Shen et al.[3]investigated the buckling and post-buckling behavior of perfect and imperfect stiffened cylindrical shells under combined external pressure and axial compression by using t
Trang 1Nonlinear buckling and post-buckling analysis of eccentrically stiffened
functionally graded circular cylindrical shells under external pressure
Vietnam National University, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Received 7 May 2012
Received in revised form
7 August 2012
Accepted 11 September 2012
Available online 27 November 2012
Keywords:
Functionally graded material
Stiffened cylindrical shells
Post-buckling
Nonlinear
a b s t r a c t
The nonlinear buckling and post-buckling behavior of functionally graded stiffened thin circular cylindrical shells subjected to external pressure are investigated by the analytical approach in this paper The shells are reinforced by eccentrically rings and stringers attached to the inside and material properties of shell and stiffeners are assumed to be continuously graded in the thickness direction Fundamental relations, equilibrium equations are derived based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense Approximate three-terms solution of deflection is more correctly chosen and explicit expression
to finding critical load and post-buckling pressure-deflection curves are given by using the Galerkin’s method The numerical results show the effectiveness of stiffeners in enhancing the stability of shells
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1 Introduction
Stiffened cylindrical shells with the functionally graded
mate-rial (FGM) properties more and more are widely used in modern
engineering In most of these applications, shell is subjected to
compressive loads and it may be buckled Therefore, the research
on nonlinear stability of these structures has received
consider-able attentions by scientists Van der Neut[1] pointed out the
importance of the eccentricity of stiffeners in the buckling of
isotropic cylindrical shells under axial compressive load Baruch
and Singer[2] showed the effect of eccentricity of stiffeners on
the general instability of stiffened cylindrical shells under
hydro-static pressure They concluded that the behavior of eccentricity
effect dependents very strongly on the geometry of the shell Shen
et al.[3]investigated the buckling and post-buckling behavior of
perfect and imperfect stiffened cylindrical shells under combined
external pressure and axial compression by using the boundary
layer theory The singular perturbation technique to determine
the buckling loads and the post-buckling equilibrium paths is
applied in their work Bushnell [4] considered the nonlinear
equilibrium of perfect locally deformed stringer-stiffened panels
under combined in-plane loads Reddy and Starnes [5] studied
the buckling of circumferentially or axially stiffened laminated
cylindrical shells subjected to simply supported end condition by
using the layerwise theory and the smeared stiffener approach
Based on the Donnell equations and the perturbation technique,
the general solution for nonlinear buckling of non-homogeneous axial symmetric ring- and stringer-stiffened cylindrical shells is given by Ji and Yeh[6] The post-buckling analysis of stiffened braided thin shells subjected to combined loading of external pressure and axial compression by perturbation method is reported by Zeng and Wu [7] The post-buckling analysis of pressure-loaded functionally graded cylindrical shells without stiffeners based on the classical shell theory with von Karman– Donnell-type of kinetic nonlinearity is presented by Shen [8] Jiang et al.[9]studied the buckling of stiffened circular cylindrical panels subjected to axial uniform compressive load by using the differential quadrature element method The cylindrical panel and the stiffeners are treated separately there Li and Shen[10] presented the investigation on a post-buckling analysis of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environment They used the higher order shear deformation shell theory and the singular perturbation technique to determine interactive buckling loads and post-buckling equilibrium paths Huang and Han[11] presented the research on nonlinear post-buckling of un-stiffened FGM cylindrical shells under uniform radial pressure by using the nonlinear large deflection theory of cylindrical shell In that work, the nonlinear buckling shape observed in experiment is taken into account Sadeghifar et al [12] investigated the buckling of stringer-stiffened laminated cylindrical shells with non-uniform eccentricity based on the Love’s first-order shear deformation theory The critical loads are calculated using the Rayleigh–Ritz procedure Stamatelos et al.[13]presented the results on the local buckling and post-buckling behavior of isotropic and orthotropic
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n
Corresponding author.
E-mail address: lekhahoa@gmail.com (L.K Hoa).
Trang 2stiffened panels based on the classical lamination plate theory and
two-dimensional Ritz displacement function for arbitrary edge
sup-ports Recently, Najafizadeh et al.[14]with the stability equations
given in terms of displacement investigated the mechanical buckling
behavior of functionally graded stiffened cylindrical shells reinforced
by rings and stringer subjected to axial compressive loading The
stiffeners and skin, in their work, are assumed to be made of
functionally graded materials and its properties vary continuously
through the thickness direction Bich et al [15] investigated the
nonlinear static post-buckling of eccentrically stiffened functionally
graded plates and shallow shells with homogeneous stiffeners
The review of the literature signifies that there is no work on
the analytical solution for externally pressurized stiffened FGM
cylindrical shells In this paper, the nonlinear buckling and
post-buckling behaviors of eccentrically stiffened functionally graded
thin circular cylindrical shells under uniform external pressure
are investigated by approximate three-terms solution of
deflec-tion The material properties of shell and stiffeners are assumed to
be continuously graded in the thickness direction The expression
of deflection including the linear buckling shape sin m px=L
sin ny=R
and the nonlinear buckling shape sin2mpx=L
are more correctly chosen The resulting equations are solved by
the Galerkin’s method to obtain closed-form expressions to
determine critical buckling loads and nonlinear post-buckling
loads–deflection curves The influences of various parameters
such as dimensional parameters, buckling modes, volume fraction
index of materials and number of stiffeners on the stability of
shell are considered in detail
2 Eccentrically stiffened functionally graded cylindrical
shells
Consider a thin circular cylindrical shell with mean radius R,
thickness h and length L subjected to uniform radial load of
intensity q Assume that two butt-ends of shell are only deformed
in their planes and they still are circular[11] The middle surface
of the shells is referred to the coordinates x, y, z as shown in
Fig 1a Further, assume that the shell is stiffened by closely
spaced circular rings and longitudinal stringers attached to inside
of the shell skin, and the stiffeners and skin are made of
functionally graded materials varying continuously through the thickness direction of the shell with the power law as follows[14]
Esh¼EmþEcm
2z þ h 2h
, Ecm¼EcEm, k Z0, h
2rzrh
2: ð1Þ
Es¼EcþEmc
2zh 2hs
k 2
, Emc¼EmEc, k2Z0, h
2rzrh
2þhs ð2Þ
Er¼EcþEmc
2zh 2hr
k 3
, k3Z0, h
2rzrh
nsh¼ns¼nr¼n¼const, where k, k2and k3are volume fractions indexes of shell, stringer and ring, respectively and subscripts c, m, sh, s and r denote ceramic, metal, shell, longitudinal stringers and circular ring, respectively It is evident that, from Eqs (1)–(3), a continuity between the shell and stiffeners is satisfied Note that the thickness of the stringer and the ring are respectively denoted
by hsand hr, andEc, Emare Young’s modulus of the ceramic and metal, respectively The coefficientnis Poison’s ratio
According to the non-linear strain–displacement relations of cylindrical shells, the mid-surface strain components are[19]
e0
x¼u;xþ1
2ðw,xÞ
2, e0
y¼v;yw
Rþ
1
2ðw,yÞ
2, g0
xy¼u;yþv;xþw;xw;y, ð4Þ
in which u ¼ u x,yð Þ, v ¼ v x,yð Þ, and w ¼ w x,yð Þ are the displace-ments of the middle surface points along x, y and z axes, respectively
The strain components across the shell thickness at a distance
z from the mid-plane are in the form
ex¼e0
xþzkx, ey¼e0þzky, gxy¼g0
xyþ2zkxy,
kx¼ w;xx, ky¼ w;yy, kxy¼ w;xy, ð5Þ where kx, kyand kxyare the change of curvatures and twist of shell, respectively
Using Eq.(5), the compatible equation is written as
e0 x,yyþe0 y,xxg0 xy,xy¼ 1
Rw;xxþw
2
R
z
x y
L
hr
br
dr
h
er
h
bs
hs
es
ds
Trang 3The Hooke’s stress–strain relations are applied for shell
ssh
x ¼ Esh
1n2exþney
,
ssh
y ¼ Esh
1n2eyþnex
,
ssh
xy¼ Esh
and for stiffeners
ss
¼Esex,
sr
Taking into account the contribution of stiffeners by the
smeared stiffeners technique and omitting the twist of stiffeners
and integrating the stress–strain equations and their moments
through the thickness of the shell, the expressions for force and
moment resultants of an eccentrically stiffened FGM cylindrical
shell are expressed by[14,19]
Nx¼C11e0
xþC12e0þC14kxþC15ky,
Ny¼C12e0
xþC22e0þC24kxþC25ky,
Nxy¼C33g0
Mx¼C14e0
xþC24e0þC44kxþC45ky,
My¼C15e0
xþC25e0þC45kxþC55ky,
Mxy¼C63g0
where the stiffness parameter Cijis given by
C11¼ E1
1n2þE1sbs
ds , C12¼ nE1
1n2,
C14¼ E2
1n2þE2sbs
ds
, C15¼ nE2
1n2,
C22¼ E1
1n2þE1rbr
dr
, C24¼ nE2
1n2,
C25¼ E2
1n2þE2rbr
dr
, C33¼ E1
2 1 þð nÞ,
C36¼ E2
1 þn, C44¼
E3
1n2þE3sbs
ds
,
C45¼ nE3
1n2, C55¼ E3
1n2þE3rbr
dr ,
C63¼ E2
2 1 þð nÞ, C66¼ E3
in which
E1¼
Z h=2
h=2
EshðzÞdz ¼ EmhþEcmh
k þ 1,
E2¼
Z h=2
h=2
zEshðzÞdz ¼ kEcmh
2
2 k þ1ð Þðkþ 2Þ,
E3¼
Z h=2
h=2
z2EshðzÞdz ¼Emh
3
12 þEcmh
4 k þ 1ð Þ
1
k þ 2þ
1
k þ3
,
E1s¼
Zh=2 þ h s
h=2
EsðzÞdz ¼ EchsþEmc
hs
k2þ1,
E2s¼
Zh=2 þ h s
h=2
zEsðzÞdz ¼Ec
2hhs
hs
hþ1
þEmchsh 1
k2þ2
hs
hþ
1 2k2þ2
,
E3s¼
Zh=2 þ h s
h=2
z2EsðzÞdz ¼Ec
3h
3 s
3 4
h2
h2sþ
3 2
h
hs
þ1
!
þEmch3s 1
k2þ3þ
1
k2þ2
h
hs
4 kð 2þ1Þ
h2
h2s
,
E1r¼
Z h=2 þ h r
h=2
ErðzÞdz ¼ EchrþEmc
hr
k3þ1,
E2r¼
Z h=2 þ h r
h=2
zErðzÞdz ¼Ec
2hhr
hr
hþ1
þEmchrh 1
k3þ2
hr
h þ
1 2k3þ2
,
E3r¼
Z h=2 þ h r
h=2
z2ErðzÞdz ¼Ec
3h
3 r
3 4
h2
h2r þ
3 2
h
hr
þ1
!
þEmch3r 1
k3þ3þ
1
k3þ2
h
hr
4 kð 3þ1Þ
h2
h2r
where bsand br denote widths of stiffeners, respectively Also,
ds and dr are the distances between two stringers and rings, respectively, and the eccentricities es and er represent the dis-tance from the shell middle surface to the centroid of the stiffeners cross section (Fig 1b)
For using later, the reverse relations are deduced from
Eq.(8a)as
e0
x¼Cn
22NxCn
12NyþCn
14kxþCn
15ky,
e0¼ Cn
12NxþCn
11NyþCn
24kxþCn
25ky,
g0
xy¼Cn
33NxyCn
where
D¼C22C11C2
12, Cn
22¼C22=D, Cn
12¼C12=D,
Cn
14¼ðC12C24C22C14Þ=D, Cn
15¼ðC12C25C22C15Þ=D,
Cn
11¼C11=D, Cn
24¼ðC12C14C11C24Þ=D,
Cn
25¼ðC12C15C11C25Þ=D, Cn
33¼ 1
C33
, Cn
36¼C36
C33
Substituting Eq.(11)into Eq.(8b)yields
Mx¼Dn
14NxþDn
24NyþDn
44kxþDn
45ky,
My¼Dn
15NxþDn
25NyþDn
54kxþDn
55ky,
Mxy¼Dn
63NxyþDn
where
Dn
14¼C14Cn
22C24Cn
12, Dn
44¼C44þC24Cn
24þC14Cn
14,
Dn
24¼C24Cn
11C14Cn
12, Dn
45¼C14Cn
15þC24Cn
25þC45,
Dn
15¼C15Cn
22C25Cn
12, Dn
54¼C15Cn
14þC25Cn
24þC45,
Dn
25¼C25Cn
11C15Cn
12, Dn
55¼C15Cn
15þC25Cn
25þC55,
Dn
63¼C63Cn
33, Dn
66¼C66C63Cn
The equilibrium equations of cylindrical shell based on the classical shell theory are given by[14,17]
Nx,xþNxy,y¼0,
Nxy,xþNy,y¼0,
Mx,xxþ2Mxy,xyþMy,yyþNy
R þNxw;xxþ2Nxyw;xyþNyw;yyþq ¼ 0
ð15Þ The first two of Eq.(15)are identically satisfied by introducing
a stress functionjðx,yÞas
Nx¼j;yy, Ny¼j;xx, Nxy¼ j;xy: ð16Þ Introduction of Eqs.(13) and (16)into the third of Eq.(15), and taking into account Eq.(5), gives the following equation
a11w;xxxxþa12w;xxyyþa13w;yyyyþa14j;xxxxþa15j;xxyy
þa16j;yyyyþ1
Rj;xxþj;yyw;xxþj;xxw;yy2j;xyw;xyþq ¼ 0 ð17Þ where
a11¼ Dn
44, a12¼ Dn
45þ2Dn
66þDn
54
, a13¼ Dn
55,
a14¼Dn
24, a15¼ Dn
142Dn
63þDn
25
, a16¼Dn
Eq (17)includes two dependent unknown functions wandj and to find a second equation relating to these two functions
Trang 4the geometrical compatibility Eq (6) is used For this aim,
substituting Eq.(11)into Eq.(6), gives
b11j;xxxxþb12j;xxyyþb13j;yyyyþb14w;xxxxþb15w;xxyyþb16w;yyyy
w2;xyþw;xxw;yyþ1
where
b11¼Cn
11, b12¼Cn
332Cn
12, b13¼Cn
22,
b14¼ Cn
24, b15¼ Cn
14þCn
25þCn
36
, b16¼ Cn
Eqs.(17) and (19)are nonlinear governing equations used to
investigate the nonlinear stability of eccentrically stiffened FGM
cylindrical shells under uniform radial loads
3 Buckling analysis
Assume that the cylindrical shell is simply supported at the
edges x ¼ 0 and x ¼ L The deflection of radial loaded shell can be
expressed by[11,16]
w ¼ w x,yð Þ ¼f0þf1sinaxsinby þf2sin2ax, ð21Þ
in whicha¼mp=L, b¼n=R and m, n are the half waves numbers
along x-axis and waves numbers along y-axis, respectively
The first term of w in Eq.(21)represents the uniform deflection
of points belonging to two butt-ends x¼0 and x ¼L, the
second term-a linear buckling shape, and the third-a nonlinear
buckling shape
As can be seen that the simply supported boundary condition
at x¼ 0 and x¼L is fulfilled on the average sense
Substituting Eq.(21)into Eq.(19)yields
b11j;xxxxþb12j;xxyyþb13j;yyyy¼B01cos 2ax
þB02cos 2by þ B03sinaxsinby þB04sin 3axsinby ð22Þ
where
B01¼ 8b14a42
Ra2
f2þ1
2a2b2f21
, B02¼1
2a2b2f21,
B03¼ b14a4
þb15a2b2þb16b4a2
R
f1þf1f2a2b2
,
B04¼f1f2a2
The general solution of this equation is given by
j¼B1cos 2ax þB2cos 2by þ B3sinaxsinby
þB4sin 3axsinby1
wheres0y is the negative average circumferential stress and
B1¼a1f2þa2f21, B2¼a3f21, B3¼a4f1f2þa5f1, B4¼a6f1f2,
ð25Þ
in which
a1¼4b14a4a2=R
=8b11a4
, a2¼a2b2=32b11a4
,
a3¼a2b2=32b13b4
, a4¼ a2b2= b11a4
þb12a2b2þb13b4
,
a5¼ b14a4
þb15a2b2þb16b4a2=R
= b11a4
þb12a2b2þb13b4
,
a6¼a2b2=81b11a4þ9b12a2b2þb13b4
ð26Þ
In order to establish a load–deflection curve, first of all,
introducing w andjinto the left side of Eq.(17), then applying
Galerkin’s method in the ranges 0ryr2pR and 0rxrL, gives
s0y¼Rq
f21¼ D01þD04f22þD05f21
2s0yhb2
D06f2þD07f21þD08f21f2¼0, ð29Þ where
D01¼ 1
2 a11a4
þa12a2b2þa13b4 h
þa5 a14a4
þa15a2
b2þa16b4a2
R
,
D03¼a2
b2ða2þa3Þ, D04¼ a2b2
2 ða4a6Þ,
D05¼ 1
2 a4 a14a4þa15a2b2þa16b4a2
R
þa2b2a52a2b2a1
,
D06¼8a2 4a11a28a2a14a1þ2a1
R
,
D07¼4 a5b216a2a14a2þ4a2
R
a2
,
In addition to three Eqs.(27)–(29), the cylindrical shell must also satisfy the circumferential closed condition[11,16] as
Z2 p R
0
Z L
0
v;ydxdy ¼
Z 2 p R
0
Z L
0
e0þw
R
1
2ðw,yÞ
2
dxdy ¼ 0
Using Eqs.(11), (16) and (24), this integral becomes
8Cn
11s0yhþ4
R 2f0þf2
Substituting Eq.(28)into Eq.(29)withs0y¼Rq=h, leads to
q ¼ 2
Rb2 D01þD04f
2
þD05f2 D03D06f2
D07þD08f2
ð32Þ
Expression(32)is used to determine the critical loads and to analyze the post-buckling load–deflection curves of nonlinear buckling shape of stiffened FGM cylindrical shells
If f2¼0, i.e the nonlinear buckling shape is ignored, Eq.(32) becomes
q ¼2D01
Rb2 ¼
1
Rb2ha11a4þa12a2b2þa13b4
þ
b14a4þb15a2b2þb16b4a2=R
b11a4þb12a2b2þb13b4
na14a4þa15a2b2þa16b4a2=R i
ð33Þ
Eq (33) is used to find critical loads in case linear buckling shape
From Eq.(21), it is obvious that the maximal deflection of the shells
locates at x ¼ iL=ð2mÞ, y ¼ jpR=ð2nÞ, where i, j are odd integer numbers
Solving f1 and f0from Eqs (28) and (31)with respect to f2, then substituting them into Eq.(34), we obtain
Wmax¼f2
2þC n
11Rs0yhRb
2
8D03
D01þD04f2þD05f21
2s0yhb2
þ b2s0yh2D012D04f222D05f2
2D03
!1=2
ð35Þ
Combining Eq (32) with Eq (35), the effects of inhomoge-neous and dimensional parameters on the post-buckling load-maximal deflection curves of shells can be analyzed
Trang 54 Numerical results
4.1 Validation of the present approach
To verify the present study, an isotropic cylindrical shell under
external pressure q is considered with the following geometric
and material properties as[2,5,18]
E ¼ 30 106Psi,n¼0:3,
h ¼ 1 in, R ¼ 82:1693 in, L ¼ 372:9745 in,
es¼er¼1:653 in, hs¼hr¼2 e rh=2
,
ds¼2pR=ns¼2pR=516, dr¼L=nr¼L=373,
bs¼0:1471dsh=hs, br¼0:1471drh=hr,
where ns, nr are the number of stringer and ring of shell,
respectively
The critical buckling loads calculated for stiffened and
un-stiffened shells are listed inTable 1to compare with the results
given by Shen[18]using asymptotic perturbation technique and
with the results reported by Reddy[5] based on Donnell’s shell
theory As can be seen, good agreement is obtained in this
comparison
Fig 2a and b shows the comparisons of the present
post-buckling paths with the results which was also analyzed by
Huang and Han[11]using the nonlinear large deflection theory
and the Ritz energy method for un-stiffened FGM cylindrical
shells under external pressure
It is clear that the present results coincide with the ones of the work[11]
In each subsection below, to illustrate the present approach for nonlinear buckling and post-buckling analysis of stiffened FGM cylindrical shells under external pressure, consider a ceramic-metal shell consisting of zirconia and aluminum with the follow-ing properties[14]Ec¼151 GPa, Em¼70 GPa and Poisson’s ratio
nis assumed to be 0.3 Also assume that k2¼k3¼1=k for all of the examples considered hereafter
4.2 Effects of buckling modes
Assume that the material and geometrical parameters of shell are taken by
k¼k2¼k3¼1, h ¼ 0:305 103m, R ¼ 60:643 103m, L ¼ 387:35 103m, hs¼0:076 103m, bs¼21:155 103 m,
hr¼0:127 103m, br¼1:27 103m
The number of stringers as well as rings are equal to 15 Using
Eq (32)and programming the Matlab software, the results are given inTable 2andFig 3
Table 2 shows that the buckling load is minimum qcr¼ 25:1349 KPa corresponding to the buckling mode (m, n)¼(1, 4) Also, the same result qcris represented onFig 3 As can be seen the lowest point of the envelope curves is regarded as the nonlinear buckling mode (m, n)¼(1, 4) Further, during the first stage, the buckling load reduces when the ratio f2/h increases At the second stage, when the ratio f2/h reaches a determined value, the buckling loads increases with the increase of this ratio
4.3 Effects of inhomogeneous and geometric parameters
Based on Eqs (32) and (35), with the database given in the Subsection 4.2, the effects of the volume fraction indexes
k, k2, k3, of the radius-to-thickness ratios R=h and of the length-to-radius ratios L=R on qWmax=h
relation curves of stiffened FG cylindrical shell are considered
Fig 4 shows the post-buckling equilibrium paths under various values of k ¼ 0 , 0:2, 0:6, 1 , 5, 10 þ1, and R ¼ L=2,
h ¼ R=500 As expected, the critical load qcr decreases with the increase of k This property corresponds to the real property of material, because the higher value of k corresponds to a metal-richer shell which usually has less stiffness than a ceramic-richer one
Fig 5plots the qW max=h
post-buckling curves versus R=h which is chosen to be 200, 300, 400, 500 It is observed that the
Table 1
Comparisons of buckling load q for isotropic cylindrical shell under external
pressure.
Singer [2]
Reddy and Starnes [5]
Shen [18] Present
calculate by
Eq (33)
103.3271 (1,4) Stringer stiffened
(inside)
Ring stiffened
(inside)
Orthogonal
stiffened
(inside)
a
The numbers in the parenthesis denote the buckling modes (m, n).
0
50
100
150
200
250
300
Wmax/h
present Ref [11]
k=1, L/R=2, R/h=200, m=1, n=7
ν=0.3
0 10 20 30 40 50 60 70
Wmax/h
present Ref [11]
k=1, R/h=500, L/R=1, m=1, n=13
ν=0.3
Trang 6buckling load q decreases markedly with the increase of R=h ratio,
i.e the more the shell is thin the more the value of critical load
is small
Effects of ratio L=R on the post-buckling curves of shell are
represented onFig 6
As can be observed, the capacity of mechanical load q bearing
of the FGM shells is considerably reduced with the increase of L=R
Besides,Figs 5 and 6also show the wave numbers n increases
with the increase of R=h, but decrease with the increase of L=R
Thus, both R=handL=R considerably affect the nonlinear buckling
mode of shells
4.4 Comparing the critical buckling loads of stiffened and un-stiffened FGM cylindrical shells
The effects of stiffeners on the critical buckling loads and post-buckling behavior of FGM cylindrical shells are considered in this subsection
Using the database in Subsection 4.2, Table 3 and Fig 7 compare the critical buckling loads qcr of stiffened FGM shell with the ones of un-stiffened FGM shell when m¼1, and
k, k2¼ k3¼1=kandn vary As can be seen, the critical buckling loads for the FGM stiffened cylindrical shells are generally upper than the corresponding values for the FGM un-stiffened
Table 2
Effects of buckling modes on critical load q cr (KPa).
20
30
40
50
60
70
80
90
100
110
f2/h
n=8
n=7 n=6 n=5 n=4
n=3
qcr=25.1349 KPa
Fig 3 Effects of buckling modes n (m¼ 1) on (qf 2 =h) curves.
0
2
4
6
8
10
12
14
16
18
20
Wmax/h
1: k=0, 2: k =0.2, 3: k= 0.6,
4: k =1, 5: k =5, 6: k =10,
7: k =∞
(m, n) = (1, 9)
2
1
3
4
6
7
5
Fig 4 Effects of k on (qW max =h) curves.
0 20 40 60 80 100 120
Wmax/h
L/R=2, m=1, k=k2=k3=1,
hs=0.076mm, bs=21.155mm,
hr=0.127mm, br= 1.27mm
1: R/h=200, n=7
2: R/h=300, n=8
3: R/h=400, n=9
4: R/h=500, n=9
1
2
3
4
Fig 5 Effects of R/h ratio on post-buckling curves of shell.
0 5 10 15 20 25
Wmax/h
k=k2=k3=1, m=1, R/h=500,
hs=0.076mm, bs=21.155mm,
hr=0.127mm, br= 1.27mm
1
2
3
4
1: L/R=1, n=13
2: L/R=2, n=9
3: L/R=3, n=7
4: L/R=4, n=7
Fig 6 Effects of L/R ratio on post-buckling curves of shell.
Trang 7cylindrical shells The prime reason is that the presence of stiffeners makes the shells to become stiffer
The same results are presented inTable 4andFig 8for FGM un-stiffened and stiffened shells Herein, the thickness of shell varies from 0.305 mm to 0.762, the wave numbers n are chosen to
be 2, 3, 4, 5 and m¼1, k ¼ k2¼k3¼1 Also as can be seen that the critical buckling loads are increased by increasing the thickness
of shell
Tables 3 and 4show that the critical loads of FGM un-stiffened shells are the smallest Critical loads of stringer stiffened shell are smaller than ring stiffened shell Finally, the critical loads of FGM orthogonal stiffened shell are the greatest
4.5 Effects of number of stiffeners
To investigate the effects of number of stiffeners, the database given in theSubsection 4.2is used here.Fig 9andTable 5illustrate the effects of number of stiffener (ns¼nr¼10, 20, 30, 40 and 50)
on buckling loads and qWmax=h curves As can be seen, these curves become lower when the number of stiffeners decreases and buckling loads increases when the number of stiffeners increases That means the percentage increase in the buckling load rises continuously with the increment of the number of stiffeners This increase is about 19% for orthogonal stiffened shell, in comparison
ns¼nr¼10 with ns¼nr¼50 (inTable 5)
5 Concluding remarks
In this paper, the thin FGM cylindrical shells reinforced by eccentrically rings and stringers attached to the inside and material properties of shell and stiffeners varying continuously graded in the thickness direction are considered
0
10
20
30
40
50
60
70
Wmax/h
Fig 7 Effects of k on (qW max =h) curves.
Table 4
Comparing the critical buckling loads of stiffened and un-stiffened FGM shells vs h.
(4) 0.533 (3) 0.610e (3)
0.686 (3) 0.762 (3)
Un-stiffened shell 22.9867 85.2212 119.6377 162.3639 214.8787
Stringer stiffened shell 23.3630 86.6765 121.0893 163.8110 216.3212
Ring stiffened shell 24.7623 87.5988 122.5735 165.9090 219.0910
Orthogonal stiffened
shell
25.1349 89.0446 124.0163 167.3479 220.5256
0
20
40
60
80
100
120
140
160
180
200
Wmax/h
Stiffened
Unstiffened
Stiffened:
hs=0.076mm; bs=21.155mm
hr=0.127mm; br=1.27mm
1: h=0.305mm, n=4
2: h=0.381mm, n=3
3: h=0.457mm, n=3
4: h=0.533mm, n=3 L=387.35mm;
R=60.643mm;
k=k2=k3=1
1
2
3
4
Fig 8 Effects of thickness h on (qW =h) curves.
20 25 30 35 40 45 50
Wmax/h
1
2
3
4
5
1: ns=nr=10 2: ns=nr=20 3: ns=nr=30 4: ns=nr=40 5: ns=nr=50
k=k 2 =k 3=1,
m=1, n=4
Orthogonal Stiffened shell
hs=0.076mm, bs=21.155mm
hr=0.127mm, br=1.27mm
L=387.35mm R=60.643mm h=0.305mm
Fig 9 Effects of number of stiffeners on (qW max =h) curves.
Table 5 Effects of number of stiffeners on buckling loads q (KPa), (m, n) ¼ (1, 4).
Stringer stiffened shell 23.2418 23.4803 23.7038 23.9134 24.1104 Ring stiffened shell 24.1355 25.3813 26.5962 27.6673 ( n ) 28.3651 ( n )
Orthogonal stiffened shell
24.3889 25.8684 27.2992 28.6843 30.0264
where ( n
) indicates m ¼1, n¼3.
Table 3
Comparing the critical buckling loads of stiffened and un-stiffened FGM shells vs k.
Stringer stiffened 33.2200 26.1480 23.3630 15.4000
Orthogonal stiffened 36.3245 28.2585 25.1349 16.8392
Trang 8Analytical approach to investigate the nonlinear buckling and
post-buckling behavior of eccentrically stiffened FGM cylindrical
shells under external pressure based on the smeared stiffeners
technique and the classical shell theory with geometrical
non-linearity in von Karman sense is studied
Approximate three-term solution of deflection including the
linear and nonlinear buckling shape is more correctly chosen and
the close-form expressions to determine critical buckling loads
and nonlinear post-buckling load–deflection curves are obtained
by using Galerkin’ method
Effects of various parameters such as dimensional parameters,
buckling modes, volume fraction index of materials and number
of stiffeners on the stability of shell are considered in detail
Results show the effectiveness of stiffeners in enhancing the
stability of shells
Major purpose of this study is to analyze the global buckling
and post-buckling behavior of FGM stiffened cylindrical shells
For local buckling analysis, the approach of Stamatelos et al.[13]
may be used
Acknowledgments
The research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under Grant
number 107.01-2012.02 The authors are grateful for this financial
support
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