supporting this work by Grant No. 463855/5.
References
[1] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube
reinforced composites: A review. Compos Struct 2015; 120: 90-97.
[2] Abdollahzadeh Shahrbabaki E, Alibeigloo A. Three-dimensional free vibration of carbon
nanotube-reinforced composite plates with various boundary conditions using Ritz method.
Compos Struct 2014; 111: 362–70.
[3] Zhang LW, Lei ZX, Liew KM. Free vibration analysis of functionally graded carbon
nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-
32
Ritz method. Compos Struct 2015; 120: 189-99.
[4] Ghorbanpour Arani A, Rousta Navi B, Mohammadimehr M. Surface stress and
agglomeration effects on nonlocal biaxial buckling polymeric nanocomposite plate
reinforced
by
CNT
using
various
approaches.
Adv
Compos
Mater
2015.
DOI:10.1080/09243046.2015.1052189.
[5] Yin L, Qian Q, Wang L, Xia W. Vibration analysis of microscale plates based on modified
couple stress theory. Acta Mech Solida Sin 2010; 23: 386-93.
[6] Hasani Baferani A, Saidi AR, Ehteshami H. Accurate solution for free vibration analysis of
functionally graded thick rectangular plates resting on elastic foundation. Compos Struct
2011; 93: 1842-53.
[7] Wang B, Zhou S, Zhao J, Chen X. A size-dependent Kirchhoff micro-plate model based on
strain gradient elasticity theory. Eur J Mech A-Solid 2011; 30: 517-24.
[8] Jomehzadeh E, Noori HR, Saidi AR. The size-dependent vibration analysis of micro-plates
based on a modified couple stress theory. Physica E 2011; 43: 877-83.
[9] Ramezani S. A shear deformation micro-plate model based on the most general form of
strain gradient elasticity. Int J Mech Sci 2012; 57: 34-42.
[10] Ke LL, Wang YS, Yang J, Kitipornchai S. Free vibration of size-dependent Mindlin
microplates based on the modified couple stress theory. J Sound Vib 2012; 331: 94-106.
[11] Thai HT, Choi DH. Size-dependent functionally graded Kirchhoff and Mindlin plate models
based on a modified couple stress theory. Compos Struct 2013; 95: 142-53.
[12] Zhang LW, Liew KM. Geometrically nonlinear large deformation analysis of functionally
graded carbon nanotube reinforced composite straight-sided quadrilateral plates. Comput.
Methods Appl. Mech. Eng. 2015; 295: 219-239.
[13] Sahmani S, Ansari R. On the free vibration response of functionally graded higher-order
shear deformable microplates based on the strain gradient elasticity theory. Compos Struct
2013; 95: 430-42.
[14] Thai HT, Kim SE. A size-dependent functionally graded Reddy plate model based on a
modified couple stress theory. Compos Part B-Eng 2013; 45: 1636-45.
[15] Kim J, Reddy JN. Analytical solutions for bending, vibration, and buckling of FGM plates
using a couple stress-based third-order theory. Compos Struct 2013; 103: 86-98.
[16] Zhang LW, Lei ZX, Liew KM, Yu JL. Static and dynamic of carbon nanotube reinforced
33
functionally graded cylindrical panels. Compos Struct 2014; 111: 205-12.
[17] Thai H, Choi DH. Analytical solutions of refined plate theory for bending, buckling and
vibration analyses of thick plates. Appl Math Model 2013; 37: 8310-23.
[18] Reddy JN, Berry J. Nonlinear theories of axisymmetric bending of functionally graded
circular plates with modified couple stress. Compos Struct 2012; 94: 3664-8.
[19] Mozafari H, Ayob A. Effect of Thickness Variation on the Mechanical Buckling Load in
Plates Made of Functionally Graded Materials. Procedia Technol 2012; 1: 496-504.
[20] Zhu P, Zhang LW, Liew KM. Geometrically nonlinear thermomechanical analysis of
moderately thick functionally graded plates using a local Petrov-Galerkin approach with
moving Kriging interpolation. Compos Struct 2014; 107: 298-314.
[21] Jarali CS, Patil SF, Pilli SC. Hygro-thermo-electric properties of CNT epoxy
nanocomposites with agglomeration effects. Mech Adv Mater Struct 2014; 22: 428-39.
[22] Lal A, Singh BN, Anand S. Nonlinear bending response of laminated composite spherical
shell panel with system randomness subjected to hygro-thermo-mechanical loading. Int J
Mech Sci 2011; 53: 855-66.
[23] Lei ZX, Liew KM, Yu JL. Buckling analysis of functionally graded carbon nanotubereinforced composite plates using the element-free kp-Ritz method. Compos Struct 2013;
98: 160-68.
[24] Mayandi K, Jeyaraj P. Bending, buckling and free vibration characteristics of FG-CNTreinforced polymer composite beam under non-uniform thermal load. P I Mech Eng L-J Mat
2013; 26: 1-16.
[25] Ghorbanpour Arani A, Maghamikia Sh, Mohammadimehr M, Arefmanesh A. Buckling
analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical
and finite element methods. J Mech Sci Technol 2011; 25 (3): 809-20.
[26] Rafiee M, He XQ, Liew KM. Non-linear dynamic stability of piezoelectric functionally
graded carbon nanotube-reinforced composite plates with initial geometric imperfection. Int
J Non-Linear Mech 2014; 59: 37-51.
[27] Zhang LW, Cui WC, Liew KM. Vibration analysis of functionally graded carbon nanotube
reinforced composite thick plates with elastically restrained edges. Int J Mech Sci 2015; 103:
9-21.
34
[28] Shooshtari A, Rafiee M. Vibration characteristics of nanocomposite plates under thermal
conditions including nonlinear effects. Int J Appl Res Mech Eng 2014; 1: 60-69.
[29] Mohammadimehr M, Saidi AR, Ghorbanpour Arani A, Arefmanesh A, Han Q. Buckling
analysis of double-walled carbon nanotubes embedded in an elastic medium under axial
compression using non-local Timoshenko beam theory. Proc IMech-E Part C: J Mech Eng
Sci 2011; 225: 498-506.
[30] Liew KM, Lei ZX, Yu JL, Zhang LW. Postbuckling of carbon nanotube-reinforced
functionally graded cylindrical panels under axial compression using a meshless approach.
Comput Method Appl Mech Eng 2014; 268: 1-17.
[31] Ghorbanpour Arani A, Vossough H, Kolahchi R, Mosallaie Barzoki AA. Electro-thermo
nonlocal nonlinear vibration in an embedded polymeric piezoelectric micro plate reinforced
by DWBNNTs using DQM. J Mech Sci Technol 2012; 26: 3047-57.
[32] Bodaghi M, Saidi AR. Levy-type solution for buckling analysis of thick functionally graded
rectangular plates based on the higher-order shear deformation plate theory. Appl Math
Model 2010; 34: 3659-73.
[33] Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Darabi MA. Nonlinear vibrations
of functionally graded Mindlin microplates based on the modified couple stress theory.
Compos Struct 2014; 114: 124-34.
[34] Zhang LW, Lei ZX, Liew KM. Vibration characteristic of moderately thick functionally
graded carbon nanotube reinforced composite skew plates. Compos Struct 2015; 122: 17283.
[35] Sahmani S, Ansari R. On the free vibration response of functionally graded higher-order
shear deformable microplates based on the strain gradient elasticity theory. Compos Struct
2013; 95: 430-42.
[36] Alzahrani EO, Zenkour AM, Sobhy M. Small scale effect on hygro-thermo-mechanical
bending of nanoplates embedded in an elastic medium. Compos Struct 2013; 105: 163-72.
[37] Mohammadimehr M, Salemi M. Bending and buckling analysis of functionally graded
Mindlin nano-plate model based on strain gradient elasticity theory. Indian J Sci Res 2014;
2: 587-598.
35
[38] Alibeigloo A. Static analysis of functionally graded carbon nanotube-reinforced composite
plate embedded in piezoelectric layers by using theory of elasticity. Compos Struct 2013;
95: 612-22.
[39] Bodaghi M, Saidi AR. Stability analysis of functionally graded rectangular plates under
nonlinearly varying in-plane loading resting on elastic foundation. Arch Appl Mech 2011;
81: 765-80.
[40] Zhang LW, Zhu P, Liew KM. Thermal buckling of functionally graded plates using a local
Kriging meshless method. Compos Struct 2014; 108: 472-92.
[41] Jomehzadeh E, Noori HR, Saidi AR. The size-dependent vibration analysis of micro-plates
based on a modified couple stress theory. Physica E 2011; 43: 877-83.
[42] Ghorbanpour Arani A, Rahnama Mobarakeh M, Shams Sh, Mohammadimehr M. The effect
of CNT volume fraction on the magneto-thermo-electro-mechanical behavior of smart
nanocomposite cylinder. J Mech Sci Technol 2012; 26: 2565-72.
[43] Shen L, Shen H, Zhang C. Nonlocal plate model for nonlinear vibration of single layer
graphene sheets in thermal environments. Comp Mater Sci 2010; 48: 680-85.
[44] Mohammadimehr M, Saidi AR, Ghorbanpour Arani A, Arefmanesh A, Han Q. Torsional
buckling of a DWCNT embedded on Winkler and Pasternak foundations using nonlocal
theory. J Mech Sci Technol 2010; 24: 1289-99.
[45] Lei ZX, Zhang LW, Liew KM. Free vibration analysis of laminated FG-CNT reinforced
composite rectangular plates using the kp-Ritz method. Compos Struct 2015; 127: 245–59.
[46] Lei ZX, Zhang LW, Liew KM, Yu JL. Dynamic stability analysis of carbon nanotubereinforced functionally graded cylindrical panels using the element-free kp-Ritz method.
Compos Struct 2014; 113: 328-338.
[47] Rahmati AH, Mohammadimehr M. Vibration analysis of non-uniform and nonhomogeneous boron nitride nanorods embedded in an elastic medium under combined
loadings using DQM. Physica B 2014; 440: 88-98.
[48] Zhang LW, Song ZG, Liew KM. Nonlinear bending analysis of FG-CNT reinforced
composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz
method. Compos Struct 2015; 128: 165–75.
[49] Zhang LW, Liew KM. Large deflection analysis of FG-CNT reinforced composite skew
plates resting on Pasternak foundations using an element-free approach. Compos Struct
36
2015; 132: 974-983.
[50] Zhang LW, Lei ZX, Liew KM. Buckling analysis of FG-CNT reinforced composite thick
skew plates using an element-free approach. Compos. Part B. 2015; 75: 36-46.
[51] Zhang LW, Huang D, Liew KM. An element-free IMLS-Ritz method for numerical solution
of three-dimensional wave equations. Comput. Methods Appl. Mech. Eng. 2015; 297: 116139.
[52] Zhang LW, Li DM, Liew KM. An element-free computational framework for elastodynamic
problems based on the IMLS-Ritz method. Eng. Anal. Bound. Elem. 2015; 54: 39-46.
[53] Mohammadimehr M, Rousta Navi B, Ghorbanpour Arani A. Free vibration of viscoelastic
double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT,
sinusoidal shear deformation theory and meshless method. Compos Struct 2015; 131: 65471.
[54] Zhang LW, Song ZG, Liew KM. State-space Levy method for vibration analysis of FG-CNT
composite plates subjected to in-plane loads based on higher-order shear deformation theory.
Compos Struct 2015; 134: 989:1003.
[55] Lei ZX, Liew KM, Yu JL. Free vibration analysis of functionally graded carbon nanotubereinforced composite plates using the element-free kp-Ritz method in thermal environment.
Compos Struct 2013; 106: 128-38.
[56] Lei ZX, Zhang LW, Liew KM. Elastodynamic analysis of carbon nanotube-reinforced
functionally graded plates. Int J Mech Sci 2015; 99: 208-17.
[57] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotube-reinforced
composite plates using finite element method with first order shear deformation plate theory.
Compos Struct 2012; 94: 1450-60.
[58] Mohammadimehr M, Rousta Navi B, Ghorbanpour Arani A. Modified strain gradient Reddy
rectangular plate model for biaxial buckling and bending analysis of double-coupled
piezoelectric polymeric nanocomposite reinforced by FG-SWNT. Compos. Part B: Eng.
2016; 87: 132-148.
37
Appendix A
The variation of strain energy for MSGT Reddy microcomposite plate reinforced by SWCNTs
subjected to hydro-thermo-mechanical loadings is written as follows [58]:
2

 x
  u 0 w 0 w 0 
1
3
3  w 0
U   M xx0 


M

c
M

c
M



xx
1
xx
1
xx

A
x x 
x
x 2
 x

2
 y
 v 0 w 0 w 0 
1
3
3  w 0
M yy0 


M

c
M

c
M

1
yy 
1
yy
  yy
y y 
y
y 2
 y
2
  u 0 v 0 w 0 w 0 w 0 w 0 
3  w 0
M 




  2c1M xy
x
x y
x y 
x y
 y

 M xy1  c1M xy3   y x  x y    M xz0  3c1M xz2   x  wx 0  


0
xy
M
0
yz
 3c1M
2
yz
  2 x  2 y

w 0 
1
3
   y  y    Px  c1Px   x 2  x y




 

  2 u 0  2v 0  2w 0 w 0  2w 0 w 0  2w 0 w 0
P 





2
x y
x 2 x
x 2 x
x y y
 x
3
2
 2w 0 w 0 
 3w 0 
 2v 0
3   w 0
0   u 0

c
P


P


 1 x

y 
3
x y y 
x y 2 
y 2
 x
 x y
0
x
 2w 0 w 0  2w 0 w 0  2w 0 w 0  2w 0 w 0 




y 2 y
y 2 y
x y x
x y x 
  2 y  2 x
  3w 0  3w 0 
1
3
c1Py3 


P

c
P

  y
1 y 
3
 y 2
y x 2 
x y
 y

38

 

(A-1)
2
  x  y 
 2w 0 
2   w 0
  Pz0  3c1Pz2  



  3c1Pz 
2
y 
y 2 
 x
 x
 2   2 u 0  2w 0 w 0  2w 0 w 0  1   2 u 0
 2v 0
0
T xxx




2

 
2
x 2 x
x 2 x  5  y 2
x y
 5  x
 2   2x 
 2w 0 w 0 w 0  2w 0 w 0  2w 0  
1
2



T


xxx 

2 
x y y
x y 2
x
y 2  
 5  x 
 2   2 x
 2 y
1   2 x
w 0   

3

2

6
c



T



1
x
xxx   c1 

2
5  y 2
x y
x   

 5  x
 2 y
 3w 0  c1   2 x
 3w 0  


2

3

 
x 3  5  y 2
x y
x y 2  
 2   2v 0  2w 0 w 0  2w 0 w 0  1   2v 0
 2 u 0
0
T yyy




2

 
2
y 2 y
y 2 y  5  x 2
x y
 5  y
 2   2 y 
 2w 0 w 0 w 0  2w 0 w 0  2w 0  
1
2



  T yyy  
2 
x y x
y x 2
y
x 2  
5

y



2
2
 2   2 y

1    y
 x
w 0   
3

2

6
c



T


1
y
   yyy   c1 
2
5  x 2
x y
y   
5

y



 3w 0  c1   2x
 2 x
 3w 0   T zzz0   2w 0  2w 0

2
3



y 3  5  x 2
x y
y x 2   5  x 2
y 2
2
2
 6
 2w 0  x  y 
2   w 0

c
T




 5 1 zzz  x 2
y 2
x
y 


 y
  x  y  2w 0 
 2w 0 
2   x



6
c
T


2

1 xyz 
 y
x
y x 
x
y x 

 y
 x
 x
2
x
y
0
2T xyz
4  0   2 u 0  2v 0 3  2v 0 3  2w 0 w 0 3  2w 0 w 0





T xxy  2
5
x 2
4 y 2
4 y 2 y
4 y 2 y
 x y
2
 2w 0 w 0
 2w 0 w 0  2w 0 w 0  2w 0 w 0  
2


 
x y x
x y y
x 2 y
x 2 y  
4 1
T xxy
5
4 3
T xxy
5
2
2
 3 w 0 3
 2 x   y 3   y 
 c1 y  2


 c1

y
2
y x
x 2
4 y 2 
2
 3  3w
  2 x  2 y 3  2 y
 3w 0
0

3
c

c
2


 c1
1
1
 y x
y 3
x 2y
x 2
4 y 2
 4

 x 1  y
4 0
2

T xxz  3c1T xxz   2
5
2 y
 x

  
 
  2w 0 1  2w 0  

0
2

T

6
c
T

 
  xxz
1 xxz  
2
4 y 2  

 x
39
following
(A-1)
4 0
T yyx
5
2
  2v 0  2 u 0 3  2 u 0 3  2w 0 w 0 3  2w 0 w 0





2
y 2
4 x 2
4 x 2 x
4 x 2 x
 x y
 2w 0 w 0
 2w 0 w 0  2w 0 w 0  2w 0 w 0  
2


 
x y y
x y x
y 2 x
y 2 x  
4 1
T yyx
5
4 3
T yyx
5
 3 w 0 3
 2 y  2 x 3  2 x 
 c1 x  2


 c1

x
2
y x
y 2
4 x 2 
2
 3  3w
  2 y  2 x 3  2 x
 3w 0
0
 3c1 2  c1  2


 c1
 y x
x 3
y x
y 2
4 x 2

 4
  y 1  x
4 0
2

T yyz  3c1T yyz   2
5
2 x
 y

  
 
  2w 0 1  2w 0  

0
2

T

6
c
T



 

yyz
1 yyz 
2
4 x 2  

 y
1  0   2 u 0  2 u 0
 2v 0
 2w 0 w 0
 2w 0 w 0

2
3
3

T zzx  3
2
5
y 2
x y
x 2 x
x 2 x
 x
2
 2w 0 w 0
 2w 0 w 0  2w 0 w 0  2w 0 w 0  
2


 
x y y
x y x
y 2 x
y 2 x  
3
  2 x
 2 y  2 x
 3w 0 
3   w 0
1
3
3c1T zzx


T

c
T
3

2






zzx
1 zzx 
3
2
x y 2 
y x
y 2
 x
 x

 

w 0   1  0   v 0  v 0
 u 0
 w 0 w 0
1 
24c1T zzx

2
3

  x 
   T zzy  3
2
2
x   5 
x
x y
y 2 y

 y
2
3
2
2
2
 2w 0 w 0
 2w 0 w 0
 2w 0 w 0  2w 0 w 0  2w 0 w 0  

2

2


 
y 2 y
x y x
x y y
x 2 y
x 2 y  
3
1 zzy
3c T
2
  2 y
  3w 0  3w 0 
 2 x   y
1
3

2


  T zzy  c1T zzy   3
3
2
y x 2 
y x
x 2
 y
 y

 

2
w 0   1  0
1 
2  w 0
2  y
24c1T zzy



Y

3
cY
 Y xx0  3cY

xx
1 xx 
1 xx 
 y


y   2 
y x
x




 1 0
 x 
2   y


1 zz  
  2 Y zz  3cY
y 
 x

2
 y  
1
 2w 0 
2   w 0
2   x
 Y xy0  3cY

 Y xy0  3cY

 
1 xy  
1 xy  
2
2 
2
x 
y  
 x
 y
  2 y  2 x 
1  0   2v 0  2 u 0 
w 0  
1
3
1 
Y


Y

cY

 6cY


 xz 

  xz
1 xz  
1
xz
y

 
2
2
 x

2   x
x y 

x

y

y

 


  2 y  2 x 
1  0   2v 0  2 u 0 
w 0   
1
3
1 


Y

cY


6
cY


Y yz 





yz
1 yz 
1 yz 
x
  dA
2   x y
y 2 
y 2 
x   

 x y

2
1 0
2  w 0
2  x
Y

3
cY
 Y yy0  3cY


yy
1
yy
1 yy 

2
y x
y
40
following
(A-1)
Appendix B
Using stresses and dimensionless parameters, the dimensionless governing equations of motion
for MSGT Reddy microcomposite plate reinforced by FG-SWCNTs with considering hydrothermo-mechanical loadings are obtained as following form [58]:
For  u 0 :
4
4
2
2   u
2
D
l

D
l
5,0 0
5,0 1  

4
13 
5
  
4
 D1,0   2u  1  8
1
2
2   u 


D
l

D
l
5,0 1
5,0 2  

 2

4 
4
   
 1    2  15
  2u 
1
1 
4
1
 4u 
2
2
2 
 2 D 5,0  2   2  2D 5,0 l 0  D 5,0 l 1  D 5,0 l 2   2 2    2D 5,0l 0 2
2
3
4
    
   2 1 
1 
4
  2v 
4
1
1  4v  1
 1  v
D 5,0l12  D 5,0l 2 2   3


D

D



3,0
5,0 

3
15
4
221  3  1
  1 
  
5
5
1
4
1 
8
  w 
  w 
 5 2c1D 5,3l 0 2  c1D 5,3l12   5   3 2  4c1D 5,3l 0 2  c1D 5,3l12   2 3 
1
5
5
    1 2 
    


 c D  2c1D 5,3  
3
1 
12
1  1 3,3
  w  
2   w 
 3  c1D1,3  D 5,1c1l1   3   2  12
   2  
1 
5
    2 1  D 5,1c1l12

 5

5
2
 1  12
1 
4
  x 
2
2   w
2
2
c
D
l

c
D
l

D
c
l

c
D

D
1
5,3
0
1
5,3
1
5,1
1
1
1
1,3
1,1

 4 

2 
2 
24 1 
5
     1  5
   
4
4
1


2D 5,1l 0 2  2c1D 5,3l 0 2  D 5,1l12  c1D 5,3l12  c1D 5,3l 2 2  

  4 
1
3
3
4
 2 2
 2 x 2 
1 2  1
2
   
 D 5,1l 2

4

3
41
(B-1)
 1 4
2
  2  D 5,1c1l1 

5
 2
1
8

D 5,1l12  D 5,1l 2 2   4
2

  x 
3
    x  1 15
4
c1D 5,3  D 5,1c1l 2 2  D 5,1  
 4

2 
4 
1
2
    2  8
2
2   

c
D
l

c
D
l

1 5,3 1
1 5,3 2 
4
 15

2
1 
8
3
   y 
2
2


 c1D 3,3  D 5,1  D 5,1c1l1  c1D 5,3  D 5,1c1l 2  D 3,1 
12 
5
2
  
4
4
1


2D 5,1l 0 2  2c1D 5,3l 0 2  D 5,1l12  c1D 5,3l12  c1D 5,3l 2 2   4

  y 
1
15
15
4
 3 


1 2  1
 3 
2


 D 5,1l 2

4

1
4
1
 4
 4
2
2
2
2
2
1  D 5,1l1  D 5,1l 2  2D 5,1l 0  c1D 5,3l1  c1D 5,3l 2      y 
15
4
15
4

   3 
123 
2


 2c1D 5,3l 0



4
4
4
2
2
2
2   x
2
D
l

c
D
l

2
c
D
l

D
l
1 5,3 1
1 5,3 0
5,1 1  
 5,1 0
4
14 
5
5
  
1 
1

3
1
I 0 2u 
1

4
1
(c1I 3  I 1 ) 2  x 
1

5
1
c1I 3 2
w

42
following
(B-1)
For v 0 :
4
1
 4u
1  4u  1

2
2
2  1
2
D
l

D
l

D
l

   D 3,0  D 5,0 
5,0 1
5,0 2   3
 5,0 0
3
15
4
122  3  2

  2 
4
D 2,0   2v  2  8
  2u  1 
4
2
2   v 
2

2
D
l

D
l



5,0 0
5,0 1  
 2    D 5,0 l1 
3 
4 





5






15


 1


2 
2 
4
  2v 
1 
4
1
 4v 
   v  2
2
2
2 
D 5,0 l 2   4   2 D 5,0  2   2  2D 5,0 l 0  D 5,0 l 1  D 5,0 l 2   2 2  
4
3
4
    1
    
   1 2 
1
2
5
4
2
2   w
2
c
D
l

c
D
l
1 5,3 1  
 1 5,3 0
5
25 
5
  
5


1 
8
2
2   w

4
c
D
l

c
D
l

1 5,3 0
1 5,3 1  

3 2 
2
3 
5
    
 2 1 
3
3

1 
12
1 
12
2   w 
2   w
c
D

D
c
l

c
D

2
c
D

D
c
l
2,3
5,1 1 1  
1 5,3
5,1 1 1  

 1 3,3
3  1
3 
2
2
2 
5
5
    1 2 
    
5

1 
4
1 
8
2
2   w
2
2
c
D
l

c
D
l

1 5,3 1  
 1 5,3 0
 c1D 3,3  D 5,1  D 5,1c1l1 
4
4
1 2 
5






5

 1 2
1 
2
3
1 
4
   x 
2
2
2
2
c1D 5,3  D 5,1c1l 2  D 3,1 
  3  2D 5,1l 0  2c1D 5,3l 0  D 5,1l1 
2
15
   2 1 
4
4
1
1
1  4
1
  x 
c1D 5,3l12  c1D 5,3l 2 2  D 5,1l 2 2  

D 5,1l12  D 5,1l 2 2 
3 
3 
15
4
4
4
    21  15
2D 5,1l 0 2 
(B-2)
4
4
1
     1  12
c1D 5,3l12  c1D 5,3l 2 2  2c1D 5,3l 0 2   3 x   2  D 5,1c1l12 
15
4
     2  5
  2 y
c1D 2,3  D 2,1  
  2

 1 
  4 y
4
4
2
2
2
2
  4  2D 5,1l 0  c1D 5,3l1  2c1D 5,3l 0  D 5,1l1  
4
5
5
  
 2 
2
1 4
3
  y 
2
2
 2  D 5,1c1l1  D 5,1c1l 2  D 5,1 c1D 5,3  

2 
1  5
2
   
  4 y 
1 8
1
8
1
2
2
2
2
D 5,1l1  D 5,1l 2  c1D 5,3l1  c1D 5,3l 2  

4 
14  15
4
15
4
   
4
4


2D 5,1l 0 2  2c1D 5,3l 0 2  D 5,1l12  c1D 5,3l12   4

 y 
1
3
3

 2 2 
2 2
1
1 2  1
2
2
    
  c1D 5,3l 2  D 5,1l 2

 4
4

2
1
1

w
 4 I 0 2v  4 (c1I 3  I 1 ) y  c1I 3
1
1
214 
43



For w 0 :
3
3
1 
12
1  12
 u
 u
2
2

c
D

D
c
l

2
c
D


D
c
l

c
D


1 5,3 
5,1 1 1
1 1,3 
2
2 
3
22  1 3,3 5 5,1 1 1
   1  5
 
5
1 8
 5u
2
2   u
2
2 
D
c
l

2
D
c
l

D
c
l

4
D
c
l





5,3
1
1
5,3
1
0
5,3
1
1
5,3
1
0
5
2 2
2
3
14  5
  2 1  5
  
1 4
5
1 4
1
2
2   u
D
c
l

2
D
c
l
 2
5,3 1 1
5,3 1 0 
4 
4
2  5
   1
1 4

24  5
 12
 v
2

  D5,1c1 l1  2c1 D5,3  c1 D3,3 
2
 5
 
3
  v 1  12
 v
 2   D5,1c1 l12  c1 D2,3  3 
5
  2  5
 
5
3
D5,3 c1 l12  2 D5,3 c1 l0 2 
5
1 8
5v
2
2   v
2
2 
D
c
l

2
D
c
l

D
c
l

4
D
c
l





5,3
1
1
5,3
1
0
5,3
1
1
5,3
1
0
4
3
2
14  5
22 12  5
 
  
1 4
6
4 2
1 6w 

2
2
2  1  w

2
c
D
l

c
D
l

 1 5,6 0
1
5,6 1  

6
6
5
26  6 

  1 
8
1

2 2
2
2 2
2
2 
 24 D5,4 c1 l1  15 D5,0 l1  18 D5,4 c1 l0  c1 D1,6  4 D5,0 l2   1  4 w 
 Kww 

 4
4 
  32 c D l 2  9 D c 2 l 2  3 c D l 2
  1  
 5 1 5,2 1 4 5,4 1 2 2 1 5,2 2

8
1

2 2
2
2 2
2
2
 24 D5,4 c1 l1  15 D5,0 l1  18 D5,4 c1 l0  c1 D2,6  4 D5,0 l2   1  4 w 


 4
4 
  32 c D l 2  9 D c 2 l 2  3 c D l 2
  2  


1 5,2 1
5,4 1 2
1 5,2 2
4
2
 5

6
6
1
 w
1
 w 
 12 2
2
2
2 
 2 4

  c1 D5,6 l1  6c1 D5,6 l0   4 2
4
2
2 1  2 4 
 5
  2 1  
9
96
8


2 2
2 2
2
2
 36 D5,4 c1 l0  2 D5,4 c1 l2  5 c1 D5,2 l1  5 D5,0 l1
 1
4w


 2 2
2
2
 4c 2 D  1 D l 2  48 D c 2 l 2  3c D l 2  2c 2 D  2 1  

1
5,6
5,0 2
5,4 1 1
1 5,2 2
1
3,6 

2

44
(B-3)
  k s D5,0  D3,0,   D3,0,  9k s c12 D5,4  D1,0,

2
 1  w  
  96
2 
2 2
2 2
 2
  D5,2 c1 l1  K G  D1,0,   9 D5,2 c1 l2  6k s c1 D5,2   1  
 5

  k s D4,0  D4,0,   D4,0,  9k s c12 D4,4  D2,0,

2

 1  w  

2 2
  96 D c 2 l 2  K  D
  2  2 


5,2 1 1
G
2,0,   9 D5,2 c1 l2  6k s c1 D4,2   2
 5

 1  5 x 1  5 y 
4 2
4

2
2
2
2
2
2
D
c
l

c
D
l

2
c
D
l

D
c
l
 5

5,4
1
0
1
5,6
1
1
5,6
0
5,4
1
1

  5
5
5
5
2  5 

  1 
5
 1  5 x
8 2
8
1  y 

2
2
2
2
2
4
D
c
l

c
D
l

D
c
l

4
c
D
l



1
5,6 1
5,4 1 1
1
5,6 0   2 3
 5,4 1 0
2
3
5
5
2312  3 2 

  2 1  
5
5
4
4
1  y 

 1  x
  2 D5,4 c1l0 2  c12 D5,6 l12  2c12 D5,6 l0 2  D5,4 c1l12   4


4
4
4 
5
5

  2 1   2 1  
9
16

2
2 2
2 2
2
2
 6 D5,2 c1l0  4 D5,4 c1 l2  24 D5,4 c1 l1  c1 D1,4  c1 D1,6  15 D5,0 l1   1  3 
x


 3
3 
 18 D c 2 l 2  1 D l 2  12c D l 2
  1  


5,4 1 0
5,0 2
1 5,2 1

4

9
16

2
2 2
2 2
2
2
 6 D5,2 c1l0  4 D5,4 c1 l2  24 D5,4 c1 l1  c1 D2,4  c1 D2,6  15 D5,0 l1   1  3 y 


  3
3 
 18 D c 2 l 2  1 D l 2  12c D l 2
  2  


5,4 1 0
5,0 2
1 5,2 1

4



 c1 D3,4  24 D5,4 c12 l12  6 D5,2 c1l0 2  2c12 D5, 6  


3
3

 8 D l 2  18 D c 2 l 2  12c D l 2  1 D l 2    1   x  1   y  
5,0 1
5,4 1 0
1 5,2 1
5,0 2
2
2
2
2 

5
  2 1   2 1  
4
9

 D5,4 c12 l2 2  2c1 D5,4  c12 D3,6

4

96

2 2
2
2 2   1  x 

  k s D5,0  9 D5,2 c1 l2  9k s c1 D5,4  6k s c1 D5,2  D5,2 c1 l1  
5

  1  
 y 
96

2 2
2
2 2  1

  k s D4,0  9 D5,2 c1 l2  9k s c1 D4,4  6k s c1 D4,2  D5,2 c1 l1  
5

  2  
 1  x
 u v 
1  y 
I 3c1 

 4

  c1  I 4  I 6c1   5


1 2  
   
 1 
 1  2w
1  2w 
I 6c12  6


2
1422  2 
 1  
q
1
4
1
I 0w 
1
4
1
45
following
(B-3)
For x :
1 4
 u

4
 
4
4
5
D 5,1l 12  D 5,3c1l 12  2D 5,3c1l 0 2  2D 5,1l 0 2 
3 
1  5
4
1  1
8
1
8
2
2
2
2   u

D
c
l

D
l

D
l

D
c
l


5,3 1 2
5,1 1
5,1 2
5,3 1 1 
4
24  4
15
4
15
 
1 
1
4
4
3
2 D 5,1l 0 2  D 5,3c1l 2 2  D 5,3c1l 12 
2 
12 
1
12
1 
5
  D1,1  c1D1,3 
1 

3 
4
1  15
1 

2 
D 5,3c1l 12 
4
12  15
4
15
D 5,3c1l 12 
4
4
1
 u
D 5,1l 12  2D 5,3c1l 0 2  D 5,1l 2 2  2 2 
3
4
  
2
2
3
  u 1  4
 u
2
2

D
c
l

D
c
l

D
c

D


5,1 1 1
5,1 1 2
5,3 1
5,1 
2
2
22  5
2
 
 
D 5,1c1l 12 
1
4
 v

3
  
4
1
4
D 5,1l 12  2D 5,3c1l 0 2  D 5,1l 2 2  D 5,3c1l 2 2  2D 5,1l 0 2 
4
15
1
4
D 5,1l 12  2D 5,3c1l 0 2  D 5,1l 2 2 
4
1
 v
D 5,3c1l 2 2  2D 5,1l 0 2 

3
4
 
2
1 3
8
 v
2
2

D
c
l

D
c

c
D

D
c
l

D

D


5,1 1 2
5,3 1
1 3,3
5,1 1 1
3,1
5,1 
1  2
5
 
4
4 2
2
2
2
2
2  w
 2c1D 5, 4 l 0  c1D 5, 4 l 1  2c1 D 5,6 l 0  c1 D 5,6 l1  5 
 
5
5
 
16
9


12c1D 5, 2 l 12  24c12 D 5, 4 l 12  6c1D 5, 2 l 0 2  D 5,0 l 12  c12 D 5, 4 l 2 2  18c12 D 5, 4 l 0 2   3

1
w
15
4

 3 
3
1
1  2
2
 
 c1 D1,6  D 5,0 l 2  c1D1, 4



4
8
1


c1D 3, 4  6c1D 5, 2 l 0 2  2c12 D 5,6  D 5,0 l 12  24c12 D 5, 4 l 12  c12 D 3,6  D 5,0 l 2 2  3

1
w
5
4



2
9 2
12 
 2
2
2
2
2

 12c1D 5, 2 l 1  2c1D 5, 4  18c1 D 5, 4 l 0  c1 D 5, 4 l 2



4
1
96
 w
2 2
2
2 2

 6k s c1D 5, 2  D 5, 2c1 l 1  9c1 k s D 5, 4  9D 5, 2c1 l 2  k s D 5,0 
1 
5
 
1 
5
5
1
1  8
8 2
 5w
2
2
2
2
2 

c
D
l

4
c
D
l

c
D
l

4
c
D
l


1 5, 4 1
1 5, 4 0
1
5,6 1
1
5,6 0 
3
2
1322  5
5
  
46
(B-4)
5
4
4 2
2
2
2
2   w

 2c1D 5, 4 l 0  c1D 5, 4 l 1  2c1 D 5,6 l 0  c1 D 5,6 l 1 
4
124 
5
5
 
1 
2
96


2 2
2
2 2
 6k s c1D 5, 2  D 5, 2c1 l 1  9c1 k s D 5, 4  9D 5, 2c1 l 2  k s D 5,0   x 
5


4
1 4
4 2
8
2
2
2
2
2
2
2   x
D
l

2
D
l

4
c
D
l

c
D
l

2
c
D
l

c
D
l


5, 2 1
5, 2 0
1 5, 4 0
1
5,6 1
1
5,6 0
1 5, 4 1 
4
14  5
5
5
 
4
1  16
1
8
1 2
8 2
1
2
2
2
2
2
2   x

c
D
l

c
D
l

D
l

c
D
l

c
D
l

D
l


1 5, 4 1
1 5, 4 2
5, 2 1
1
5,6 2
1
5,6 1
5, 2 2 
4
24  15
2
15
4
15
4
 
1
1
1
 2

2c1 D 5,6 l 0 2  4c1D 5, 4 l 0 2  c12 D 5,6 l 2 2  c1D 5, 4 l 2 2  D 5, 2 l 2 2  2D 5, 2 l 0 2   4

1

4
2
4

 2 x2 
2 2
4 2
8
1 2  4
2
2
2
  
 D 5, 2 l 1  c1 D 5, 6 l 1  c1D 5, 4 l1

3

3
3
9
1
 88

c1D 5, 2 l 12  c12 D 5, 4 l 2 2  2D 5,0 l 0 2  12c1D 5, 2 l 0 2  2c1D1, 4  D 5,0 l 2 2  D1, 2   2

1 5
 x
4
4



2
32
1  3
 2
2
2
2
2
2
2
2

 c1D 5, 2 l 2  D 5,0 l 1  c1 D1,6  24c1 D 5, 4 l 1  18c1 D 5, 4 l 0

2

15
32
52


9c1D 5, 2 l 2 2 c12 D 5,6  D 5, 2  12c12 D 5, 4 l 2 2  c1D 5, 2 l 12  c12 D 5, 4 l 12  D 5,0 l 2 2   2

1
 x
5
5



2
2  4
 2
2

 D 5,0 l 1  2c1D 5, 4

3

1
1
8
4
 2

2c D l 2  c 2 D l 2  D l 2  c D l 2  D l 2 
4
1  1 5,6 0 4 1 5,6 2 4 5, 2 2 15 1 5, 4 1 15 5, 2 1    y



1
132  4 2
 3
2
2
2
2

 c1 D 5,6 l 1  4c1D 5, 4 l 0  c1D 5, 4 l 2  2D 5, 2 l 0

 15

2
1
1
8
4
 2

2c1 D 5,6 l 0 2  c12 D 5,6 l 2 2  D 5, 2 l 2 2  c1D 5, 4 l 12  D 5, 2 l12   4

 y
1
4
4
15
15



3
1
12  4 2
 3
2
2
2
2

 c1 D 5,6 l 1  4c1D 5, 4 l 0  c1D 5, 4 l 2  2D 5, 2 l 0

 15

2
15
 68 2

2
2
2
2
2
2
2
 5 c1 D 5, 4 l 1  12c1D 5, 2 l 0  c1 D 3,6  2 c1D 5, 2 l 2  18c1 D 5, 4 l 0  2D 5,0 l 0  

 2
1 
28
39 2
2
2
2
  y 
D 3, 2  D 5,0 l 1  D 5, 2  2c1D 3, 4  2c1D 5, 4  8c1D 5, 2 l 1  c1 D 5, 4 l 2 

 
12
15
4


 c12 D 5,6  3 D 5,0 l 2 2

4


1
1
1
w
  3  I 1  c1I 3 u  4  I 2  2c1I 4  c12 I 6   x  5 c1  I 4  c1I 6 
1
1
1

47
following
(B-4)
For y :
2
1
3
8
 u
2
2
D 5,3c1  D 5,1c1l 2  c1D 3,3  D 5,1c1l1  D 3,1  D 5,1 

2 
2
5
 
4
1  4
4
1
1
2
2
2
2
2
2   u
 D 5,1l 1  D 5,3c1l 1  2D 5,3c1l 0  D 5,1l 2  D 5,3c1l 2  2D 5,1l 0  3 
23  15
15
4
4
  
4
1
1
2
2   u

 2D 5,1l 0  D 5,3c1l 1  D 5,1l 1  2D 5,3c1l 0  D 5,1l 2  D 5,3c1l 2 
3
2 12 
15
15
4
4
 
1 
4
2
4
2
2
2
4
4
2
2
2   v
 D 5,1l 1  D 5,3c1l 1  2D 5,3c1l 0  2D 5,1l 0  4 
23  5
5
 
1 4
2
4
2  1
1
8
8
2
2
2
2  v
 D 5,1l 2  D 5,3c1l 2  D 5,1l 1  D 5,3c1l 1  4 
14  4
4
15
15
 
1
4
4
1
 4v
2
2
2
2
2 
 2D 5,1l 0  D 5,3c1l 2  D 5,3c1l 1  D 5,1l 1  2D 5,3c1l 0  D 5,1l 2  2 2 
2 12 
4
3
3
4
  
1 
2
2
2
3
  v 2  4
v
2
2
 c1D 2,3  D 2,1  D 5,1c1l 1  2  2  D 5,1c1l 1  D 5,1c1l 2  D 5,3c1  D 5,1  2 
2 
5
1  5
2
 
 
1 
12
2
5
4
4 2
2
2
2
2  w
 2c1D 5, 4 l 0  c1D 5, 4 l 1  2c1 D 5,6 l 0  c1 D 5,6 l 1  5 
25 
5
5
 
1 
2
16
9


12c1D 5, 2 l 12  24c12 D 5, 4 l 12  6c1D 5, 2 l 0 2  D 5,0 l 12  c12 D 5, 4 l 2 2  3

1
w
15
4

 3 
3
1
2 
2
2
2
2
 
 18c1 D 5, 4 l 0  c1 D 2,6  D 5,0 l 2  c1D 2, 4



4
8


2
2
c
D

6
c
D
l

2
c
D

D 5,0 l 12  24c12 D 5, 4 l12  c12 D 3,6  3
1
3,
4
1
5,
2
0
1
5,6
1 
w
5



2
9 2
2 1  1
 2
2
2
2
2
2

  D 5,0 l 2  12c1D 5, 2 l 1  2c1D 5, 4  18c1 D 5, 4 l 0  c1 D 5, 4 l 2 
 4

4
1 
96
 w
2 2
2
2 2

 6k s c1D 4, 2  D 5, 2c1 l 1  9c1 k s D 4, 4  9D 5, 2c1 l 2  k s D 4,0 
2 
5
 
1  8
8 2
 5w
2
2
2
2
2 

c
D
l

4
c
D
l

c
D
l

4
c
D
l


1 5, 4 1
1 5, 4 0
1
5,6 1
1
5,6 0 
3
2
2312  5
5
  
1 
4
5
 w

4
 
5
4
5
2c1D 5, 4 l 0 2  c1D 5, 4 l 12  2c12 D 5,6 l 0 2  c12 D 5,6 l 12 
4 
2 1 
1
1
8
4
 2

2c1 D 5,6 l 0 2  c12 D 5,6 l 2 2  D 5, 2 l 2 2  c1D 5, 4 l 12  D 5, 2 l12  4

1

4
4
15
15

 3 x 
3
1
2 1  4 2
2
2
2
2
  
  c1 D 5,6 l 1  4c1D 5, 4 l 0  c1D 5, 4 l 2  2D 5, 2 l 0

 15

2
48
(B-5)
1
1
8
4
 2

2c D l 2  c 2 D l 2  D l 2  c D l 2  D l 2
1  1 5,6 0 4 1 5,6 2 4 5,2 2 15 1 5,4 1 15 5,2 1   4 x



1
213  4 2
 3
2
2
2
2

  c1 D 5,6l1  4c1D 5,4l 0  c1D 5,4l 2  2D 5,2l 0

2
 15

68 2
 15

2
2
2
2
2
  2 c1D 5,2l 2  18c1 D 5,4l 0  2D 5,0l 0  5 c1 D 5,4 l1


 2
1 
28
 x
2
2
2
12c1D 5,2l 0  c1 D 3,6  D 5,0l1  D 5,2  2c1D 3,4  2c1D 5,4 



12
15



 8c D l 2  D  39c 2 D l 2 c 2 D  3 D l 2

1 5,2 1
3,2
1
5,4 2
1
5,6
5,0 2


4
4


96


2 2
2
2 2
 6k s c1D 4, 2  D 5, 2c1 l 1  9c1 k s D 4, 4  9D 5, 2c1 l 2  k s D 4,0   y 
5


1 4
4
5
  y

4
 
4
8
5
D 5, 2 l 12  2D 5, 2 l 0 2  4c1D 5, 4 l 0 2  c12 D 5,6 l 12  2c12 D 5,6 l 0 2  c1D 5, 4 l 12 
4 
2  5
 4 y
1  8 2
16
1
8
1 2
1
2
2
2
2
2
2 

 c1 D 5,6 l 1  c1D 5, 4 l 1  c1D 5, 4 l 2  D 5, 2 l1  c1 D 5,6 l 2  D 5, 2 l 2 
4
14  15
15
2
15
4
4
 

1
1
1

2c 2 D l 2  4c1D 5, 4 l 0 2  c12 D 5,6 l 2 2  c1D 5, 4 l 2 2  D 5, 2 l 2 2  4
 y
1  1 5,6 0
4
2
4

 2 2
2 2
4
4 2
8
1 2 
2
2
2
2
  
 2D 5, 2 l 0  D 5, 2 l1  c1 D 5,6 l1  c1D 5, 4 l1



3
3
3
9
1
 88

c D l 2  c 2 D l 2  2D 5,0 l 0 2  12c1D 5, 2 l 0 2  2c1D 2, 4  D 5,0 l 2 2  2
 y
1  5 1 5, 2 1 4 1 5, 4 2
4



2
32
2  3
 2
2
2
2
2
2
2
2

  c1D 5, 2 l 2  D 5,0 l 1  c1 D 2,6  24c1 D 5, 4 l1  18c1 D 5, 4 l 0  D 2, 2 
 2

15
32
52
 2

c1 D 5,6  D 5, 2  12c12 D 5, 4 l 2 2  c1D 5, 2 l12  c12 D 5, 4 l12  9c1D 5, 2 l 2 2   2

 y
1
5
5



2
4
1 
 2
2
2

 D 5,0 l 2  D 5,0 l1  2c1D 5, 4



3

1
1
w
  24  I 1  c1I 3 v  4  I 2  2c1I 4  c12I 6   y 
c I  c1I 6 
4 1 4
1
1
21

49
following
(B-5)
Appendix C
Various boundary conditions such as simply supported, clamped, and free boundary conditions
are considered as follows:
Simply supported boundary conditions:
x  0, a
2
5
3
5
3
5
0
1 T yyy
0
0
0
u  Px0  T xxx
 T yyx
 T zzx
0
v , x  M xy0 
1 Px0

2 y
5 x
0
4 T xxy 4 T yyx 1 T zzx
1 T zzy 1 Y xz0




5 x
5 y
5 y
5 x
2 x
0

0
0
1 Y yz

0
4 y
0
P 3 1 Py
2 T 3
3 T yyy 6 T xxy
 c1 x  c1
 3c1Pz2  c1 xxx  c1
 c1
x
2 y
5
x
10
y
5
y
3
w  c1M
3
xx
3
3
3
3
1 0 6
4 0
3 T yyx 1 0
3 T zzx
2
2
2
 T zzz
 c1T zzz
 T xxz
 6c1T xxz

c

T

6
c
T

c
 5 1 x 5  yyz 1 yyz  5 1 x
5
5
5
3
3 T zzy 1 0
2
3
3
1
2
3
3
3
3
 c1
 Y xy  3cY
c1T xxx
 c1T yyx
 c1T zzx
  c1Py3
1 xy   c1Px 
10
x
2
5
5
5
2
3
6
3
3
3
3
 c1T yyy
 c1T xxy
 c1T zzy
0
10
5
10
2 1
2
3 1
3 1
3
3
3
3
Px1  c1Px3  T xxx
 c1T xxx
 T yyx
 T zzx
 c1T zzx
 c1T yyx

5
5
5
5
5
1 1
1
4 1
4
1 1
1
3
3
3
3
  T yyy
 c1T yyy
 T xxy
 c1T xxy
 T zzy
 c1T zzy
 Y xz1  cY

1 xz   0
5
5
5
5
5
2
1
3
1
Px1
Px3 1 Py 1 Py
2 T xxx
1
3
0
2
y  M xx  c1M xx 
 c1

 c1
 Pz  3c1Pz 
x
x
2 y
2 y
5 x
(C-1)
1
3
1
3
T yyy
T yyy
2 T xxx
2 0 6
4 T xxy 8 0
2
2
 c1


 T zzz  c1T zzz 
 T xxz  3c1T xxz

5
x
y
y
5
5
5 y
5

1
3
1
3
T zzy
3 T yyx 8 0
2 T xxy 3 1
1  T zzy
2
3
 T yyz  3c1T yyz

c

T

c
T


c
 5 1 y 5  zzx 1 zzx  5  x 1 x
5 x
5


1
Y yz3
1 0
1  Y yz
2
Y

3
cY


c
 xy 1 xy  4  y 1 y
2

3
 3 T yyx

c
0

1
5

x

50



y  0,b
0
0
1 Py 1 T xxx
4 T xxy 4 T yyx 1 T zzx
1 T zzy 1 T yyy 1 Y xz0







2 x
5 y
5 x
5 y
5 y
5 x
5 x
4 x
0
M xy0 
0
0
0
0
1 Y yz
 u, y  0
4 x
2 0
2 0
3 0
3 0
v  Py0  T xxx
 T yyy
 T xxy
 T zzy
0
5
5
5
5
3
3
P 3
1 P 3
3 T 3
2 T
3 T
3
w  c1M yy
 c1 x  c1 y  3c1Pz2  c1 xxx  c1 yyy  c1 xxy
2 x
y
10
x
5
y
5
y
0

3
1 0 6
1 0
6 T yyx 4 0
3 T zzx
2
2
2
 T zzz
 c1T zzz
 T xxz
 6c1T xxz

c

T

6
c
T

c
 5 1 x 5  yyz 1 yyz  10 1 x
5
5
5
3
3 T
1
2
3
3
1
2
3
3
3
3
 c1 zzy  Y xy0  3cY
c1T yyy
 c1T xxy
 c1T zzy
  c1Px3
1 xy   c1Py 
5
y
2
5
5
5
2
3
6
3
3
3
3
 c1T xxx
 c1T yyx
 c1T zzx
0
10
5
10
1 1
1
4 1
4
2 1
1
3
3
3
3
x   T xxx
 c1T xxx
 T yyx
 c1T yyx
 T zzx
 c1T zzx
 Y yz1  cY

1 yz   0
5
5
5
5
5
2
1
3
1
3
1
P
P
1 Px 1 Px
1 T xxx
1
M yy
 c1M yy3  y  c1 y 
 c1
 Pz0  3c1Pz2 
y
y
2 x
2 x
5 x
3
(C-2)
3
1 T xxx
2 T yyy 2 T yyy 6
3 T x xy 3 T xxy 4 T yyx
2
 c1

 c1
 c1T zzz

 c1

5
x
5 y
5
y
5
5 y
5
x
5 x
1
3
1
3
1
2 0
4 T yyx 2 0
1  T 1
T 3 
2
2
 T xxz
 3c1T xxz
 c1
 T yyz  3c1T yyz
  zzx  c1 zzx 


5
5
x
5
5  x
x 
1
3
1
 1 0
T zzy
Y yz3 
3  T zzy
1  Y yz
2
 
 c1
 c1
  Y xy  3cY


1 xy  
5  y
y  2
4  x
x 
2 1
2
3 1
3 1
3
3
3
3
 Py1  c1Py3  T yyy
 c1T yyy
 T xxy
 T zzy
 c1T zzy
 c1T xxy
0

5
5
5
5
5
3
Clamped boundary conditions:
x  0, a u  v  w  x  y  u , x  v , x  w , x  w , xx  w , xy  x , x  y , x  0
y  0,b u  v  w  x  y  u , y  v , y  w , y  w , yy  w , xy  x , y  y , y  0
51
(C-3)
Free boundary conditions:
x  0, a
M xy
M xx3
1
M  M  c1
 c1
 M xz0  3c1M xz2  M xx
 c1M xx3  M 1xy  c1M xy3  M xx3  0
x
y
3
0
xy
0
xx
y  0,b
M xy0  M yy0  c1
(C-4)
M yy3
y
 c1
M xy3
y
1
1
 M yz0  3c1M yz2  M xy
 c1M xy3  M yy
 c1M yy3  M yy3  0
52
Figure Captions
Fig. 1
Schematic of micro composite Reddy plate embedded in an elastic foundation
Fig. 2
Distribution types of SWCNTs in micro composite Reddy plate
Fig. 3
Effects of various boundary conditions on a- Dimensionless deflection bDimensionless critical buckling load
Fig. 4
Effects of material length scale parameters on a- Dimensionless deflection bDimensionless critical buckling load
Fig. 5
Effects of SWCNT volume fraction of the micro composite plate using FSDT and TSDT
on a- Dimensionless deflection b- Dimensionless critical buckling load
Fig. 6
Effects of SWCNT distribution types on a-Dimensionless deflection b- Dimensionless
critical buckling load
Fig. 7
Effects of elastic foundation parameters on a- Dimensionless deflection bDimensionless critical buckling load
Fig. 8
Effects of environment parameters on a-Temperature changes on the dimensionless
deflection b-Moisture changes on the dimensionless deflection c-Temperature and
moisture changes on dimensionless critical buckling load
Fig. 9
Effects of axial loading on dimensionless critical buckling load of micro composite
plate
Fig. 10
Effects of volume fraction on the dimensionless natural frequencies of micro
composite plate for FSDT and TSDT
Fig. 11
Effects of SWCNT distribution types on the dimensionless natural frequency of micro
composite plate
Fig. 12
The dimensionless natural frequency of micro composite plate for SGT, MCST and CT
Fig. 13
The dimensionless natural frequency of micro composite plate against elastic
foundation parameters
Fig. 14
The dimensionless natural frequency of micro composite plate with
moisture and temperature changes
Fig. 15
First six vibration mode shapes of micro composite Reddy plate with clamped
boundary conditions in all edges
53
different
z
y
a
b
(KG)
h
x
(Kw)
Fig. 1: Schematic of micro composite plate embedded in an elastic foundation
54
Fig. 2: Distribution types of SWCNTs in micro composite Reddy plate
55
1
x 10
-3
scsc
cccc
sccc
sssf
scsf
0.9
0.8
-4
x 10
3.65
3.64
0.7
3.63
W
max
0.6
3.62
0.5
39.95
40
40.05
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
a/h
a-Dimensionless deflection
56
40
45
50
350
300
scsc
ssss
sfsf
sssf
scsf
250
N
cr
285
200
284.5
284
150
283.5
100
283
39.9
50
5
10
15
20
25
30
40
35
40.1
40
45
50
a/h
b-Dimensionless critical buckling load
Fig. 3: Effects of various boundary conditions on a- Dimensionless deflection b- Dimensionless critical buckling
load
57
x 10
1.8
-4
MSGT
MCST
CT
1.6
1.4
W
max
1.2
1
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
40
45
50
4.5
5
a/h
a-Dimensionless deflection
600
MSGT
MCST
CT
550
500
450
N
cr
400
350
300
250
200
150
100
1
1.5
2
2.5
3
a/b
3.5
4
b-Dimensionless critical buckling load
58
Fig. 4: Effects of material length scale parameters on a- Dimensionless deflection b- Dimensionless critical
buckling load
1
x 10
-3
V*
0.8
0.7
max
0.6
W
=0.11 TSDT
CNT
*
VCNT=0.14
V* =0.17
CNT
*
V
=0.11
CNT
*
VCNT=0.14
V* =0.17
CNT
0.9
TSDT
TSDT
FSDT
FSDT
FSDT
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
a/h
a-Dimensionless deflection
59
40
45
50
500
450
400
350
N
cr
300
250
*
VCNT=0.11 TSDT
200
V*
150
100
50
0
5
=0.14 TSDT
CNT
*
V
=0.17 TSDT
CNT
*
VCNT=0.11 FSDT
V* =0.14 FSDT
CNT
*
VCNT=0.17 FSDT
10
15
20
25
30
35
40
45
50
a/h
b-Dimensionless critical buckling load
Fig. 5: Effects of SWCNT volume fraction of the micro composite plate using FSDT and TSDT on
Dimensionless deflection b- Dimensionless critical buckling load
60
a-
x 10
-4
FG-X
UD
FG-V
FG-O
W
max
2
1
0
15
20
25
30
35
40
45
50
a/h
a-Dimensionless deflection
360
FG-X
UD
FG-V
FG-O
340
320
300
N
cr
280
260
240
220
200
180
25
30
35
40
a/h
b- Dimensionless critical buckling load
61
45
50
Fig. 6: Effects of SWCNT distribution types on a- Dimensionless deflection b- Dimensionless critical buckling
load
3.2
x 10
-7
KG=0 (N/m)
K =103 (N/m)
3.1
G
4
K =10 (N/m)
G
3
W
max
2.9
2.8
2.7
2.6
2.5
0
2
4
6
K (N/m 3)
w
a-Dimensionless deflection
62
8
10
x 10
11
220
KG=0 (N/m)
K =103 (N/m)
200
G
K =104 (N/m)
G
N
cr
180
160
140
120
100
0
2
4
6
3
K (N/m )
w
8
10
x 10
11
b- Dimensionless critical buckling load
Fig. 7: Effects of elastic foundation parameters on a- Dimensionless deflection b- Dimensionless critical
buckling load
63
x 10
-4
3.5
 T=0
 T=20
 T=40
 T=50
x 10
-5
 H=0 %
 H=0.5%
 H=1 %
3
2.5
max
W
W
max
2
1
1.5
1
0.5
0
0
5
10
15
a/h
20
25
5
10
15
a/h
20
b-Moisture changes on the dimensionless deflection
a-Temperature changes on the dimensionless deflection
300
 T=0
 T=50
 T=0
 T=50
250
&
&
&
&
 H=0%
 H=0%
 H=3%
 H=3%
N
cr
200
150
100
50
0
2.5
3
25
3.5
4
4.5
a/b
c-Temperature and moisture changes on dimensionless critical buckling load
64
5
Fig. 8: Effects of environment parameters on a-Temperature changes on the dimensionless deflection
b-Moisture changes on the dimensionless deflection c-Temperature and moisture changes on dimensionless
critical buckling load
65
3500
f1= -1 & f2=0
f1=0 & f2= -1
3000
f = -1 & f = -1
1
2
f =1 & f = -1
2500
1
2
f = -1 & f =1
1
2
Ncr
2000
1500
400
1000
200
500
0
5
0
10
15
20
25
30
35
0
50
40
45
50
a/h
Fig. 9: Effects of axial loading on dimensionless critical buckling load of micro composite plate
66
60
55
50
45

1
40
35
V*
30
25
20
15
10
5
=0.11 TSDT
CNT
*
VCNT=0.14
V* =0.17
CNT
*
VCNT=0.11
V* =0.14
CNT
*
V
=0.17
CNT
10
15
20
25
30
35
40
TSDT
TSDT
FSDT
FSDT
FSDT
45
50
a/h
Fig. 10: Effects of volume fraction on the dimensionless natural frequencies of micro composite Reddy plate
for FSDT and TSDT
67
55
50
FG-X
UD
FG-V
FG-O

1
45
40
35
30
25
20
25
30
35
a/h
40
45
50
Fig. 11: Effects of SWCNT distribution types on the dimensionless natural frequency of micro composite plate
68
160
MSGT
MSCT
CT
140
120

1
100
80
60
40
20
1
1.5
2
2.5
3
a/b
3.5
4
4.5
5
Fig. 12: The dimensionless natural frequency of micro composite plate for MSGT, MCST and CT
69
34
KG=0 (N/m)
32
K =103 (N/m)
G
4
K =10 (N/m)
G
30

1
28
26
24
22
20
0
2
4
6
K (N/m 3)
w
8
10
x 10
11
Fig. 13: The dimensionless natural frequency of micro composite plate against elastic foundation parameters
70
100
 T=0
 T=50
 T=0
 T=50
90
80
&
&
&
&
 H=0%
 H=0%
 H=3%
 H=3%
70

1
60
50
40
30
20
10
0
2
2.5
3
3.5
a/b
4
4.5
5
Fig. 14: The dimensionless natural frequency of micro composite plate with different moisture and
temperature changes
71
0.4
1
0.2
0.5
0
-0.2
0
-0.4
-0.6
-0.5
-0.8
-1
1
-1
1
1
1
0.8
0.5
0.8
0.5
0.6
0.6
0.4
0.4
0.2
0
0
0.2
0
1
1
0.5
0.5
0
0
0
-0.5
-0.5
-1
1
-1
1
1
0.8
0.5
1
0.6
0.4
0
0.8
0.5
0.2
0.6
0.4
0
0
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
1
-1
1
0.2
0
1
1
0.8
0.5
0.8
0.5
0.6
0.6
0.4
0.4
0
0.2
0
0
0.2
0
Fig. 15: First six vibration mode shapes of micro composite Reddy plate with clamped boundary conditions in
all edges
72
Table captions
Table 1
The temperature- dependent material properties of armchair SWCNTs (10, 10) as reinforcement
[56]
Table 2
i Coefficients of SWCNTs
Table 3
Deflections of square plate using FSDT with h  2 mm and q  0.1MPa
Table 4
Dimensionless natural frequency of square plate using FSDT ( a / h  50 )
Table 5
Dimensionless critical buckling load of simply supported square plate using FSDT
( N cr  N x & N y  0 )
Table 6
The effect of various boundary conditions on the dimensionless deflection of microcomposite
Reddy plate
Table 7
The effect of various boundary conditions on the dimensionless critical buckling load of
microcomposite Reddy plate
73
Table 1: The temperature- dependent material properties of armchair SWCNTs (10, 10) as reinforcement [56]
CNT
( 12  0.175 )
Temperature (K)
CNT
E 11
(TPa )
300
CNT
 22
(1 / K )
CNT
E 22
(TPa )
CNT
G12
(TPa )
CNT
11
(1 / K )
5.6466
7.0800
1.9445
3.4584 e-6
5.1682 e-6
500
5.5308
6.9348
1.9643
4.5361 e-6
5.0189 e-6
700
5.4744
6.8641
1.9644
4.6677 e-6
4.8943 e-6
74
Table 2: i Coefficients of SWCNTs
1
3
2
0.11
0.149
0.934
0.934
0.14
0.150
0.941
0.941
0.17
0.149
1.381
1.381
*
V CNT
75
Table 3: Deflections of square plate using FSDT with h  2 mm and q  0.1MPa
CCCC
V
*
CNT
SSSS
a/h
10
Ref. [57]
Present work
Ref. [57]
Present work
UD
1.412e-3
1.351e-3
2.394e-3
2.325e-3
FG V
1.486e-3
1.425e-3
2.864e-3
2.407e-3
FG  O
1.595e-3
1.538e-3
3.378e-3
3.328e-3
FG  X
1.318e-3
1.258e-3
2.012e-3
1.942e-3
UD
0.1698
0.1681
0.7515
0.7478
FG V
0.2384
0.2370
1.082
0.815
FG  O
0.3085
0.3108
1.416
1.4256
FG  X
0.1223
0.1204
0.5132
0.508
0.17
50
76
Table 4: Dimensionless natural frequency of square plate using FSDT ( a / h  50 )
V
*
CNT
Boundary
condition
ssss
Vibration
mode
FG  X
UD
Ref. [57]
Present work
Ref. [57]
Present work
(1,1)
23.697
23.694
28.413
28.473
(2,1)
28.987
28.909
33.434
33.831
(3,1)
43.165
42.499
47.547
46.867
(4,1)
67.475
64.975
72.570
69.967
(1,1)
49.074
49.068
57.245
57.458
(2,1)
54.324
54.187
62.236
62.310
(3,1)
68.069
67.035
75.746
75.902
(4,1)
92.868
89.164
100.850
97.241
0.17
cccc
77
Table 5: Dimensionless critical buckling load of simply supported square plate using FSDT ( N cr  N x & N y  0 )
Buckling mode
Ref. [23]
Analytical results
Present work
(1,1)
39.1744
39.4761
39.6118
(2,1)
60.4874
61.6814
61.6944
(3,1)
105.391
109.656
109.094
(4,1)
152.501
157.904
157.431
78
Table 6: The effect of various boundary conditions on the dimensionless deflection of microcomposite Reddy plate
Various BC’s
a/h=5
a/h=10
a/h=20
a/h=40
a/h=50
CCCC
0.005955e-4
0.034684e-4
0.212830e-4
1.620255e-4
3.248260e-4
SCCC
0.005976e-04
0.054614e-04
0.452244e-04
3.620371e-4
7.252371e-4
SCSC
0.005993e-4
0.053240e-4
0.449507e-4
3.644958e-4
7.307905e-4
SCSF
0.010344e-4
0.053278e-4
0.449580e-4
3.647185e-4
7.314404e-4
SSSF
0.015764e-4
0.089327e-4
0.542206e-4
4.166228e-4
8.479397e-4
79
Table 7: The effect of various boundary conditions on the dimensionless critical buckling load of microcomposite
Reddy plate
Various BC’s
a/h=5
a/h=10
a/h=20
a/h=40
a/h=50
SFSF
56.2912
117.9949
208.6554
283.5979
301.1995
SSSF
56.3306
118.0022
208.9319
283.9166
301.4740
SCSF
56.3600
118.0322
209.0333
283.9940
301.5439
SSSS
56.3761
118.0091
209.1943
284.2547
301.7760
SCSC
56.4322
118.0677
209.3784
284.4188
301.9331
CCSS
56.4879
118.2862
209.6814
284.9494
302.4903
CCCS
56.5777
118.5460
210.07995
285.5589
303.1221
CCCC
56.6054
118.5748
210.1706
285.6402
303.2003
80