Journal of Science and Transport Technology Vol. 4 No. 1, 13-22
Journal of Science and Transport Technology
Journal homepage: https://jstt.vn/index.php/en
JSTT 2024, 4 (1), 13-22
Published online 22/03/2024
Article info
Type of article:
Original research paper
DOI:
https://doi.org/10.58845/jstt.utt.2
024.en.4.1.13-22
*Corresponding author:
E-mail address:
quangln@utt.edu.vn
Received: 17/02/2024
Revised: 18/03/2024
Accepted: 20/03/2024
Nonlinear dynamic buckling analysis of the
FG-GPLRC cylindrical and sinusoid panels
with porous core under time-dependent axial
compression
Nguyen Thi Phuong1,2, Vu Minh Duc1,2,3, Luu Ngoc Quang4,*
1Laboratory of Advanced Materials and Structures, Institute for Advanced
Study in Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam,
nguyenthiphuong@tdtu.edu.vn
2Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City,
Vietnam
3Institute of Transport Technology, University of Transport Technology, Hanoi,
Vietnam
4Faculty of Civil Engineering, University of Transport Technology, Hanoi,
Vietnam
Abstract: The nonlinear dynamic buckling responses of functionally graded
graphene platelet reinforced composite (FG-GPLRC) cylindrical and sinusoid
panels with porous core are presented in this paper. The governing
formulations are established by applying the nonlinear higher-order shear
deformation theory (HSDT). an approximation technique is used to determine
the stress function for complexly curved panels. Euler-Lagrange equations can
be used to achieve the nonlinear motion equation. Numerical investigations are
considered using the Runge-Kutta method for dynamic responses, and using
the Budiansky-Roth criterion for critical dynamic buckling loads. Some
discussions on the dynamic buckling responses of panels with porous core can
be achieved from the numerical examples.
Keywords: Sinusoid panel; Cylindrical panel; Dynamic buckling; Higher-order
shear deformation theory (HSDT); Nonlinear mechanics; Functionally graded
graphene platelet-reinforced composite;
1. Introduction
Cylindrical panels and Rectangular plates
are widely used in many structures in engineering
fields. Therefore, a lot of researchers have posed
different problems for these structures such as
linear and nonlinear dynamic and static buckling
behavior, vibration, and dynamic response
behavior, …
Dynamic responses and vibration of
functionally graded material (FGM) plates with
variable delamination parameters were studied by
Wang et al. [1] using the extended Chebyshev–Ritz
method. Liew et al. [2] investigated the vibration
behavior of coating-FGM-substrate cylindrical
panels subjected to a temperature gradient across
the thickness with general boundary conditions.
The static behavior of viscoelastic FGM cylindrical
panels under uniform pressure was studied by
Norouzi and Alibeigloo [3] using the 3D elasticity
theory, state space method, state space differential
quadrature method, and Fourier expansion. The
semi-analytical approaches for dynamic buckling
JSTT 2024, 4 (1), 13-22
Nguyen et al
14
and vibration behavior of FGM cylindrical panels
and doubly curved shallow shell panels were
presented [4-6] using the Donnell shell theory,
higher-order shear deformation theory (HSDT),
and the Galerkin method.
Song et al. [7] investigated the buckling and
postbuckling behavior of functionally graded
graphene platelet-reinforced composite (FG-
GPLRC) plates under biaxial compressions using
the first-order shear deformation theory and two-
step perturbation technique. Based on the classical
plate theory and the Rayleigh–Ritz technique, the
free vibrations of FG-GPLRC cantilever torsional
plates were studied by Sun et al. [8] with variations
in the transverse direction of pore and graphene.
The vibration behavior of FG-GPLRC doubly
curved shell panels was mentioned using the first-
order shear deformation theory [9], and HSDT [10].
The complexly curved panels made from
composites reinforced by graphene sheets and
GPLs were mentioned in nonlinear buckling
problems [11,12] and in nonlinear dynamic
buckling and vibration behavior [13]. Bending
behavior of sandwich beams with FGM porous
core subjected to different load types by using the
Ritz energy method [14].
For FGM structures with porous core, the
plates, cylindrical panels, and cylindrical shells
were investigated in nonlinear buckling and
postbuckling problems using the Galerkin method
[15-18]. The Ritz energy method was applied to
investigate the nonlinear thermos-mechanical
buckling and postbuckling of FG-GPLRC spherical
caps and circular plates with porous core [19].
Due to the architecture and engineering
requirements, complexly curved panels can be
designed with several engineering equipments.
The numerical-analytical algorithm for the
nonlinear problem of dynamic buckling of sinusoid
and cylindrical FG-GPLRC panels with porous
core. The HSDT with von Karman nonlinearity is
applied. By applying the like-Galerkin method, the
stress function can be determined. The nonlinear
motion equations are obtained using the Euler-
Lagrange equations. By applying the Runge-Kutta
method and the Budiansky-Roth criterion, critical
dynamic buckling loads can be achieved. The
significant influences of material, foundation, and
geometrical properties on the dynamic buckling
responses are investigated and validated.
2. Geometrical and Material designs of FG-
GPLRC cylindrical and sinusoid panels with
porous core
Fig. 1. Material configuration and geometry of the FG-GPLRC cylindrical and sinusoid panels with
porous core
JSTT 2024, 4 (1), 13-22
Nguyen et al
15
Material configuration and geometry of
sinusoid and cylindrical FG-GPLRC panels and
coordinate system are seen in Fig. 1, with the
length of edges, rise, and thickness (
a
,
b
,
h
, and
) of curved mid-surface.
The curved
y
-direction surface equation of
sinusoid panels is written by
( ) ( )
1sin .
s
Y y H y b= −
(1)
Based on Eq. (1), the radius equation can be
obtained, as
( )
( )
3/2
2 2 2 2
1
2
1
cos
sin .
s
H y b b
RbH y b
+
=
(2)
The Halpin-Tsai model is used to determine
the elastic modulus of panels, as
( ) ( )
( )
( )
( )
11
1
22
2
33
88
55 ,
88
GPL
fm
GPL
GPL
m
GPL
Vz
E z E
Vz
Vz
E
Vz


+
=−
+
+−
(3)
where
12
12
12
11
,,
2 , 2 ,
GPL m GPL m
GPL m GPL m
GPL GPL GPL GPL
E E E E
E E E E
a t b t
− + − +
==
++
==



with
GPL
E
and
m
E
are respectively the
elastic moduli of the GPL and matrix. The length,
thickness, and width of the GPL are
,
GPL
a
GPL
t
and
.
GPL
b
the GPL volume fraction
GPL
V
with
1
M GPL
VV+=
, derived by
( )
( )
( )
,
1
GPL
GPL
GPL GPL M GPL
W
Vz
WW

=+−
(4)
where
GPL
and
M
are respectively the
mass densities of the GPLs and matrix.
In this paper, the mass distribution laws of
GPL for upper and lower FG-GPLRC coatings are
chosen, as [19]
+) The upper coating
22
c
h
hz−
−




- UD type:
*;
GPL GPL
WW=
(5)
- X type:
*
842
GPL GPL
c
zh
h
WW
h

+−


−

=
(6)
- O type:
*
84
22
GPL
c
GPL W
h h
zh
W
+
= − −


−

(7)
- V type:
*
42
GPL
c
c
GPL
zh
WhW
h
+
=−



(8)
- Λ type:
*
42
GPLP
c
GL W
h
zh
Wh



+
=−
(9)
+) The lower coating
22
c
hh
z




- UD type:
*;
GPL GPL
WW=
(10)
- X type:
*
842GPP
c
GL L
zh
WhW
h

−
=+


−

(11)
- O type:
*
84
22
GPL
c
GPL W
h h
zh
W
−
= − +


−

(12)
- V type:
*
24
cGPGPL L
c
hW
h
z
Wh
−
=−



(13)
- Λ type:
*
24
GPLP
c
GL W
h
hz
Wh



−
=−
(14)
The coefficient of thermal expansion and
Poisson ratio of the panels are determined as
( )
( )
( )
( )
1,
1.
m GPL GPL GPL
m GPL GPL GPL
z V V
z V V
= − +
= − +
(15)
The porous core layer of the panels is chosen
as the same material as the FG-GPLRC matrix.
Effective elastic modulus and coefficient of thermal
expansion can be determined as [14]
( )
( )
( )
0
0
1 cos
,
1 1 1 cos
,
22
cm
cm
cm
cc
E E e z h
e z h
hh
z

= −

=

= − − −

−
(16)
where
0
e
is the porosity coefficient (
0
01e
).
3. Physical relations and shell theory
Hooke's law for FG-GPLRC panels with
porous core can be applied as
JSTT 2024, 4 (1), 13-22
Nguyen et al
16
11 12
12 22
66
44
55
0 0 0
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
,
x
y
xy
xz
yz
x
y
xy
xz
yz
QQ
QQ
Q
Q
Q
T
T








=








− 


− 







(17)
where
T
is the temperature change, and
the expressions for reduced stiffnesses
ij
Q
of the
panel can be applied as
( )
( )
( ) ( )
( )
( )
( )
11 22 12
22
66 55 44
,,
11
.
21
ff
f
E z E z z
Q Q Q
zz
Ez
Q Q Q z
= = =
−−
= = = 
+


(18)
The HSDT and the von Karman
nonlinearities are applied to establish the basic
formulas, and the relations between strains at mid-
surface with displacements and rotations are
presented, as
( )
2*
, , , ,
0
2*
0 , , , ,
**
0, , , , , , , ,
1 , 3 , ,
1 , 3
1 , , 3
2
2,
,
x x x x
x
y y y y y
xy x y x y y x x y
x x x x xx x
y y y y
xy y x x y xy
u w w w
v w w w w R y
v u w w w w w w
w

 ++


= + + −
+ + + +




+
= = −
+
,,
, , ,
,,
02
2
2
0,,
,
2
4
, 3 , .
3
x
yy y y
y x xy x y
x x x x
xz xz
yz
yz y y y y
w
w
ww
ww
h



+


+ +


+ +
 
= = − =
+ +




(19)
where geometrical imperfection is denoted
by
*
w
.
The expressions of internal forces and
moments are derived as follows
11 12
12 22
66
11 12 11 12
12 22 12 22
66 66
11 12 11 12
12 22 12 22
66 66
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0
x
y
xy
x
y
xy
x
y
xy
NLL
NLL
NL
MF F A A
MF F A A
FA
M
A A D D
T
A A D D
T
AD
T








=









01
01
0
12
12
1
4
3
4
3
3
0
.
0
0
x
y
xy
x
y
xy
x
y
xy














−

















(20)
The shear force expressions and the higher-
order shear force expressions are expressed by
0
44 44
0
55 55
2
44 44
2
55 55
00
00 .
00
00
x xz
y yz
x xz
y yz
QCB
QCB
SBE
SBE



=




(21)
The stiffness components of panels in Eq.
(20) are determined by
( ) ( )
( )
( ) ( )
2246
2
224
2
, , , 1, , , ,
, 1,2,6
, , 1, , ,
h
p p p p
ij
ij ij ij ij h
h
p p p
ij
ij ij ij h
L F A D Q z z z dz
ij
C B E Q z z dz
−
−
=
=
=
(22)
JSTT 2024, 4 (1), 13-22
Nguyen et al
17
( )
( )
( )
( )
1 2 4
23
11 12
2
, 4,5
,,
1, , ,
h
h
ij
T Q Q z z dz
−
=
=
+
The strain compatibility equation of the
panels taking into account the imperfection can be
established, as
( )
2
0 , 0 , 0 , , , ,
,
* * *
, , , , , ,
2.
x yy xy xy y xx xx yy xy
xx
xx yy xx yy xy xy
w w w
w
w w w w w w
Ry
− + = − +
− − − +
(23)
The stress function
( )
,xy
is introduced as
, , ,
, , .
y xx x yy xy xy
N N N= = = −
(24)
The strain compatibility equation (23) is
rewritten by using Eqs. (24) and (20), as follows
( )
( )
* * * *
22 , 66 12 , 11 ,
,2*
, , , , ,
**
, , , ,
2
2 0,
xxxx xxyy yyyy
xx
xy xx yy xx yy
xy xy xx yy
L L L L
ww w w w w
Ry
w w w w
+ + +
+ − + +
− + =
(25)
where
2*
11 22 12 11 22
* * *
12 12 22 11 66 66
,,
, , 1 ,
L L L L L
L L L L L L
= − =
= − = =
4. Solutions, Boundary conditions, and Euler-
Lagrange equations
The panels are considered with four simply
supported and freely movable edges, as
00,
0,
0, 0,
0,
00, 0,
0, 0,
0,
, 0, 0,
0, 0, 0,
0, 0, 0,
0, 0, 0.
x x x y x xa
xa
xy x x a x a
xa
y y y y
y a y a
xy x
y a y a
ya
N N hP T
N M w
N N M T
Nw
=
=
==
=
==
==
=
= = − = =
= = =
= = = =
= = =
(26)
The deflection, rotations, and imperfect
deflection of the considered panels are modeled by
approximate solutions satisfying the boundary
conditions (26), as
*
sin sin ,
sin sin ,
cos sin ,
sin cos ,
xx
yy
w W x y
w h x y
xy
xy
=
=
=

=
(27)
where
is the imperfection size of the
panels, the number of half waves in the
x
and
y
directions
m
and
n
, with
, m a n b = =
The stress function is chosen including linear
and nonlinear terms, in the form [11-13]
1 2 3
22
00
cos 2 cos 2 sin sin
11
.
22
yx
x y x y
N x N y
= +
+
+
+
(28)
Substituting Eq. (27) and Eq. (28) into the
nonlinear deformation compatibility equation (25),
the like-Galerkin method is used as [11-13]
00
00
00
cos 2 0,
cos 2 0,
sin sin 0,
ba
ba
ba
x dxdy
y dxdy
x y dxdy
=
=
 =



(29)
leads to
( )
( )
1 12 11
2 22 21
33 3
2,
2,
,
V W W h WV
V W W h WV
V W
= + +
= +
=
+
(30)
The Lagrange function is used to obtain the
motion equations, as
.
Total t in ext
U U U U= − +
(31)
The strain energy is presented, as
( )
2
2 0 0
1.
2
xy xy xz xz
hba yz yz x x
in yy
h
xy
U dxdydz
T
−
+


+ +

=
+


− +


(32)
The work done can be determined, as
( )
2*
0 0 , , ,
00
1 2 , ,
00
1
2
1.
2
ba
ext x x x x x
ba
xx yy
U N w w w dxdy
w K w K w w dxdy

= − −




− − +





(33)
with
1
K
(N/m3) and
2
K
(N/m) are the
stiffnesses of foundations.