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Trang 1Research on nonlinear torsional buckling and post-buckling of
eccentrically stiffened functionally graded thin circular cylindrical shells
Vietnam National University, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Received 6 November 2012
Received in revised form 28 February 2013
Accepted 10 March 2013
Available online 22 March 2013
Keywords:
Functionally graded material
A Discontinuous reinforcement
B Buckling
B Elasticity
C Analytical modelling
a b s t r a c t
This paper is presented to solve the nonlinear buckling and post-buckling problem of functionally graded stiffened thin circular cylindrical shells only under torsion by the analytical approach The shells are rein-forced by rings and stringers attached to their inside and the material properties of shell and the stiffen-ers are assumed to be continuously graded in the thickness direction Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense are derived Approximate three-term solution of deflection is chosen more correctly and the explicit expression to finding critical load and post-buckling torsional load–deflection curves are given The effects of various parameters and the effectiveness of stiffeners on the stability of shell are shown
Ó 2013 Elsevier Ltd All rights reserved
1 Introduction
Cylindrical shell is one of the important structures used widely
in engineering applications When shells are subjected to
compres-sive loads, they may be buckled As a result, an investigation of
buckling and post-buckling of these shells is a necessary
fundamen-tal problem and has been attracted attention of many researchers
Concerning the buckling problem of thin-walled tubes under
tor-sion, pioneer approximate solutions were obtained by Donnell
[1] The post-buckling of cylinders under torsion and axial
compres-sion was studied by Loo[2] Nash[3]presented the approximate
solutions on the buckling of initially imperfect torsion-loaded
cylin-drical shells by applying the Ritz method Yamaki[4]obtained the
approximate solutions on the post-buckling behavior of shells
un-der torsion that the results were found to be in reasonable
agree-ment with experiagree-ment ones Shaw et al.[5] solved the problem
on the imperfect laminated cylindrical shells in torsion and axial
compression Their analysis were based on Donnell-type nonlinear
kinematic relations and laminated cylindrical shell theory Lennon
and Das[6]analyzed the torsional buckling behavior of stiffened
cylinders under combined loading The effects of stiffeners on
post-buckling behavior in torsion was investigated in that paper
Mao and Lu[7] studied an elastic plastic buckling of cylindrical
shells under torsion with a deep thick-shell model in which the
ef-fect of the factor (1 + z/R) and the efef-fect of the mechanical boundary
conditions are considered Using singular perturbation technique,
Zhang and Han[8]investigated the buckling and post-buckling of imperfect cylindrical shells subjected to torsion based on the von Karman–Donnell-type nonlinear differential equations By the above same method, Shen and Xiang[9] analyzed the buckling and post-buckling of an anisotropic laminated cylindrical shells un-der torsion or unun-der combined axial compression and torsion based
on the classical shell theory with von Karman–Donnell-type of kinematic nonlinearity the extension–twist, extension–flexual and flexual–twist couplings are considered Paimushin [10] reported details of local and global buckling of cylindrical shells der combined loading He showed the existence of previously un-known torsional, flexural, and torsional–flexural buckling modes for cylindrical shells which were subjected to simultaneous com-pression and external pressure Takano[11]studied the buckling
of thin and moderately thick anisotropic cylinders under combined torsion and axial compression His investigation showed that the buckling loads of a cylindrical shell are affected not only by anisot-ropy and transverse shear stiffness but also by shell length Some significant results on the mechanics of composite shells and curved beams have been obtained Fraternali and Reddy[12] presented the penalty model for the analysis of laminated compos-ite shells Their method offers the possibility to easily obtain accu-rate interlaminar stresses By the same method, a one-dimensional theory and a finite element model for the stress analysis of curved composite beams are investigated by Ascione and Fraternali[13], Fraternali and Bilotti[14]and Fraternali and Feo[15]
A new class of composite material known as functionally graded materials (FGMs) has been received considerable attention re-cently Shen[16], based on the higher order shear deformation
1359-8368/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved.
⇑Corresponding author Tel.: +84 989358315.
E-mail address: lekhahoa@gmail.com (L.K Hoa).
Contents lists available atSciVerse ScienceDirect
Composites: Part B
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p o s i t e s b
Trang 2theory, obtained the results of stability problem of torsion-loaded
functionally graded shells in thermal environments A singular
per-turbation technique is employed to determine buckling shear load
and post-buckling equilibrium paths Huang and Han[17]studied
the nonlinear buckling of torsion-loaded FGM un-stiffened
cylin-drical shells by using the nonlinear large deflection shell theory
and Ritz method The nonlinear buckling shape observed in
exper-iment is taken into account in their work Sofiyev and Kuruoglu
[18] investigated the torsional vibration and buckling of
un-stiffened cylindrical shell with functionally graded coatings
sur-rounded by an elastic medium The modified Donnell type dynamic
stability and compatibility equations with linear
strain–displace-ment relation of three-layered of cylindrical shell and Galerkin
method are used to determine the expressions for torsional
buck-ling load and torsional frequency parameter Li and Wang [19]
investigated an elastic stability of a simply supported FGM
sand-wich circular cylindrical shell under torsion loading by
semi-ana-lytical method The governing equations for static buckling of the
structure in terms of displacements were formulated using the
Flu-gge thin shell theory in which the strain–displacement relation is
linear Bich et al.[20]presented the buckling of un-stiffened FGM
conical panels subjected to mechanical loads by using the
equilib-rium and linear stability equations in terms of displacement
components Galerkin method was applied to obtain closed-form
relations of bifurcation type buckling loads
For dynamic analysis of FGM shells, many studies have been
focused on the characters of vibration and behavior of buckling
of un-stiffened shells Sofiyev and Schnack[21]studied the
stabil-ity of FGM cylindrical shells under linearly increasing dynamic
tor-sional loading The modified Donnell type dynamic stability
equation and Galerkin method were used However, the
geometri-cal relation is linear and the approximate solution was chosen by
one-term Bich et al [22] presented an analytical approach to
investigate the nonlinear static and dynamic unsymmetrical
responses of un-stiffened FGM shallow spherical shells under
external pressure incorporating the effect of temperature The
clas-sical shell theory is used and Galerkin method is applied Bich and
Nguyen[23]studied the nonlinear vibration of FGM un-stiffened
cylindrical shells based on improved Donnell equations ignoring
the shallowness of shell Their results shown that the Volmir’s
assumption can be used for nonlinear dynamic analysis with an
acceptable accuracy
For stiffened cylindrical shell, the stability problem is also very
interest subject Van der Neut[24]pointed out the importance of
the eccentricity of stiffeners in the buckling of isotropic cylindrical
shells under axial compressive load Barush and Singer [25]
showed the effect of eccentricity of stiffeners on the general
insta-bility of stiffened cylindrical shells under hydrostatic pressure
They concluded that the behavior of eccentricity effect dependents
very strongly on the geometry of the shell The researches on this
problem have been continued for many year to obtain more precise
and reasonable solution
Recently, Najafizadeh et al.[26]with the linear stability
equa-tions in terms of displacements studied buckling of FGM
cylindri-cal shell reinforced by rings and stringers under axial compression
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.[27]presented an
ana-lytical approach to investigated the nonlinear post-buckling of
eccentrically stiffened FGM plates and shallow shells based on
the classical shell theory in which the stiffeners are assumed to
be homogeneous Dung and Hoa[28]obtained the results on the
nonlinear buckling and post-buckling analysis of eccentrically
stiff-ened FGM circular cylindrical shells under external pressure The
material properties of shell and stiffeners are assumed to be
continuously graded in the thickness direction Galerkin method
was used to obtain closed-form expressions to determine critical buckling loads Bich et al.[29]obtained the results on the nonlin-ear dynamic analysis of eccentrically stiffened FGM cylindrical panels The governing equations of motion were derived by using the smeared stiffeners technique and the classical shell theory with von Karman geometrical nonlinearity The same authors [30] investigated the nonlinear vibration dynamic buckling of eccentri-cally stiffened imperfect FGM doubly curved thin shallow shells based on the classical shell theory The nonlinear critical dynamic buckling load is found according to the Budiansky–Roth criterion The review of the literature signifies that there are very little re-searches on nonlinear stability of eccentrically stiffened FGM shells and there is no work on the analytical solution for torsion-loaded stiffened FGM cylindrical shells Following the idea of works [26,28], in this paper the nonlinear buckling and post-buckling behaviors of eccentrically stiffened functionally graded thin circu-lar cylindrical shells subjected to uniform torsional load are inves-tigated The present novelty is that the shells under torsional load are reinforced by rings and stringers attached to their inside and the material properties of shell and the stiffeners continuously are graded in the thickness direction The theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense, are derived In addition, an approximate three-term solution of deflection including the linear buckling shape sin (mpx/L) sin -n(y kx)/R and the nonlinear buckling shape sin2(mpx/L) are more correctly chosen The resulting equations are solved by 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 stiffener, twist angle, dimensional parameters, buckling modes, and volume frac-tion index of materials on the stability of shell are clarified in detail
2 Eccentrically stiffened functionally graded cylindrical shells Consider a thin circular cylindrical shell with mean radius R, thickness h and length L only subjected to uniform torsional loads Assume that two butt-ends of shell are only deformed in their planes and they still are circular[32] The middle surface of the shells is referred to the coordinates (x, h, z), y = Rh 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[26,28]:
Esh¼ Emþ Ecm
2z þ h 2h
k
; Ecm¼ Ec Em;
k P 0; h
26z 6
h
Es¼ Ecþ Emc
2z h 2hs
k 2
; Emc¼ Em Ec;
k2P0; h
26z 6
h
Er¼ Ecþ Emc
2z h 2hr
k 3
; k3P0; h
26z 6
h
2þ hr; ð3Þ
msh¼ms¼mr¼m¼ const;
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
Trang 3It 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 hs, and hr; and Ec, Emare
Young’s modulus of the ceramic and metal, respectively The
coeffi-cientmis Poison’s ratio
To account for the effect of large deflection, the von Karman
type nonlinear kinematic relation for the strain components across
the shell thickness at a distance z from the middle surface are of
the form[31]
ex¼e0
xþ zkx; ey¼e0
yþ zky; cxy¼c0
xyþ 2zkxy;
kx¼ w;xx; ky¼ w;yy; kxy¼ w;xy; ð4Þ
in which
e0
x¼ u;xþ1
2w
2
;x; e0
y¼v;yw
Rþ
1
2w
2
;y;
c0
xy¼ u;yþv;xþ w;xw;y; ð5Þ
where u = u(x, y),v=v(x, y) and w = w(x, y) are the displacements of
the middle surface points along x, y and z axes, respectively, and kx,
ky and kxy are the change of curvatures and twist of shell,
respectively
The compatible equation deduced from Eqs.(5)is written by
e0
x;yyþe0
y;xxc0
xy;xy¼ 1
Rw;xxþ w
2
;xy w;xxw;yy: ð6Þ
Hooke’s law for cylindrical shell is defined as
rsh
x ¼ Esh
1 m2ðexþmeyÞ;
rsh
y ¼ Esh
1 m2ðeyþmexÞ;
rsh¼ Esh
2ð1 þmÞcxy;
ð7Þ
and for stiffeners
rs
x¼ Esex;
rr
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[26,31]
Nx¼ C11e0
xþ C12e0
yþ C14kxþ C15ky;
Ny¼ C12e0
xþ C22e0
yþ C24kxþ C25ky;
Nxy¼ C33c0
xyþ C36kxy;
ð9Þ
Mx¼ C14e0
xþ C24e0
yþ C44kxþ C45ky;
My¼ C15e0
xþ C25e0
yþ C45kxþ C55ky;
Mxy¼ C63c0
xyþ C66kxy;
ð10Þ
where the stiffness parameters Cijare given by
C11¼ E1
1 m2þE1sbs
ds
; C12¼ mE1
1 m2; C14¼ E2
1 m2þE2sbs
ds
;C15¼ mE2
1 m2;
C22¼ E1
1 m2þE1rbr
dr
; C24¼ mE2
1 m2;C25¼ E2
1 m2þE2rbr
dr
; C33¼ E1 2ð1 þmÞ;
C36¼ E2
1 þm; C44¼ E3
1 m2þE3sbs
ds
;C45¼ mE3
1 m2;C55¼ E3
1 m2þE3rbr
dr
;
C63¼ E2 2ð1 þmÞ; C66¼
E3
1 þm;
ð11Þ
in which
E1¼
Z h=2
h=2
EshðzÞdz ¼ Emh þEcmh
k þ 1;
E2¼
Z h=2
h=2
zEshðzÞdz ¼ kEcmh
2
2ðk þ 1Þðk þ 2Þ;
E3¼
Z h=2
h=2
z2EshðzÞdz ¼Emh
3
12 þ Ecmh
3 1 4ðk þ 1Þ
1
k þ 2þ
1
k þ 3
;
E1s¼
Z h=2þh s
h=2
EsðzÞdz ¼ Echsþ Emc
hs
k2þ 1;
E2s¼
Z h=2þh s
h=2
zEsðzÞdz ¼Ec
2hhs
hs
hþ 1
þ Emchsh 1
k2þ 2
hs
hþ
1 2k2þ 2
Fig 1 Geometry and coordinate system of a stiffened FGM circular cylindrical shell.
Trang 4Z h=2þh s
h=2
z2EsðzÞdz ¼Ec
3h
3 s
3 4
h2
h2sþ
3 2
h
hs
þ 1
!
þ Emch3s 1
k2þ 3þ
1
k2þ 2
h
hs
4ðk2þ 1Þ
h2
h2s
;
E1r¼
Z h=2þh r
h=2
ErðzÞdz ¼ Echrþ Emc
hr
k3þ 1;
E2r¼
Z h=2þh r
h=2
zErðzÞdz ¼Ec
2hhr
hr
hþ 1
þ Emchrh 1
k3þ 2
hr
hþ
1 2k3þ 2
;
E3r¼
Z h=2þh r
h=2
z2ErðzÞdz ¼Ec
3h
3 r
3 4
h2
h2rþ
3 2
h
hr
þ 1
!
þ Emch3r 1
k3þ 3þ
1
k3þ 2
h
hr
4ðk3þ 1Þ
h2
h2r
where the bsand brdenote widths of stiffeners, respectively Also, ds
and drare the distances between two stringers and rings,
respec-tively, and the eccentricities esand errepresent the distance from
the shell middle surface to the centroid of the stiffeners cross
sec-tion (Fig 1b)
For later use, the reverse relations obtained from Eqs.(9)are as
e0
x¼ C22Nx C12Nyþ C14kxþ C15ky;
e0
y¼ C12Nxþ C11Nyþ C24kxþ C25ky;
c0
xy¼ C33Nxy C36kxy;
ð13Þ
where
D¼ C22C11 C212; C
22¼ C22=D; C
12¼ C12=D;
C
14¼ ðC12C24 C22C14Þ=D;C
15¼ ðC12C25 C22C15Þ=D;
C
11¼ C11=D; C
24¼ ðC12C14 C11C24Þ=D;
C
25¼ ðC12C15 C11C25Þ=D; C
33¼ 1
C33
; C
36¼C36
C33
:
ð14Þ
Substituting Eqs (13) into Eqs (10), the moment resultants
become
Mx¼ D14Nxþ D24Nyþ D44kxþ D45ky;
My¼ D
15Nxþ D
25Nyþ D
54kxþ D
55ky;
Mxy¼ D
63Nxyþ D
66kxy;
ð15Þ
where
D
14¼ C14C22 C24C12; D
44¼ C44þ C24C24þ C14C14;
D
24¼ C24C
11 C14C
12;D
45¼ C14C
15þ C24C
25þ C45;
D15¼ C15C22 C25C12; D54¼ C15C14þ C25C24þ C45;
D
25¼ C25C
11 C15C
12; D
55¼ C15C
15þ C25C
25þ C55;
D
63¼ C63C
33;D
66¼ C66 C63C
36:
ð16Þ
The equilibrium equations of cylindrical shell based on the
clas-sical shell theory are given by[31,32]
Nx;xþ Nxy;y¼ 0;
Nxy;xþ Ny;y¼ 0;
Mx;xxþ 2Mxy;xyþ My;yyþNy
R þ Nxw;xxþ 2Nxyw;xyþ Nyw;yy¼ 0:
ð17Þ
The first two of Eqs.(17)are identically satisfied by introducing
a stress functionu(x, y) as
Nx¼u;yy; Ny¼u;xx; Nxy¼ u;xy: ð18Þ
Introduction of Eqs.(15) and (18)into the third of Eqs.(17), tak-ing into account Eq.(4), yields the following equation:
a11w;xxxxþa12w;xxyyþa13w;yyyyþa14u;xxxxþa15u;xxyyþa16u;yyyy
þ1
Ru;xxþu;yyw;xxþu;xxw;yy 2u;xyw;xy¼ 0; ð19Þ
in which
a11¼ D44; a12¼ D45þ 2D66þ D54
; a13¼ D55;
a14¼ D24; a15¼ D14 2D63þ D25
; a16¼ D15: ð20Þ
Eq.(19)includes two dependent unknown functions w andu and to find a second equation relating to these two functions the geometrical compatibility Eq.(6)is used For this aim, substituting
Eq.(13)into Eq.(6), obtains
b11u;xxxxþ b12u;xxyyþ b13u;yyyyþ b14w;xxxxþ b15w;xxyyþ b16w;yyyy
w2
;xyþ w;xxw;yyþ1
where
b11¼ C11; b12¼ C33 2C12; b13¼ C22;
b14¼ C24; b15¼ C 14þ C25þ C36
; b16¼ C15: ð22Þ
Eqs.(19) and (21)are the nonlinear governing equations used to investigate the nonlinear stability of eccentrically stiffened FGM cylindrical shells under uniform torsion loads
3 Solution of the problem Consider a torsion-loaded cylindrical shell and it is simply sup-ported at two butt-ends x = 0 and x = L The deflection of shell in this case can be expressed by[17,32]
w ¼ wðx; yÞ ¼ f0þ f1sinax sin bðy kxÞ þ f2sin2ax; ð23Þ
in whicha= mp/L, b = n/R and m is the number of axis half waves and n is the number of circumferential waves The first term of w
in Eq (23)represents the uniform deflection of points belonging
to two butt-ends x = 0 and x = L, 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
x = 0 and x = L is fulfilled on the average sense
Substituting Eq.(23)into Eq.(21)obtains
b11u;xxxxþ b12u;xxyyþ b13u;yyyy¼ B01cos 2ax þ B02cos 2bðy kxÞ
þ B03cos b y þ a
b k
x
þ B04cos b y a
bþ k
x
þ B05 cos b y 3a
bþ k
x
cos b y þ 3a
b k
x
; ð24Þ
where
B01¼ 2f2a2
4b14a2
1 R
þ1
2f
2a2
b2
; B02¼1
2f
2a2
b2;
B03¼ f1
1
2b14½ða2
þ b2
k2Þ2þ ð2abkÞ2 1
2
1
R b15b2
ða2
þ b2
k2Þ þ1
2b16b4
þabk 2b14ða2
þ b2
k2Þ þ1
R b15b2
þ1
2f1f2a2
b2;
B04¼1
2f1 b14½ða2
þ b2k2Þ2þ ð2abkÞ2 þ 1
R b15b2
ða2
þ b2k2Þ
b16b4þ 2abk2b14ða2
þ b2k2Þ þ1
R b15b2
1
2f1f2a2b2;
B05¼1
2f1f2a2
b2;
ð25Þ The general solution of Eq.(24)for torsion-loaded shell is of the form
Trang 5u¼ B1cos 2ax þ B2cos 2bðy kxÞ þ B3cos b y þ a
b k
x
þ B4cos b y a
bþ k
x
þ B5cos b y 3a
bþ k
x
þ B6cos b y þ 3a
b k
x
wheresis torsional load intensity and the coefficients Biare defined
by
B1¼ B01
16b11a4¼ A11f2þ A12f2;
B2¼ B02
16b4½b11k4þ b12k2þ b13¼ A21f
2;
b4 b11 a
b k 4
þ b12 a
b k 2
þ b13
¼ A31f1þ A32f1f2;
b4 b11 a
bþ k 4
þ b12 a
bþ k 2
þ b13
¼ A41f1þ A42f1f2;
b4 b11 3a
bþ k 4
þ b12 3a
bþ k 2
þ b13
b4 b11 3a
b k 4
þ b12 3a
b k 2
þ b13
ð27Þ
in which
A ¼a2þ b2k2; A11¼4b14a2 1=R
8b11a2 ;A12¼ b
2 32b11a2;
A21¼ a2
32b2
½b11k4þ b12k2þ b13;
A31¼
1 b14½A2þ ð2abkÞ2 þ 1 b15b2
A b16b4
þabkh2b14A þ1 b15b2i
b4 b11 a
b k 4
þ b12 a
b k 2
þ b13
2b2
b11 a
b k 4
þ b12 a
b k 2
þ b13
A41¼b14½A
2
þ ð2abkÞ2 þ 1 b15b2
A b16b4þ 2abkh2b14A þ1 b15b2i 2b4
b11 a
bþ k 4
þ b12 a
bþ k 2
þ b13
2b2
b11 a
bþ k 4
þ b12 a
bþ k 2
þ b13
2b2
b11 3a
bþ k 4
þ b12 3a
bþ k 2
þ b13
2b2
b11 3a
b k 4
þ b12 3a
b k 2
þ b13
ð28Þ
In order to establish a torsional load–deflection curve, first of
all, introducing w anduinto the left side of Eq.(19), then
apply-ing Galerkin’s method in the ranges 0 6 y 6 2pR and 0 6 x 6 L,
lead to
2shb2
kþ D1þ D2f2þ D3f2
þ D4f2
D5f2 D6f2þ D7f2f2¼ 0; ð30Þ
where
D1¼a11½ða2
þ b2k2Þ2þ ð2abkÞ2 þa12b2ða2
þ b2k2Þ þa13b4
A31b4 a14
a
b k
4
þ a15 1
Rb2
b k
2
þa16
þ A41b4 a14
a
bþ k
4
þ a15 1
Rb2
bþ k
2
þa16
;
D2¼ A32b4 a14 a
b k
4
þ a15 1
Rb2
a
b k
2
þa16
þa2
b2ðA31þ A41 2A11Þ þ A42b4
a14
a
bþ k
4
þ a15 1
Rb2
bþ k
2
þa16
;
D3¼ 2ðA21þ A12Þa2
b2; D4¼ ðA32þ A42 A5þ A6Þa2
b2;
D5¼ 8a2 2a11a2þ 4a14a21
R
A11
;
D6¼ 8a2A12 4a14a21
R
þ 2ðA31 A41Þa2
b2;
D7¼ 2a2b2ðA32þ A42 A5þ A6Þ:
ð31Þ
In addition to Eqs.(29) and (30), the cylindrical shell must also satisfy the circumferential closed condition[17,32]as
Z2pR
0
Z L
0 v;ydx dy ¼
Z2pR
0
Z L
0
e0
yþw
R
1
2w
2
;y
dx dy ¼ 0: ð32Þ
Using Eqs.(13), (18), (23) and (26), this integral becomes
2f0þ f21
4Rf
2
Eliminating f2 from Eqs.(29) and (30) leads to the equation representings–f1relation as
s¼ D1þ D2D6f
2
D5þ D7f2
þ D3f2þ D4 D6f
2
2
D5þ D7f2
ð2hb2kÞ: ð34Þ
Eq (34) is used to analyze the post-bucklings–f1 curves of stiffened FGM cylindrical shells
When f1?0, Eq.(34)becomes
Eq.(35)is used to find upper critical loads in case linear buck-ling shape
From Eq.(23), it is obvious that the maximal deflection of the shells
locates at x = iL/(2m), y = jpR/(2n) + ikL/(2m), where i, j are odd inte-ger numbers
Solving f2and f0from Eqs.(30) and (33)with respect to f1, then substituting them into Eq.(36), obtains
Wmax¼ D6f
2
2 D 5þ D7f2 þ1
8Rf
2
1b2þ f1: ð37Þ
Combining Eq.(34)with Eq.(37), the effects of inhomogeneous and dimensional parameters on the post-buckling load–maximal deflection curves of shells can be analyzed
The angle of twist is defined[8,32]as
w¼ 1
2pRL
Z 2pR 0
ZL 0
@u
@yþ
@v
@x
dx dy
¼ 1
2pRL
Z 2pR 0
ZL 0
c0
xy w;xw;y
Using Eqs.(13) and (18), this integral becomes
Trang 6w¼ 1
2pRL
Z 2pR
0
Z L
0
C36w;xy C33u;xy w;xw;y
dx dy:
Substituting w andufrom Eqs.(23) and (26)into this equation
obtains
w¼ C33sh þ1
4b
When f1= 0, Eq.(39)shows that the relation between twist
an-gle and shear stress is linear When f1–0, combining Eq.(34)with
Eq.(39), thes–wrelation of shells will be studied
4 Numerical results and discussion
4.1 Comparison results
To validate the present study, three comparisons on critical
tor-sion load are made with results from open literatures
Tables 1 and 2compare the results of this paper for un-stiffened
isotropic cylindrical shell under torsion load with the results given
by Shen[16]using the higher order shear deformation shell theory
and with experimental results of Nash[33]and Ekstrom[34] As
can be seen that good agreements are obtained in these
comparisons
Fig 2shows the comparisons of the present post-buckling paths
with the results which they were analyzed by Huang and Han[17]
using the nonlinear large deflection theory for un-stiffened FGM
cylindrical shells under torsion load As can be observed, the
present results coincide with the ones of the work[17] In addition, the present critical value scr= 204.12 MPa corresponding to the lowest point of the envelope curve (see Fig 2) is much close to the one of Ref.[17]scr= 204.15 MPa obtained by using the Ritz en-ergy method
In the following subsections, the materials used[26]are Zirco-nia with Ec= 151 GPa and Aluminum with Em= 70 GPa Also as-sume that k2= k3= 1/k andm= 0.3
4.2 Nonlinear critical torsional load finding procedure Consider a stiffened FGM shell with the material and geometri-cal parameters: k = 1, k2= k3= 1, L = 387.35 103m, L/R = 1, R/
h = 100, hs/h = hr/h = 1/2, bs= hs, br= hr The number of stringers as well as rings is equal to 20
Based on Eq.(34)with various combinations of the modes (m, n, k) the critical loadscrof stiffened FGM shell may be found As can
be seen, fromTable 3, the critical loadscr= 265.0121 MPa corre-sponding to m = 1, n = 9 and k = 0.55 Graphically, according to Ref.[17], one also can define the critical condition as the possible lowest point ofs–Wmax/h curves (Fig 3) Thus, the specific solution procedures are exhibited as follows: by using Eqs.(34) and (37), a series ofsversus Wmax/h, the curves can be drawn under various combinations of (m, n, k) From the lowest of these curves, an enve-lope curve is obtained The lowest point of the enveenve-lope curve is regarded as the critical condition By mentioned procedure, in this
corresponding to the buckling mode (m, n, k) = (1, 9, 0.55)
Table 1
Comparisons of critical torsion loadscr(psi) for un-stiffened isotropic cylindrical shell.
h = 0.0172 in.
Table 2
Comparisons of critical torsion loadscr(psi) for un-stiffened isotropic cylindrical shell.
h = 0.0075 in.
200
210
220
230
240
250
present Huang
1/h
Zirconia/Ti-6Al-4V
T=300K, k=1,
R/h=100, L/R=2
1: (n,λ)=(7,0.3) 2: (n,λ)=(8,0.4) 3: (n,λ)=(9,0.5)
1
2
3
τcr=204.12 MPa
10 20 30 40
50
Lower (34) Upper (35) Nonlinear of Huang Linear of Huang
Mcr
2 2 2
2 1
L Z
Rh
π τ ν
=
= −
Z
Trang 74.3 Effect of the mode (m, n, k) on the critical torsional load
In this subsection, by using Eqs (34) and (35), the effect of
mode on the critical loads of stiffened FGM is presented inTable 4
It is seen that, the lower and upper critical loads depend clearly
on the mode In addition, the circumferential wave number n
in-creases with inin-creases of R/h ratio or L/R ratio decreasing
4.4 Effects of geometric parameters
Based on Eqs (34) and (37) with the database given in
Sec-tion4.2, the effects of the radius-to-thickness ratios R/h and of
the length-to-radius ratios L/R ons–Wmax/h post-buckling curves
of stiffened FGM cylindrical shell are considered
Fig 4plots the post-buckling curves versus R/h = (100; 200;
300; 400 and 500) It is observed that the torsional buckling load
s decreases markedly with the increase of R/h ratio This result
agrees with the actual property of structure, i.e the shell is thinner
the value of critical load is smaller This remark is also illustrated in Table 5
Effect of L/R ratio also is analyzed inTable 5andFig 5 As can be seen the critical torsional loads of shells decreases considerably when L/R ratio increases
Thus, both the cases, the critical torsion load is very sensitive with the change of R/h or L/R
4.5 Effects of volume fraction index Using the database in Section4.2, the effects of index volume k
on the critical buckling loads and post-buckling behavior are given
Table 3
Critical buckling load versus (m, n, k).
7 321.2948(0.36) a
606.3897(1.12) 936.3017(1.10) 1445.007(1.24)
8 278.0196(0.53) 532.2178(0.85) 839.2876(0.94) 1301.552(1.08)
9 265.0121(0.55) 485.9032(0.72) 775.4175(0.84) 1201.095(0.95)
10 271.2489(0.56) 464.4975(0.66) 737.0675(0.76) 1133.255(0.86)
11 289.8609(0.57) 462.1628(0.64) 718.7535(0.71) 1091.230(0.79)
12 317.1922(0.58) 474.0187(0.63) 716.2710(0.68) 1069.487(0.74)
13 351.2284(0.59) 496.6100(0.62) 726.6889(0.66) 1063.989(0.70)
14 390.8035(0.59) 527.6847(0.61) 747.6301(0.64) 1071.617(0.68)
a The number of k.
250
280
310
340
370
400
max/
1: (n, λ)=(7, 0.36) 2: (n, λ)=(8, 0.53) 3: (n, λ)=(9, 0.55) 4: (n, λ)=(10, 0.56) 5: (n, λ)=(11, 0.57)
1
5
2
3
4
τcr=265.0121 MPa
Fig 3 Critical buckling load (m = 1).
Table 4
Effect of mode on the critical buckling load (m = 1).
Lower critical load calculated by Eq (34)
Upper critical load calculated by Eq (35)
Lower critical load calculated by Eq (34)
Upper critical load calculated by Eq (35)
a
0 2 4 6 8 10 12 14 16 18 20 0
50 100 150 200 250 300 350
max/
1: R/h=100, (5, 0.26) a
2: R/h=200, (6, 0.21) 3: R/h=300, (7, 0.21) 4: R/h=400, (7, 0.18) 5: R/h=500, (8, 0.19)
1
2
3
4
5
Fig 4 Effects of R/h ratio on post-buckling curves of shell, m = 1, L/R = 1 a
Bulking mode (n, k).
0 2 4 6 8 10 12 14 16 18 20 100
150 200 250 300 350
max/
1: L/R=1, (9,0.5) 2: L/R=1.5, (8,0.43) 3: L/R=2, (8,0.42) 4: L/R=2.5, (7,0.37) 5: L/R=3, (7,0.37)
1
2
3
4
5
Fig 5 Effects of L/R ratio ons–W max /h curves (m = 1, R/h = 100).
Trang 8inFigs 6 and 7for stiffened FGM shell and un-stiffened FGM shell
with m = 1, k2= k3= 1/k but n and k vary
It can be observed, the critical torsional loads of shells with or
without stiffener decrease with the increase of k This property
appropriate to the real characteristic of material, because the
high-er value of k corresponds to a metal-richhigh-er shell which usually has
less stiffness than a ceramic-richer one
4.6 Effects of number of stiffeners
To investigate the effects of number of stiffeners, the database is
used here taken from database in Section4.2with hs= hr= h, bs=
hs, br= hr.Fig 8andTable 6illustrate the effects of number of
stiff-ener (ns= nr= 10, 20, 30, 40 and 50) on the critical torsional loads
As expected, these curves become higher when the number of
stiffeners increases and critical torsion loads decrease when the
number of stiffeners decreases The prime reason is that the
pres-ence of stiffeners makes the shells to become stiffer.Table 6also
shows that the percentage increase in the buckling load rises
continuously with the increment of the number of stiffeners This
increase is about 35.8% for orthogonal stiffened shell, in compari-son ns= nr= 10 with ns= nr= 50
4.7 Comparison of critical torsion loads of stiffened and un-stiffened FGM cylindrical shells
Using the database in Section4.6, the comparison between the critical torsional loadsscrof stiffened FGM and un-stiffened FGM shell is given
Table 7shows that the critical torsional loads of FGM stiffened cylindrical shells are generally upper than the corresponding values of the FGM un-stiffened cylindrical shells In addition, the critical torsional loads of FGM un-stiffened shells are the smallest, the critical torsional loads of stringer stiffened shell are smaller than ring stiffened shell, and finally the critical loads of FGM ring-stringer stiffened shell are the greatest Thus a presence of stiffener enhances the stability of shell
4.8 Effects of k and Z ons–wpost-buckling curves With the database in Section4.6and Z ¼ L2
=ðRhÞ ¼ 300 given by [16],Fig 9shows the effects of volume fraction k on post-buckling
s–wcurves for un-stiffened and stiffened FGM cylindrical shells Comparing these curves, it can be seen that the, they become to
be more down in the increase of k Fig 10illustrates the effects
of Batdorf shell parameter Z ¼L 2
Rh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 m2
p
on the post-buckling
s–wcurves of shells Similar to above case, the critical torsional loads of shells in this case also decrease when Z increases In addi-tion, the post-bucklings–wcurves are nonlinear and slope down-ward immediately after buckling With further increase in twist angle, the torsional load exhibits an increase after reaching the minimum post-buckling load
5 Conclusions The shells stiffened by eccentrically rings and stringers attached
to the inside and material properties of shell and stiffeners varying continuously graded in the thickness direction are investigated in
Table 5
Effects of L/R and R/h on critical torsional load for FGM stiffened cylindrical shells.
100
150
200
250
300
350
400
450
500
Stiffened Unstiffened
max/
)
)
)
)
1 2
3
4
Fig 6 Effects of k ons–W max /h curves (m = 1, n = 9, s
Stiffened, u Unstiffened).
10-2 10-1 100 101 102
0
50
100
150
200
250
300
350
400
1
2
3
k
1: R/h=100, (9,0.55) 2: R/h=200, (12,0.47) 3: R/h=300, (14,0.44)
Fig 7 Effects of k on torsional load (m = 1).
280 300 320 340 360 380 400 420 440 460 480
max/
1: ns=nr=10, (9,0.62) 2: ns=nr=20, (8,0.68) 3: ns=nr=30, (8,0.71)
4: ns=nr=40, (8,0.73) 5: ns=nr=50, (8,0.75)
5
4
3
2
1
Fig 8 Effects of number of orthogonal stiffeners (m = 1).
Trang 9this paper An analytical approach to analyze the nonlinear
buck-ling and post-buckbuck-ling behavior of eccentrically stiffened FGM
cylindrical shells under torsion based on the classical shell theory
and the smeared stiffeners technique with geometrical
nonlinear-ity in von Karman sense is presented The results obtained show
some remarks as:
i The expression of deflection with three-term including the linear and nonlinear buckling shape is more correctly chosen
ii The close-form expressions to determine critical buckling loads and nonlinear post-buckling load–deflection curves are obtained
Table 6
(m = 1, k = 1) Effects of number of stiffeners on the critical torsional loadsscr(MPa).
Table 7
Comparison of critical torsional loads for stiffened and un-stiffened FGM shells.
280.4184(9, 0.50) 250.3686(9, 0.50) 204.3720(9, 0.51) 163.3540(9, 0.51)
a
The numbers in the parentheses denote the buckling mode (n, k).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
50
100
150
200
250
300
(a) for un-stiffened shell
1
2
3
4
1: k=0, (8,0.43) 2: k=0.5, (8,0.42) 3: k=1, (8,0.42) 4: k=5, (8,0.42)
Z L2/( )Rh =300 R/h=100
0 50 100 150 200 250 300 350 400 450
(b) for stiffened shell
1
2
3
4
1: k=0, (7,0.52) 2: k=0.5, (7,0.50) 3: k=1, (7,0.49) 4: k=5, (7,0.49)
Z=L2/( )Rh =300 R/h=100
Fig 9 Effects of k ons–wcurves FGM cylindrical shell.
0
50
100
150
200
250
(a) unstiffened shell
ψ(deg)
1: Z=300, (8,0.42) 2: Z=500, (7,0.36) 3: Z=1000, (7,0.36) 4: Z=1500, (6,0.31) 5: Z=2000, (6,0.31)
1
2
3
4
5
k=1
R/h=100
0 0.5 1 1.5 2 2.5 3 3.5 4 0
50 100 150 200 250 300
ψ(deg)
k=1
R/h=100
1: Z=300, (7,0.49) 2: Z=500, (7,0.46) 3: Z=1000, (6,0.36) 4: Z=1500, (6,0.36) 5: Z=2000, (6,0.36)
1
3
4
5
(b) stiffened shell
2
Fig 10 Effects of Z ons–wcurves for cylindrical shell.
Trang 10iii Both the post-buckling mode and the post-buckling paths of
torsion-loaded stiffened FGM cylindrical shells can be well
predicted by using the nonlinear large deflection theory
iv The stiffener system strongly enhances on the stability and
load-carrying capacity of FGM cylindrical shells
v The critical torsion load is affected significantly when
mate-rial distribution was varied by changing the values of the
power law exponent k Both the critical torsional load and
the post-buckling carrying capacity decrease greatly when
the radius-to-thickness or length-to-radius ratio increase
vi The critical torsional load decreases with the increase of
twist angle
Acknowledgements
This research is funded by Vietnam National Foundation for
Sci-ence and Technology Development (NAFOSTED) under Grant No
107.01-2012.02 The authors are grateful for this financial support
References
[1] Donnell LH Stability of thin-walled tubes under torsion NACA Report No 479;
1933.
[2] Loo TT Effects of large deflection and imperfections on the elastic buckling of
cylinders under torsion and axial compression Proc Second US Natl Congr Appl
Mech 1954;5:345–57.
[3] Nash WA Buckling of initially imperfect cylindrical shells subjected to torsion.
J Appl Mech 1957:125–30.
[4] Yamaki N Postbuckling behavior of circular cylindrical shells under torsion.
Ing Arch 1975:79–89.
[5] Shaw D, Simitses GJ, Sheinman I Imperfect laminated cylindrical shells in
torsion and axial compression Acta Astronaut 1983;10(5–6):395–400.
[6] Lennon RF, Das PK Torsional buckling behavior of stiffened cylinders under
combined loading Thin-Walled Struct 2000;38:229–45.
[7] Mao R, Lu G A study of elastic–plastic buckling of cylindrical shells under
torsion Thin-walled Struct 2002;40:1051–71.
[8] Zhang X, Han Q Buckling and post-buckling behaviors of imperfect cylindrical
shells subjected to torsion Thin-walled Struct 2007;45:1035–43.
[9] Shen HS, Xiang Y Buckling and post-buckling of anisotropic laminated
cylindrical shells under combined axial compression and torsion Compos
Struct 2008;84:375–86.
[10] Paimushin VN Torsional, flexural, and torsional–flexural buckling modes of a
cylindrical shell under combined loading Mech Solid 2007;42:437–46.
[11] Takano A Buckling of thin and moderately thick anisotropic cylinders under
combined torsion and axial compression Thin-Walled Struct 2011;49:304–16.
[12] Fraternali F, Reddy JN A penalty model for the analysis of laminated composite
shells Int J Solids Struct 1993;30:3337–55.
[13] Ascione L, Fraternali F A penalty model for the analysis of composite curved
beams Comput Struct 1992;45:985–99.
[14] Fraternali F, Bilotti G Non-linear elastic stress analysis in curved composite beams Comput Struct 1997;62:837–69.
[15] Fraternali F, Feo L On a moderate rotation theory of thin-walled composite beams Compos Part B Eng 2000;31:141–58.
[16] Shen HS Torsional buckling and postbuckling of FGM cylindrical shells in thermal environments Int J Non-Linear Mech 2009;44:644–57.
[17] Huang H, Han Q Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment Euro J Mech A/Solids 2010;29: 42–8.
[18] Sofiyev AH, Kuruoglu N Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium Composites Part B 2013;45:1133–42.
[19] Li S, Wang S Elastic stability of circular cylindrical shells with FGMS layers under end torsion force Lixue Xuebao/Chin J Theoret Appl Mech 2010;42(6):1172–9.
[20] Bich DH, Phuong NT, Tung HV Buckling of functionally graded conical panels under mechanical loads Compos Struct 2012;94(4):1379–84.
[21] Sofiyev AH, Schnack E The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading Eng Struct 2004;26(10):1321–31.
[22] Bich DH, Dung DV, Hoa LK Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects Compos Struct 2012;94(9):2952–60.
[23] Bich DH, Nguyen NX Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations J Sound Vib 2012;331(25):5488–501.
[24] Van der Neut A The general instability of stiffened cylindrical shells under axial compression Rep S314 National Aeronautical Research Institute Amsterdam; 1947.
[25] Baruch M, Singer J Effect of eccentricity of stiffeners on the general instability
of stiffened cylindrical shells under hydro-static pressure J Mech Eng Sci 1963;5:23–7.
[26] Najafizadeh MM, Hasani A, Khazaeinejad P Mechanical stability of functionally graded stiffened cylindrical shells Appl Math Model 2009;33:1151–7.
[27] Bich DH, Nam VH, Phuong NT Nonlinear post-buckling of eccentrically stiffened functionally graded plates and shallow shells Vietnam J Mech 2011;3:131–47.
[28] Dung DV, Hoa LK Nonlinear buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure Thin-Walled Struct 2013;63:117–24.
[29] Bich DH, Dung DV, Nam VH Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels Compos Struct 2012;94(8):2465–73.
[30] Bich DH, Dung DV, Nam VH Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells Compos Struct 2013;96:384–95.
[31] Brush DO, Almroth BO Buckling of bars, plates and shells New York: Mc Graw-Hill; 1975.
[32] Volmir AS Stability of elastic systems Science ed Moscow; 1963 [in Russian] [33] Nash WA An experimental analysis of the buckling of thin initially imperfect cylindrical shells subject to torsion Proc Soc Exp Stress Anal 1959;16(2):55–68.
[34] Ekstrom RE Buckling of cylindrical shells under combined torsion and hydrostatic pressure Exp Mech 1963;3:192–7.