Buckling and postbuckling of axially-loaded CNT-reinforced composite cylindrical shell surrounded by an elastic medium in thermal environment

s ore sensitive to change of environment temperature. Specifically, decrease in axial load-
deflection curves due to high temperature is more pronounced for FG-V type of CNT distribution. 
Next, the effects of radius-to-thickness ratio on the postbuckling behavior of FG-CNTRC 
cylindrical shells surrounded by Winkler elastic foundation and loaded by axial compression are 
examined in Fig. 5. It is clear that load-deflection equilibrium paths are rapidly reduced when 
ratio is increased. Furthermore, number of full wave in circumferential direction is increased 
as CNTRC cylindrical shell becomes thinner. 
Finally, the effects of surrounding elastic 
foundations on the postbuckling behavior of 
FG-CNTRC cylindrical shells subjected to axial 
compression are given in Fig. 6. It is evident 
that surrounding elastic foundations have 
beneficial influences on the nonlinear stability 
of axially-loaded CNTRC cylindrical shells. 
More specifically, although severity of snap-
through response is not milder, both buckling 
load and postbuckling equilibrium path are 
pronouncedly enhanced due to the embrace of 
elastic foundations, especially Pasternak type 
foundations. 
Fig. 6. Effects of surrounding elastic media on the 
postbuckling behavior of FG-CNTRC cylindrical 
shells under axial compression. 
/R h
/R h
n
Fig. 5. Effects of radius-to-thickness ratio o th
p ckling of CNTRC cylindrical shell sur-
rounded by Winkler foundation
Fig. 4. Effects of CNT distribution and thermal 
environments on the postbuckling of CNTRC 
cylindrical shells. 
Fig. 5. Effects of radius-to-thickness ratio on the 
postbuckling of CNTRC cylindrical shell 
surrounded by Winkler foundation. 
Again, the simultaneous effects of CNT distribution and thermal environments on the 
postbuckling behavior of CNTRC cylindrical shells are considered in Fig. 4. It is realized that FG-V 
shell is more sensitive to change of environment temperature. Specifically, decrease in axial load-
deflection curves due to high temperature is more pronounced for FG-V type of CNT distribution. 
Next, the effects of radius-to-thickness ratio on the postbuckling behavior of FG-CNTRC 
cylindrical shells surrounded by Winkler elastic foundation and loaded by axial compression are 
examined in Fig. 5. It is clear that load-deflection equilibrium paths are rapidly reduced when 
ratio is increased. Furthermore, number of full wave in circumferential direction is increased 
as CNTRC cylindrical shell becomes thinner. 
Finally, the effects of surrounding elastic 
foundations on the postbuckling behavior of 
FG-CNTRC cylindrical shells subjected to axial
co pression are given in Fig. 6. It is evident 
that surrounding elastic foundations have 
beneficial influences on the nonlinear stability 
of axially-loaded CNTRC cylindrical shells. 
More specifically, although severity of snap-
through response is not milder, both buckling 
load and postbuckling equilibrium path are 
pronouncedly enhanced due to the embrace of 
elastic foundations, especially Pastern k type 
foundations. 
Fig. 6. Effects of surrounding elastic media on the 
postbuckling behavior of FG-CNTRC cylindrical 
shells under axial compression. 
/R h
/R h
n
Fig. 6. Effects of surrounding elastic media on
the postbuckling behavior of FG-CNTRC cylin-
drical shells under axial compression
Fin lly, the effects of surrounding
elastic foundations on the postbuckling
behavior of FG-CNTRC cylindrical shells
subjected to axial compression are given
in Fig. 6. It is evident that su rounding
elastic foundations have beneficial influ-
ences on the nonlinear stability of axially-
loaded CNTRC cylindrical shells. More
specifically, although severity of snap-
through response is not milder, both buck-
ling load n postbuckling equilibrium
path are pronouncedly enhanced due to
the embrace of elastic foundations, espe-
cially Pasternak type foundations.
5. CONCLUDING REMARKS
Based on an analytical approach with three-term solution of deflection and Galerkin
method, nonlinear buckling and postbuckling behaviors of simply supported thin CN-
TRC circular cylindrical shells surrounded by elastic media and subjected to uniform
axial compression have been presented. The results show that CNT volume fraction has
very sensitive effects on the buckling load, postbuckling strength and snap-through re-
sponse of CNTRC cylindrical shells. FG-X type shells have the best postbuckling be-
havior in general, and FG-V type shells have relatively high equilibrium paths in small
region of postbuckling response in particular. The study also indicates that elevated tem-
perature has deteriorative effects on buckling resistance and postbuckling load carrying
capabilities of CNTRC cylindrical shells, and these effects are more pronounced in small
44 Hoang Van Tung, Pham Thanh Hieu
region of deflection. As a final remark, although intensity of snap-through instability
is not reduced, surrounding elastic foundations, especially Pasternak type foundations,
have significant and beneficial influences on buckling resistance and postbuckling re-
sponse of axially-loaded CNTRC cylindrical shells.
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.02-2017.11.
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APPENDIX A
The coefficients aj3 (j = 1÷ 4) and ak4 (k = 1÷ 3) in Eqs. (22b) and (22c) are
a13 = a11β4m + a21δ
4
n + a31β
2
mδ
2
n + k1 + k2
(
β2m + δ
2
n
)
+
β2m
a12β4m + a22β2mδ2n + a32δ4n
(
β2m
R
− a42β4m − a52β2mδ2n − a62δ4n
)(
1
R
+ a41δ2n
)
,
a23 =
β2mδ
2
n
a12β4m + a22β2mδ2n + a32δ4n
[
2
β2m
R
+ (a41 − a52) β2mδ2n − a42β4m − a62δ4n
]
+
δ2n
16a12
(
4
R
− 16a42β2m
)
,
a33 =
β4m
16a32
+
δ4n
16a12
,
a43 =
β4mδ
4
n
a12β4m + a22β2mδ2n + a32δ4n
+
β4mδ
4
n
81a12β4m + 9a22β2mδ2n + a32δ4n
,
a14 = 4a11β4m +
3
4
k1 + β2mk2 −
1
4a12R
(
4a42β2m −
1
R
)
,
a24 =
β2mδ
2
n
2 (a12β4m + a22β2mδ2n + a32δ4n)
(
β2m
R
− a42β4m − a52β2mδ2n − a62δ4n
)
+
δ2n
16a12R
,
a34 =
β4mδ
4
n
2 (a12β4m + a22β2mδ2n + a32δ4n)
+
β4mδ
4
n
2 (81a12β4m + 9a22β2mδ2n + a32δ4n)
.
(A1)
The details of coefficients fi1 (i = 1÷ 4), f j2 (j = 1÷ 7) and fk3 (k = 1÷ 7) in the
Eqs. (25)–(27) are defined as follows
f11 =
1
2R2h
(1− ν12ν21) e¯21 + K1E
m
0
2R4h
, f21 =
n2
8R3h
(1− ν12ν21) e¯21 ,
f31 =
ν12e¯21
Rh e¯11
, f41 =
1
Rh e¯11
(e¯11e¯21T − ν12e¯21e¯11T) ,
f12 = a¯11
m4pi4
R4hL
4
R
+ a¯21
n4
R4h
+ a¯31
m2n2pi2
R4hL
2
R
+ K1
Em0
R4h
+ K2
Em0
R2h
(
m2pi2
R2hL
2
R
+
n2
R2h
)
+
(
m2pi2
R3hL
2
R
− a¯42m
4pi4
R4hL
4
R
− a¯52m
2n2pi2
R4hL
2
R
− a¯62 n
4
R4h
)(
a¯41m2n2pi2L2R +m
2pi2RhL2R
a¯12m4pi4 + a¯22m2n2pi2L2R + a¯32n4L
4
R
)
,
48 Hoang Van Tung, Pham Thanh Hieu
f22 =
n2
R3h
e¯21 (1− ν12ν21) ,
f32 =
1
R4h
(
a¯12m4pi4 + a¯22m2n2pi2L2R + a¯32n4L
4
R
) [a¯41m4n4pi4+ m4n2pi4Rh
+m2n2pi2R4hL
2
R
(
m2pi2
R3hL
2
R
− a¯42m
4pi4
R4hL
4
R
− a¯52m
2n2pi2
R4hL
2
R
− a¯62 n
4
R4h
)]
+
n2
16a¯12R2h
(
4
Rh
− 16a¯42m
2pi2
R2hL
2
R
)
+
n2e¯21
2R3h
(1− ν12ν21) ,
f42 =
m4pi4
16a¯32R4hL
4
R
+
n4
16a¯12R4h
+
n4e¯21
8R4h
(1− ν12ν21) ,
f52 =
m4n4pi4
R4h
(
a¯12m4pi4 + a¯22m2n2pi2L2R + a¯32n4L
4
R
)
+
m4n4pi4
R4h
(
81a¯12m4pi4 + 9a¯22m2n2pi2L2R + a¯32n4L
4
R
) ,
f62 =
m2pi2
R2hL
2
R
+ ν12
n2e¯21
R2h e¯11
, f72 =
n2
R2h e¯11
(e¯11e¯21T − ν12e¯21e¯11T) ,
f13 = K1
Em0
R4h
+
e¯21
R2h
(1− ν12ν21) ,
f23 =
e¯21
2R2h
(1− ν12ν21) + 4a¯11m
4pi4
R4hL
4
R
+ K1
3Em
4R4h
+ K2
m2pi2Em
R4hL
2
R
− 1
4Rh a¯12
(
4a¯42
m2pi2
R2hL
2
R
− 1
Rh
)
,
f33 =
n2e¯21
8R3h
(1− ν12ν21) + n
2
16a¯12R3h
+
m2n2pi2L2R
2
(
a¯12m4pi4 + a¯22m2n2pi2L2R + a¯32n4L
4
R
) (m2pi2
R3hL
2
R
− a¯42m
4pi4
R4hL
4
R
− a¯52m
2n2pi2
R4hL
2
R
− a¯62 n
4
R4h
)
,
f43 =
m4n4pi4
2R4h
(
a¯12m4pi4 + a¯22m2n2pi2L2R + a¯32n4L
4
R
)
+
m4n4pi4
2R4h
(
81a¯12m4pi4 + 9a¯22m2n2pi2L2R + a¯32n4L
4
R
) ,
f53 =
m2pi2
R2hL
2
R
, f63 =
ν12e¯21
Rh e¯11
, f73 =
1
Rh e¯11
(e¯11e¯21T − ν12e¯21e¯11T) .
(A2)
Buckling and postbuckling of axially-loaded CNT-reinforced composite cylindrical shell surrounded. . . 49
APPENDIX B
The coefficients fi4, fi5 (i = 1÷ 5) in the Eqs. (29) are defined as
( f14, f24, f34, f44, f54)=
1
f64
( f21 f32 − f11 f42,− f21 f52, f21 f62 − f31 f42, f21 f72 − f41 f42, f21 f12) ,
( f15, f25, f35, f45, f55)=
1
f65
(2 f11 f32− f11 f22,−2 f11 f52, 2 f11 f62− f31 f22, 2 f11 f72− f41 f22, 2 f11 f12) ,
(B1)
in which
f64 = f65 = 2 f11 f42 − f21 f22 , (B2)
The coefficients f j6 (j = 1÷ 8) in the Eq. (30) are defined as
f16 = f33 f35 − f13 f34 − f63 , f26 = f53 − f43 f35 , f36 = f33 f55 − f13 f54 ,
f46 = f13 f14 + f23 − f33 f15 − f43 f55 , f56 = f13 f24 − f33 f25 + f43 f15 ,
f66 = f43 f25 , f76 = f33 f45 − f13 f44 − f73 , f86 = − f43 f45.
(B3)