Nonlinear static and dynamic buckling analysis of imperfecteccentrically stiffened functionally graded circular cylindrical thin shells under axial compression a Vietnam National Univers
Trang 1Nonlinear static and dynamic buckling analysis of imperfect
eccentrically stiffened functionally graded circular cylindrical thin
shells under axial compression
a Vietnam National University, Ha Noi, Vietnam
b University of Transport Technology, Ha Noi, Vietnam
a r t i c l e i n f o
Article history:
Received 24 November 2012
Received in revised form
20 May 2013
Accepted 3 June 2013
Available online 12 June 2013
Keywords:
Static and dynamic buckling analysis
Stiffener
Functionally graded material
Stiffened circular cylindrical shell
Critical buckling load
a b s t r a c t
An analytical approach is presented to investigate the nonlinear static and dynamic buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression Based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense, initial geometrical imperfection and the smeared stiffeners technique, the governing equations of motion
of eccentrically stiffened functionally graded circular cylindrical shells are derived The functionally graded cylindrical shells with simply supported edges are reinforced by ring and stringer stiffeners system on internal and (or) external surface The resulting equations are solved by the Galerkin procedure to obtain the explicit expression of static critical buckling load, post-buckling load–deflection curve and nonlinear dynamic motion equation The nonlinear dynamic responses are found by using fourth-order Runge–Kutta method The dynamic critical buckling loads of shells under step loading of infinite duration are found corresponding to the load value of sudden jump in the average deflection and those of shells under linear-time compression are investigated according to Budiansky–Roth criterion The obtained results show the effects of stiffeners and input factors on the static and dynamic buckling behavior of these structures
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1 Introduction
Functionally graded (FGM) plate and shell structures have became
popular in engineering designs of coating of nuclear reactors and
space shuttle The static and dynamic behavior of FGM cylindrical shell
attracts special attention of a lot of scientists in the world
In static analysis of FGM cylindrical shells, many studies have been
focused on the buckling and post-buckling of shells under mechanic
and thermal loading Shen[1]presented the nonlinear post-buckling
of perfect and imperfect FGM cylindrical thin shells in thermal
environments under lateral pressure by using the classical shell theory
with the geometrical nonlinearity in von Karman–Donnell sense By
using higher order shear deformation theory; this author[2]
contin-ued to investigate the post-buckling of FGM hybrid cylindrical shells in
thermal environments under axial loading Bahtui and Eslami [3]
investigated the coupled thermo-elasticity of FGM cylindrical shells
Huang and Han[4–7] studied the buckling and post-buckling of
un-stiffened FGM cylindrical shells under axial compression, radial
pressure and combined axial compression and radial pressure based
on the Donnell shell theory and the nonlinear strain–displacement
relations of large deformation The post-buckling of shear deformable FGM cylindrical shells surrounded by an elastic medium was studied
by Shen[8] Sofiyev[9]analyzed the buckling of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation Zozulya and Zhang[10]studied the behavior of function-ally graded axisymmetric cylindrical shells based on the high order theory
For dynamic analysis of FGM cylindrical shells, Ng et al.[11]and Darabi et al.[12] presented respectively linear and nonlinear para-metric resonance analyses for un-stiffened FGM cylindrical shells Three-dimensional vibration analysis offluid-filled orthotropic FGM cylindrical shells was investigated by Chen et al [13] Sofiyev and Schnack[14] and Sofiyev [15] obtained critical parameters for un-stiffened cylindrical thin shells under linearly increasing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with Ritz type variation method Shariyat[16,17]investigated the nonlinear dynamic buckling problems of axially and laterally preloaded FGM cylindrical shells under transient thermal shocks and dynamic buckling analysis for un-stiffened FGM cylindrical shells under complex combinations of thermo–electro-mechanical loads Geometrical imperfection effects were also included in his research Li et al [18] studied the free vibration of three-layer circular cylindrical shells with functionally graded middle layer Huang and Han[19] presented the nonlinear
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International Journal of Mechanical Sciences
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n Corresponding author Tel.: +84 98 3843 387.
E-mail address: hoainam.vu@utt.edu.vn (V.H Nam)
Trang 2dynamic buckling problems of un-stiffened functionally graded
cylind-rical shells subjected to time-dependent axial load by using the
Budiansky–Roth dynamic buckling criterion [20] Various effects of
the inhomogeneous parameter, loading speed, dimension parameters;
environmental temperature rise and initial geometrical imperfection
on nonlinear dynamic buckling were discussed Shariyat[21]analyzed
the nonlinear transient stress and wave propagation analyses of the
FGM thick cylinders, employing a unified generalized
thermo-elasticity theory
Recently, idea of eccentrically stiffened FGM structures has been
proposed by Najafizadeh et al.[22]and Bich et al.[23,24] Najafizadeh
et al.[22]have studied linear static buckling of FGM axially loaded
cylindrical shell reinforced by ring and stringer FGM stiffeners In order
to provide material continuity and easily to manufacture, the FGM
shells are reinforced by an eccentrically homogeneous stiffener
system; Bich et al.[23]have investigated the nonlinear static
post-buckling of functionally graded plates and shallow shells and
non-linear dynamic buckling of functionally graded cylindrical panels[24]
Literature on the nonlinear static and dynamic analysis of
imperfect FGM stiffened circular cylindrical shells is still very
limited In this paper, the mentioned just problem is investigated
by analytical approach The nonlinear dynamic equations of
eccentrically stiffened FGM circular cylindrical shells are derived
based on the classical shell theory with the nonlinear strain–
displacement relation of large deflection and the smeared
stiffen-ers technique By using the Galerkin method, the closed-form
expression to determine the static critical buckling load and load–
deflection curves are obtained The nonlinear dynamic responses
are found by using fourth-order Runge–Kutta method The
dynamic buckling loads of shells under step loading of infinite
duration are found corresponding to the load value of sudden
jump in the average deflection and those of shells under
linear-time compression are investigated according to Budiansky–Roth
criterion The results show that the stiffener, volume-fractions
index, initial imperfection and geometrical parameters influence
strongly to the static and dynamic buckling of shells
2 Eccentrically stiffened FGM (ES-FGM) circular cylindrical
shells
2.1 Functionally graded material
In this paper, functionally graded material is assumed to be
made from a mixture of ceramic and metal with the
volume-fractions given by a power law
Vmþ Vc¼ 1;
Vc¼ VcðzÞ ¼ 2z þ h
2h
; where h is the thickness of shell; k≥0 is the volume-fraction index;
z is the thickness coordinate and varies from −h=2 to h=2; the subscripts m and c refer to the metal and ceramic constituents respectively According to the mentioned law, the Young modulus and the mass density can be expressed in the form
EðzÞ ¼ EmVmþ EcVc¼ Emþ ðEc−EmÞ 2z þ h
2h
; ρðzÞ ¼ ρmVmþ ρcVc¼ ρmþ ðρc−ρmÞ 2z þ h
2h
Poisson’s ratio ν is assumed to be constant
2.2 Constitutive relations and governing equations Consider a functionally graded thin circular cylindrical shell with length L, mean radius R This shell is assumed to be reinforced
by closely spaced [22,25,29] homogeneous ring and stringer stiffener systems (see Fig 1) Stiffener is pure-ceramic if it is located at ceramic-rich side and is pure-metal if is located at metal-rich side, such FGM stiffened circular cylindrical shells provide continuity within shell and stiffeners and can be easier manufactured The origin of the coordinate 0 locates on the middle plane and at the left end of the shell, x; y (y ¼ Rθ) and z axes are in the axial, circumferential, and inward radial directions, respectively
According to the von Karman nonlinear strain–displacement relations [25], the strain components at the middle plane of imperfect circular cylindrical shells are of the form
ε0
x¼ ∂u
∂xþ
1 2
∂w
∂x
2
þ ∂w
∂x∂w
0
∂x ;
ε0¼ ∂v
∂y−
w
Rþ1 2
∂w
∂y
2
þ ∂w
∂y∂w
0
∂y ;
γ0
xy¼ ∂u
∂yþ ∂
v
∂xþ ∂
w
∂x∂w∂yþ ∂
w
∂y∂w
0
∂x þ ∂
w
∂x∂w
0
∂y ;
χx¼ ∂2w
∂x2; χy¼ ∂2w
∂y2; χxy¼ ∂2w
Nomenclature
h thickness of the shell
m number of half waves in axial direction
n number of wave in circumferential direction
z coordinate in thickness direction
EðzÞ; Em; Ec Young's modulus of shell, metal, ceramic,
respectively
ρðzÞ; ρm; ρc mass density of shell, metal, ceramic, respectively
L length of the shell
R radius of the shell
Es; Er Young's modulus of stringer and ring stiffeners,
respectively
Aij; Bij; Dijextensional, coupling and bending stiffness of the
un-stiffened shell, respectively
Cs; Cr coupling parameters
ss; sr spacing of the stringer and ring stiffeners, respectively
As; Ar cross-section areas of stiffeners
Is; Ir moments of inertia of stiffener cross sections relative
to the shell middle surface
zs; zr eccentricities of stiffeners with respect to the middle
surface of shell
ds; dr width of the stringer and ring stiffeners, respectively
hs; hr height of the stringer and ring stiffeners, respectively
f ¼ f ðtÞ time dependent total amplitude
f0 known imperfect amplitude
r0 compressive load per unit length
r0¼ r0=h compressive stress
rsbu static buckling stress
rscr static critical buckling stress
rscr static critical buckling loads per unit length
t; tcr time and dynamic critical time
rdcr dynamic critical buckling stress
τcr dynamic coefficient
D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 191
Trang 3where u ¼ uðx; yÞ, v ¼ vðx; yÞ and w ¼ wðx; yÞ are displacements
along x; y and z axes respectively and w0¼ w0ðx; yÞ denotes initial
imperfection of shell, which is very small compared with the shell
dimensions, but may be compared with the shell wall thickness
The strains across the shell thickness at a distance z from the
mid-surface are given by
εx¼ ε0
x−zχx;
εy¼ ε0−zχy;
γxy¼ γ0
From Eq (2) the strains must be relative in the deformation
compatibility equation
∂2ε0
x
∂y2 þ∂2ε0
∂x2−∂
2γ0
xy
∂x∂y ¼ −
1 R
∂2w
∂x2 þ ∂2w
∂x∂yþ ∂
2w0
∂x∂y
− ∂∂x2w2þ ∂2w0
∂x2
∂2w
∂y2þ ∂2w0
∂y2
Hooke's stress–strain relation is applied for the shell
ssh
1−ν2ðεxþ νεyÞ; ssh
1−ν2ðεyþ νεxÞ; ssh¼ EðzÞ
2ð1 þνÞγxy; ð5Þ and for stiffeners
ss¼ Esεx; sr
where Es; Er is Young's modulus of stringer and ring stiffeners,
respectively
The force and moment of an un-stiffened FGM circular
cylind-rical shell can be determine by
fðNx; Ny; NxyÞ; ðMx; My; MxyÞg ¼Z h =2
−h=2fsx; sy; sxygð1; zÞ dz: ð7Þ 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 shell, we obtain the expressions for force
and moment resultants of an ES-FGM circular cylindrical shell
Nx¼ A11þEsAs
ss
ε0
xþ A12ε0−ðB11þ CsÞχx−B12χy;
Ny¼ A12ε0
xþ A22þErAr
sr
ε0−B12χx−ðB22þ CrÞχy;
Nxy¼ A66γ0
Mx¼ ðB11þ CsÞε0
xþ B12ε0− D11þEsIs
s
χx−D12χy;
My¼ B12ε0
xþ ðB22þ CrÞε0−D12χx− D22þErIr
sr
χy;
Mxy¼ B66γ0
where Aij; Bij; Dij ði; j ¼ 1; 2; 6Þ are extensional, coupling and bending stiffness of the un-stiffened FGM cylindrical shell
A11¼ A22¼ E1
1−ν2; A12¼ E1ν
1−ν2; A66¼ E1
2ð1 þνÞ;
B11¼ B22¼ E2
1−ν2; B12¼ E2ν
1−ν2; B66¼ E2
2ð1 þνÞ;
D11¼ D22¼ E3
1−ν2; D12¼ E3ν
1−ν2; D66¼ E3
with
E1¼ EmþEc−Em
k þ 1
h; E2¼ ðEc−EmÞkh2
2ðk þ 1Þðk þ 2Þ;
E3¼ Em
12þ ðEc−EmÞ 1
k þ 3−k þ 21 þ 1
4k þ 4
h3;
Is¼dsh3s
12 þ Asz2
s; Ir¼drh3r
12 þ Arz2
r;
Cs¼ 7EsAszs
ss ; Cr¼ 7ErArzr
sr ;
zs¼hsþ h
2 ; zr¼hrþ h
where the coupling parameters Csand Cr are negative for outside stiffeners and positive for inside ones The spacing of the stringer and ring stiffeners is denoted by ss and sr respectively The quantities As, Ar are the cross-section areas of stiffeners and
Is, Ir, zs, zr are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of shell, respectively The width and thickness of the stringer and ring stiffeners are denoted by ds; hsand dr; hr respectively The Young modulus of stiffeners Es, Er take the value Em if the full metal stiffeners are put at the metal-rich side of the shell and conversely,
Ecif the full ceramic ones are put at the ceramic-rich side From the constitutive relations (8), one can write inversely
ε0
x¼ An
22Nx−An
12Nyþ Bn
11χxþ Bn
12χy;
ε0¼ An
11Ny−An
12Nxþ Bn
21χxþ Bn
22χy;
γ0
xy¼ An
66þ 2Bn
in which
An11¼1
Δ A11þEsAs
ss
; An
12¼A12
Δ ; An22¼1
Δ A22þErAr
sr
;
An66¼ 1
A ; Δ ¼ A11þEsAs
s
A22þErAr
s
−A2
12:
Fig 1 Configuration of an eccentrically stiffened cylindrical shell.
Trang 4Bn11¼ An
22ðB11þ CsÞ−An
12B12; Bn
22¼ An
11ðB22þ CrÞ−An
12B12;
Bn12¼ An
22B12−An
12ðB22þ CrÞ; Bn
21¼ An
11B12−An
12ðB11þ CsÞ;
Bn66¼B66
Substituting Eq.(12)into Eq.(9)leads to
Mx¼ Bn
11Nxþ Bn
21Ny−Dn
11χx−Dn
12χy;
My¼ Bn
12Nxþ Bn
22Ny−Dn
21χx−Dn
22χy;
Mxy¼ Bn
66Nxy−2Dn
in which
Dn11¼ D11þEsIs
ss −ðB11þ CsÞBn
11−B12Bn21;
Dn22¼ D22þErIr
sr −B12Bn12−ðB22þ CrÞBn
22;
Dn12¼ D12−ðB11þ CsÞBn
12−B12Bn22;
Dn21¼ D12−B12Bn11−ðB22þ CrÞBn
21;
The nonlinear equations of motion of a cylindrical thin shell based
on the classical shell theory and the assumption [12,14,26]
uo ow and vo ow, ρ1ð∂2u=∂t2Þ-0, ρ1ð∂2v=∂t2Þ-0 are given by
[4,14]
∂Nx
∂x þ ∂
Nxy
∂y ¼ 0;
∂Nxy
∂x þ ∂
Ny
∂y ¼ 0;
∂2Mx
∂x2 þ 2 ∂2Mxy
∂x∂y þ ∂
2My
∂y2 þ Nx ∂2w
∂x2 þ ∂2w0
∂x2
þ2Nxy ∂2w
∂x∂yþ ∂
2w0
∂x∂y
þ Ny ∂2w
∂y2þ ∂2w0
∂y2
þ1
RNy¼ ρ1∂2w
∂t2 ; ð16Þ where
ρ1¼Zh=2
−h=2ρðzÞdz þ ρs
As
ssþ ρr
Ar
sr ¼ ρmþ ρc−ρm
k þ 1
h þρs
As
ss
þ ρr
Ar
with
ρs¼ ρm; ρr¼ ρmfor metal stiffeners;
ρs¼ ρc; ρr¼ ρcfor ceramic stiffeners:
Considering thefirst two of Eq.(16), a stress functionφ may be
defined as
Nx¼ ∂2φ
∂y2; Ny¼ ∂2φ
Substituting Eq.(11)into the compatibility Eqs.(4)and(14)into
the third of Eq (16), taking into account Eqs (2) and (18)
neglecting small terms of higher second order with respect to
w0, yields
An11∂4φ
∂x4þ ðAn
66−2An
12Þ ∂4φ
∂x2∂y2þ An
22∂4φ
∂y4þ Bn
21∂4w
∂x4
þðBn
11þ Bn
22−2Bn
66Þ ∂4w
∂x2∂y2þ Bn
12∂4w
∂y4þ1 R
∂2w
∂x2
− ∂x∂y∂2w
−∂∂x2w2
∂2w
∂y2
−2∂x∂y∂2w∂2w0
∂x∂y
þ ∂2w
∂x2
∂2w0
∂y2 þ ∂2w
∂y2
∂2w0
ρ1∂2w
∂t2 þ Dn
11∂4w
∂x4 þ ðDn
12þ Dn
21þ 4Dn
66Þ ∂4w
∂x2∂y2þ Dn
22∂4w
∂y4−Bn
21∂4φ
∂x4
−ðBn
11þ Bn
22−2Bn
66Þ ∂4φ
∂x2∂y2−Bn
12∂4φ
∂y4−1R∂∂x2φ2−∂∂y2φ2 ∂∂x2w2 þ ∂2w0
∂x2
þ2 ∂2φ
∂x∂y ∂
2w
∂x∂yþ ∂
2w0
∂x∂y
−∂∂x2φ2
∂2w
∂y2þ ∂2w0
∂y2
Eqs.(19)and(20)are a nonlinear equation system in terms of two dependent unknowns w andφ They are used to investigate the dynamic characteristics of imperfect ES-FGM circular cylind-rical shells
3 Nonlinear static and dynamic buckling analysis Suppose that an imperfect ES-FGM cylindrical shell is simply supported and subjected to axial compressive load r0¼ r0h where
r0is the average axial stress on the shell's end sections, positive when the shells subjected to axial compression (in N/m2) Thus, the boundary conditions considered in the current study are
w ¼ 0; Mx¼ 0; Nx¼ −r0h; Nxy¼ 0; at x ¼ 0; L: ð21Þ The deflection of shell is satisfying the mentioned condition (21) is represented by
w ¼ f ðtÞ sinmπx
L sin
ny
where f ðtÞ is the time dependent total amplitude, m is the number
of half waves in axial direction and n is the number of wave in circumferential direction
The initial-imperfection w0is assumed to be the same form of the deflection as
w0¼ f0 sinmπx
L sin
ny
where f0is the known imperfect amplitude
Table 1 Comparisons of dynamic critical buckling stress r dcr (MPa) and dynamic coefficient
τ cr ¼ rdcr=r scr of perfect un-stiffened FGM cylindrical shells under linear-time compression.
rdcrðm; nÞ τ cr ¼ r dcr
r scr rdcrðm; nÞ τ cr ¼ r dcr
r scr R/h ¼500, L/R ¼2, c ¼ 100 MPa/s
k ¼0.2 194.94(2,11) 1.030 194.94(2,11) 1.030
k ¼1.0 169.94(2,11) 1.034 169.94(2,11) 1.034
k ¼5.0 149.98(2,11) 1.041 150.25(2,11) 1.040 R/h ¼500, L/R ¼2, k ¼ 0.5
c¼ 100 MPa/s 181.68(2,11) 1.032 181.67(2,11) 1.032 c¼ 50 MPa/s 179.38(2,11) 1.019 179.37(2,11) 1.019 c¼ 10 MPa/s 177.02(2,11) 1.006 177.97(1,8) 1.009 L/R ¼2, k¼ 0.2, c ¼ 100 MPa/s
R/h ¼ 800 124.67(2,12) 1.049 124.91(2,12) 1.051 R/h ¼ 600 162.18(3,14) 1.026 162.25(3,14) 1.027 R/h ¼ 400 239.56(5,15) 1.013 239.18(5,15) 1.011
Table 2 Comparisons of static critical buckling load per unit length r scr ¼ rscrh (10 6 N/m)
of perfect stiffened homogeneous cylindrical shells under axial compression.
Present Brush and Almroth [25] Difference (%)
50 rings, 50 stringers, L ¼ 1 m, R¼ 0.5 m, E ¼ 7 1010N=m 2 , υ ¼ 0:3,
d r ¼ d s ¼ 0:0025 m, h r ¼ h s ¼ 0:01 m Internal stiffeners
R/h ¼ 100 3.0725(6,7) 3.0906(6,7) 0.59 R/h ¼ 200 1.4147(6,7) 1.4328(6,7) 1.28 R/h ¼ 500 0.6924(5,6) 0.7057(5,6) 1.92 External stiffeners
R/h ¼ 100 3,9529(9,3) 3.9551(9,2) 0.06 R/h ¼ 200 2.1410(9,4) 2.1469(9,4) 0.28 R/h ¼ 500 1.2764(6,6) 1.2897(6,6) 1.04
D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 193
Trang 5Substituting Eqs (22) and (23) into Eq (19) and solving
obtained equation for unknownφ lead to
φ ¼ φ1 cos2mπx
L þ φ2 cos2ny
R −φ3 sinmπx
L sin
ny
R −r0hy
2
2; ð24Þ where denote
φ1¼ n2λ2
32m2π2An11f ðf þ 2f0Þ;
φ2¼ m2π2
32n2λ2An22f ðf þ 2f0Þ;
φ3¼ B
n
21m4π4þ ðBn
11þ Bn
22−2Bn
66Þm2n2π2λ2þ Bn
12n4λ4−L 2
Rm2π2
An11m4π4þ ðAn
66−2An
12Þm2n2π2λ2þ An
22n4λ4 f;
f ¼ f ðtÞ; λ ¼ L
Substituting the expressions (22)–(24) into Eq.(20)and applying
Galerkin method to the resulting equation yield
ρ1L4€f þ D þ B2
A
! f
þ Gf ðf þ f0Þðf þ 2f0Þ−L2
m2π2hr0ðf þ f0Þ ¼ 0; ð26Þ where
A ¼ An11m4π4þ ðAn
66−2An
12Þm2n2π2λ2þ An
22n4λ4;
B ¼ Bn21m4π4þ ðBn
11þ Bn
22−2Bn
66Þm2n2π2λ2þ Bn
12n4λ4−L
2
Rm
2π2;
D ¼ Dn11m4π4þ ðDn
12þ Dn
21þ 4Dn
66Þm2n2π2λ2þ Dn
22n4λ4;
4λ4
16An11þm4π4
16An22
!
Introduce parameters
D ¼ D
h3; B ¼B
h; A ¼ Ah; G ¼G
h; ξ ¼ f
h; ξ0¼f0
Fig 2 Effect of k on the static post-buckling of un-stiffened shells.
Fig 3 Effect of k on the static post-buckling of external ring and stringer stiffened
shells.
Fig 4 Effect of k on the static post-buckling of internal ring and stringer stiffened
shells.
Fig 5 Dynamic responses of un-stiffened shell under step loading of infinite duration.
Fig 6 Dynamic response of external rings and stringers stiffened shell under step loading of infinite duration.
Fig 7 Dynamic response of internal rings and stringers stiffened shell under step loading of infinite duration.
Trang 6the non-dimension form of Eq.(26)is written as
ρ1L4
h3 €ξ þ D þB2
A
!
ξ
þ Gξðξ þ ξ0Þðξ þ 2ξ0Þ− Lh
2
m2
π2
3.1 Static buckling and post-buckling analysis
Omitting the term of inertia, Eq.(29)leads to
r0¼ h
2
L2m2π2 D þB
2
A
! ξ
ðξ þ ξ0Þþ
h2
L2m2π2Gξðξ þ 2ξ0Þ: ð30Þ Puttingξ0¼ 0 in Eq.(30), yields
r0¼ h
2
L2m2π2 D þB
2
A
!
2
From Eq.(31), by takingξ ¼ 0 the buckling stress of perfect
ES-FGM cylindrical shells can be determined as
rsbu¼ h
2
L2m2π2 D þB
2
A
!
The static critical buckling stress of perfect ES-FGM cylindrical
shells are determined by condition rscr¼ minrsbuvs ðm; nÞ and the
static post-buckling curves of perfect and imperfect shells may be
traced by using Eqs.(30)and(31)with the same buckling mode
shape of critical buckling stress for evaluate static behavior of
these structures
3.2 Dynamic buckling analysis For dynamic buckling analysis, this paper investigates two cases as following
Case 1 Consider a cylindrical shell subjected to the axial com-pression linearly varying on time r0¼ ct in which c is a loading speed By using the Runge–Kutta method, the responses of ES-FGM cylindrical shells can be determined from Eq (29) The dynamic critical time tcr can be obtained according to Budiansky–Roth criterion [20]: for large value of loading speed, the amplitude– time curve of obtained displacement response increases sharply and this curve obtain a maximum by passing from the slope point and at the corresponding time t ¼ tcrthe stability loss occurs Here,
tcr is called critical time and the corresponding dynamic critical buckling stress rdcr¼ ctcr and dynamic coefficient τcr¼ rdcr=rscr Case 2 Assume that a shell is conducted for step loading of
infinite duration r0¼ const; ∀t The dynamic critical load is found based on the criterion mentioned in[27]: the load corre-sponding to a sudden jump in the maximum average deflection in the time history of the shell is taken as the critical buckling step load
4 Numerical results and discussions
To validate the present formulation, two comparisons on critical load are carried out with results from open literatures First, the dynamic buckling of perfect un-stiffened FGM cylind-rical shells under linear-time compression is given in Table 1, which was also analyzed by Huang and Han[19]using the energy method and classical shell theory As can be seen, the good agreements are observed
Second, the present static buckling load (Table 2) of stiffened homogeneous cylindrical shells under axial compression is com-pared with the results in the monograph of Brush and Almroth [25](based on equations in page 180) where the smeared stiffen-ers technique, equilibrium path and classical shell theory are used This comparison once again also shows that the good agreements are obtained
To illustrate the proposed approach of eccentrically stiffened FGM cylindrical shells, the stiffened and un-stiffened FGM cylind-rical shells are considered with R ¼ 0:5 m, L ¼ 0:75 m, R=h ¼ 250 The combination of materials consists of aluminum Em¼ 7
1010N/m2, ρm¼ 2702 kg/m3 and alumina Ec¼ 38 1010
N/m2,
ρc¼ 3800 kg/m3 The compressive stress of dynamic analysis is taken to be r0¼ 1010t Poisson's ratio ν is chosen to be 0.3 for simplicity The height of stiffeners is equal to 0:01 m, its width
0:0025 m The material properties are Es¼ Ec and Er¼ Ec,ρs¼ ρc
and ρr¼ ρc with internal stringer stiffeners and internal ring
Fig 8 Effect of k on the dynamic responses of un-stiffened shells under linear-time
compression.
Fig 9 Effect of k on the dynamic responses of external ring and stringer stiffened
shells under linear-time compression.
Fig 10 Effect of k on the dynamic responses of internal ring and stringer stiffened
D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 195
Trang 7stiffeners; Es¼ Em, Er¼ Em, ρs¼ ρm and ρr¼ ρm with external
stringer stiffeners and external ring stiffeners, respectively The
stiffener system includes 15 ring stiffeners and 63 stringer
stiffen-ers distributed regularly in the axial and circumferential
direc-tions, respectively
InFigs 2–4, the static post-buckling curves of un-stiffened and
stiffened shells are traced by Eqs.(30)and(31)of perfect (ξ0¼ 0)
and imperfect (ξ0¼ 0:1) cases versus three different values of
volume fraction index k (¼0.2, 1, 5) As can be seen, the
post-buckling curves are lower with increasing values of k
Further-more, the post-buckling curves of imperfect shells are lower than
those of perfect shells when deflection is small and post-buckling
curves of imperfect shells is higher than that of perfect shells
when the deflection is sufficiently large
By using the fourth-order Runge–Kutta method, Eq (29) is solved to obtain the dynamic responses of perfect (ξ0¼ 0) shells under step loading of infinite duration Dynamic responses of un-stiffened and un-stiffened shells are presented inFigs 5–7 As can be seen, there is a sudden jump in the value of the average deflection when the axial compression reaches the critical value In addition, the dynamic critical step load corresponding to internal ring and stinger stiffened shell is biggest This value is bigger than about 1.4 times in comparison with the external ring and stinger stiffened shell
perfect and imperfect un-stiffened and stiffened shells under linear-time compression Thesefigures also show that there is no
definite point of instability as in static analysis Rather, there is a
Table 3
Effect of k on critical static and dynamic buckling stress r 0 ( 10 8 N/m 2 ).
Un-stiffened
External rings and stringers
Internal rings and stringers
Table 4
Effects of number, type and position of stiffeners on critical static and dynamic buckling stress r 0 ( 10 8
N/m 2
).
ER, external rings; IR, internal rings; ES, external stringers; IS, internal stringers.
Fig 11 Effect of external ring and external stringer stiffeners on the static
post-buckling curves.
Fig 12 Effect of internal ring and internal stringer stiffeners on the static post-buckling curves.
Trang 8region of instability where the slope of ξ vs t curve increases
rapidly (in perfect shell cases) According to the Budiansky–Roth
criterion[20], the critical time t can be taken as an intermediate
value of this region Therefore, one can choose the inflexion point
of curve i.e d2ξ=dt2jt ¼ tcr¼ 0 as Huang and Han[19] This region is clearly recognized with perfect shells but it is very difficult to
Fig 13 Effect of internal stiffeners and external stiffeners on the static
post-buckling curves.
Fig 14 Effect of position of stiffeners on the static post-buckling of stiffened shells.
Fig 15 Effect of external ring and external stringer stiffeners on the dynamic
responses of shells under linear-time compression.
Fig 16 Effect of internal ring and internal stringer stiffeners on the dynamic
responses of shells under linear-time compression.
Fig 17 Effect of internal stiffeners and external stiffeners on the dynamic responses of shells under linear-time compression.
Fig 18 Effect of stiffeners position on the dynamic responses of shells under linear-time compression.
Table 5 Effect of R/h on critical static and dynamic buckling load per unit length r 0 ( 10 6
N/m).
Un-stiffened Static 6.247(6,9) 0.999(7,15) 0.250(9,21) 0.062(25,20) Dynamic r 0 ¼ const 6.247(6,9) 0.999(7,15) 0.250(9,21) 0.062(25,20) Dynamic r 0 ¼ ct 6.457(6,9) 1.062(7,15) 0.277(9,21) 0.069(25,20) External rings and stringers
Static 8.341(4,8) 2.932(4,9) 1.859(4,9) 1.292(4,8) Dynamic r0¼ const 8.341(4,8) 2.932(4,9) 1.859(4,9) 1.293(4,8) Dynamic r 0 ¼ ct 8.607(4,8) 3.049(4,9) 1.920(4,9) 1.326(4,8) Internal rings and stringers
Static 9.964(3,7) 4.070(3,7) 2.380(3,6) 1.480(3,6) Dynamic r0¼ const 9.964(3,7) 4.070(3,7) 2.381(3,6) 1.480(3,6) Dynamic r 0 ¼ ct 10.288(3,7) 4.216(3,7) 2.457(3,6) 1.528(3,6)
Fig 19 Effect of R/h on the static post-buckling of un-stiffened shells.
D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 197
Trang 9define that with imperfect shells Therefore, critical dynamic
buckling compressions of imperfect un-stiffened and stiffened
cylindrical shells cannot accurately predict by Budiansky–Roth
criterion (like a remark given by Huang and Han [19] for FGM
un-stiffened shells) Thisfigure also shows that a sudden jump in the value of deflection occurs earlier when k increases and it corresponds a smaller dynamic buckling compression
In the nextfigures, the dynamic response is traced by relation
of deflection ratio ξ versus excited load r0(where r0¼ ct)
of stiffened and un-stiffened cylindrical shells vs four different values of volume fraction index k ¼(0.2,1,5,10) With the same input parameters, the effectiveness of stiffeners is obviously proven; the critical buckling stress of stiffened shell is greater than one of un-stiffened shell.Table 3also shows that the dynamic critical stress decreases with the increase of the volume fraction index k and the buckling modes mð ; nÞ seem smaller with stiffened shells The critical parameterτcris larger than 1, it denotes that the dynamic critical buckling stress of linear-time compression case is larger than static buckling stress The largest value of and τcr is equal to 1.085 for the internal rings and stringers stiffened shell with k ¼10 and the smallestτ ¼ 1:028 corresponds to external
Fig 20 Effect of R/h on the static post-buckling of external stiffened shells.
Fig 21 Effect of R/h on the static post-buckling of internal stiffened shells.
Fig 22 Effect of R/h on the dynamic responses of un-stiffened shells under
linear-time compression.
Fig 23 Effect of R/h on the dynamic responses of internal ring and stringer
stiffened shells under linear-time compression.
Fig 24 Effect of R/h on the dynamic responses of internal ring and stringer stiffened shells under linear-time compression.
Fig 25 Effect of loading speed on the dynamic responses of un-stiffened shells.
Fig 26 Effect of loading speed on the dynamic responses of internal stiffened shells.
Trang 10rings and stringers stiffened shell with k ¼0.2 In addition, when
the shell subjected to the step loading of infinite duration, it seems
that the dynamic critical buckling compression is approximately
equal to the static critical buckling compression
Effect of the stiffener number, type and position of stiffeners on
the nonlinear critical buckling stress is given inTable 4 Clearly, the
ring or stringer stiffeners lightly influence to the critical buckling
stress of shells But, the combination of ring and stringer stiffeners
has a considerable effect on the stability of shell Especially, the
critical buckling stress of internal rings and stringers stiffened
shell is greatest and the critical buckling stress of internal rings
stiffened shell is smallest When the number of stiffeners
increases, it is evident that critical buckling stresses increase
Figs 11–14show the effect of type and position of stiffeners on
the static post-buckling of stiffened and un-stiffened shells (k ¼1,
15 rings and 63 stringers) According to the critical buckling
values, the post-buckling of un-stiffened shells is lower than one
of stiffened shells For stiffened shells, the post-buckling of
internal rings stiffened shell is the lowest and one of internal
rings and stringers stiffened shell is the highest
Figs 15–18show the effect of type and position of stiffeners on
the dynamic response of stiffened and un-stiffened shells under
linear-time compression (k¼ 1, 15 rings and 63 stringers) For one
type of stiffeners shells (Figs 15 and 16), it seem that the
amplitude responses of perfect rings stiffened shells are smallest
and those of perfect stringers stiffened shells are the biggest
In the results considered, the slope of instability region of stringer
stiffened shells is smaller
The effects of R/h on the behavior of buckling loads per unit
length (r0¼ r0h) are illustrated in Table 5 Clearly, the critical
buckling load decreases when the R/h ratio increases, the stiffeners
are more effective with thinner shells When R/h ratio increases,
the critical buckling loads of un-stiffened shells strongly decrease
(about 100 times for variation of R/h from 100 to 1000) but lightly
with stiffened shells (about 6 times for variation of R/h from 100
to 1000)
stiffened shells The post-buckling curves of shells are much
higher when R/h ratio decreases
The dynamic responses of un-stiffened and stiffened shells
under linear-time compression are presented in Figs 22–24
As can be observed, maximal amplitude responses of instability
region increase when R/h ratio increases Thesefigures also show
that the slope of instability region of thinner shells is greater
Effects of the loading speed on the dynamic responses of
un-stiffened and internal un-stiffened shells under linear-time
compres-sion are shown inFigs 25and26 Three values of loading speed
are used, i.e c ¼1010, c ¼2 1010, c ¼5 1010 Clearly, the critical
dynamic buckling loads and amplitude response increase when
the loading speed increases It mean that rapidly compressed
cylindrical shell will buckle at a higher critical stress than a very
slowly compressed cylindrical shell
5 Conclusions
A formulation of governing equations of eccentrically stiffened
functionally graded circular cylindrical thin shells based on the
classical shell theory and the smeared stiffeners technique with
von Karman–Donnell nonlinear terms is presented in this paper
By using the Galerkin method the explicit expressions of static
buckling compression, post-buckling load–deflection curve and
the nonlinear dynamic equation of ES-FGM circular cylindrical
shells are obtained, the later is solved by using the Runge–Kutta
method and the criteria for determining critical dynamic
compres-sions are applied
Some conclusions can be obtained from the present analysis: (i) Stiffeners enhance the static and dynamic stability and load-carrying capacity of FGM circular cylindrical shells
(ii) Combination of ring and stringer stiffeners has a large effect
on the stability of shell The critical buckling compression of internal rings and stringers stiffened shell is greatest (iii) In dynamic linear-time load case, there is no definite point of instability as in static analysis Rather, there is a region of instability where the slope ofξ vs t curve increases rapidly in perfect shell cases
(iv) For imperfect FGM shell, it is difficult to accurately predict the critical buckling compression
(v) The dynamic critical buckling compressions of linear-time compression case is larger than static critical buckling com-pressions (τcr is about 1.028–1.085) and the dynamic critical buckling compression of step loading of infinite duration
is approximately equal to the static critical buckling compression
(vi) Initial geometrical imperfection, radius-to-thickness ratio, position, type and number of stiffeners significantly influence
on the static and dynamic behavior of cylindrical shell Major purpose of this study is to analyze the global buckling and post-buckling behavior of FGM cylindrical shells reinforced by closely spaced stiffener system For local buckling analysis, the approach of Stamatelos et al.[28]may be used
Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2012.02
References
[1] Shen HS Postbuckling analysis of pressure-loaded functionally graded cylind-rical shells in thermal environments Eng Struct 2003;25(4):487–97 [2] Shen HS Postbuckling of axially-loaded FGM hybrid cylindrical shells in thermal environments Compos Sci Technol 2005;65(11–12):1675–90 [3] Bahtui A, Eslami MR Coupled thermoelasticity of functionally graded cylind-rical shells Mech Res Commun 2007;34(1):1–18
[4] Huang H, Han Q Buckling of imperfect functionally graded cylindrical shells under axial compression Eur J Mech—A/Solids 2008;27(6):1026–36 [5] Huang H, Han Q Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells Int J Mech Sci 2009;51 (7):500–7
[6] Huang H, Han Q Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure Int J Non-Linear Mech 2009;44(2):209–18
[7] Huang H, Han Q Research on nonlinear postbuckling of FGM cylindrical shells under radial loads Compos Struct 2010;92(6):1352–7
[8] Shen HS Postbuckling of shear deformable FGM cylindrical shells surrounded
by an elastic medium Int J Mech Sci 2009;51(5):372–83 [9] Sofiyev AH Buckling analysis of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation Mech Res Commun 2010;37 (6):539–44
[10] Zozulya VV, Zhang Ch A high order theory for functionally graded axisym-metric cylindrical shells Int J Mech Sci 2012;60(1):12–22
[11] Ng TY, Lam KY, Liew KM, Reddy JN Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading Int J Solids Struct 2001;38(8):1295–309
[12] Darabi M, Darvizeh M, Darvizeh A Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading Compos Struct 2008;83(2):201–11
[13] Chen WQ, Bian ZG, Ding HJ Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells Int J Mech Sci 2004;46(1):159–71 [14] Sofiyev AH, Schnack E The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading Eng Struct 2004;26 (10):1321–31
[15] Sofiyev AH The stability of compositionally graded ceramic–metal cylindrical shells under aperiodic axial impulsive loading Compos Struct 2005;69 (2):247–57
D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 199