Nonlinear dynamical analysis of eccentrically stiffened functionally gradedcylindrical panels Dao Huy Bicha, Dao Van Dunga, Vu Hoai Namb,⇑ a Vietnam National University, Hanoi, Viet Nam
Trang 1Nonlinear dynamical analysis of eccentrically stiffened functionally graded
cylindrical panels
Dao Huy Bicha, Dao Van Dunga, Vu Hoai Namb,⇑
a
Vietnam National University, Hanoi, Viet Nam
b
Faculty of Civil Engineering, University of Transport Technology, Ha Noi, Viet Nam
a r t i c l e i n f o
Article history:
Available online 28 March 2012
Keywords:
Functionally graded material (FGM)
Dynamical analysis
Critical dynamic buckling load
Vibration
Cylindrical panel
Stiffeners
a b s t r a c t
Based on the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened function-ally graded cylindrical panels with geometricfunction-ally imperfections are derived in this paper The character-istics of free vibration and nonlinear responses are investigated The nonlinear dynamic buckling of cylindrical panel acted on by axial loading is considered The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth Some numerical results are given and compared with the ones of other authors
Ó 2012 Elsevier Ltd All rights reserved
1 Introduction
Functionally graded materials (FGMs) are composite materials
which have mechanical properties varying continuously from one
surface to the other of structure The concept of functionally
graded material was proposed in 1984[1]and it is often used in
heat-resistance structure as elements in aerospace and nuclear
reactors[2] Today, the application of this material is getting
var-ied, so the problems of static and dynamic behaviors of structures
such as FGM plates and shells have been properly noticed
For dynamical analysis of FGM shells, many studies have been
focused on the characters of vibration and behavior of buckling
of shells Ng et al.[3]and Darabi et al.[4]presented respectively
linear and nonlinear parametric resonance analyses for FGM
cylin-drical shells Loy et al.[5]and Pradhan et al.[6]studied the free
vibration characteristics of FGM cylindrical shells By using
Galerkin technique together with Ritz type variational method,
Sofiyev[7]and Sofiyev and Schnack[8]obtained critical
parame-ters for cylindrical thin shells under linearly increasing dynamic
torsional loading, and under a periodic axial impulsive loading
By using a higher order shell theory and a finite element solving
method, Shariyat [9] investigated nonlinear dynamic buckling
problems of axially and laterally preloaded FGM cylindrical shells
under transient thermal shocks Geometrical imperfection effects
were also included in his research Using the similar method, he
also presented a dynamic buckling analysis for FGM cylindrical
shells under complex combinations of thermo–electro-mechanical loads[10] Huang and Han[11]presented nonlinear dynamic buck-ling problems of functionally graded cylindrical shells subjected to time-dependent axial load by using Budiansky–Roth dynamic buckling criterion [12] Various effects of the inhomogeneous parameter, loading speed, dimension parameters; environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling were discussed Liew et al.[13]presented the nonlinear vibration analysis for layered cylindrical panels contain-ing FGMs and subjected to a temperature gradient ariscontain-ing from steady heat conduction through the panel thickness
Ganapathi [14] studied the dynamic stability behavior of a clamped FGMs spherical shell structural element subjected to external pressure load He solved the governing equations employ-ing the Newmark’s integration technique coupled with a modified Newton–Raphson iteration scheme Sofiyev [15–17]studied the vibration and buckling of the FGM truncated conical shells under dynamic axial loading Based on first-order shear deformation the-ory, the dynamic thermal buckling behavior of functionally graded spherical caps is studied by Prakash et al.[18] Dynamic buckling of functionally graded materials truncated conical shells subjected to normal impact loads is discussed by Zang and Li[19]
For FGM shallow shells, Alijani et al.[20], Chorfi and Houmat [21]and Matsunaga[22]investigated nonlinear forced vibrations
of FGM doubly curved shallow shells with a rectangular base Non-linear dynamical analysis of imperfect functionally graded material shallow shells subjected to axial compressive load and transverse load was studied by Bich and Long[23]and Dung and Nam[24] The motion, stability and compatibility equations of these 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved.
⇑ Corresponding author.
E-mail address: hoainam.vu@utt.edu.vn (V.H Nam).
Contents lists available atSciVerse ScienceDirect
Composite Structures
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 s t r u c t
Trang 2structures were derived using the classical shell theory The
non-linear transient responses of cylindrical and doubly-cuvred
shal-low shells subjected to excited external forces were obtained and
the dynamic critical buckling loads were evaluated based on the
displacement responses using Budiansky–Roth dynamic buckling
criterion[12]
However, there are very little researches on nonlinear dynamic
problems of imperfect eccentrically stiffened functionally graded
shells Recently, Najafizadeh et al.[25] studied statical buckling
behaviors of FGM cylindrical shell Bich et al [26]have studied
the nonlinear statical postbuckling of eccentrically stiffened
func-tionally graded plates and shallow shells
Following the idea of Ref.[26], this paper establishes dynamics
governing equations and investigates nonlinear vibration and
dy-namic buckling of imperfect reinforced FGM cylindrical panel It
shows the influences of stiffener, of volume-fraction index, of
ini-tial imperfection and of geometrical parameters to the dynamic
characteristics of panels
2 Eccentrically stiffened FGM cylindrical panels (ES-FGM
cylindrical panels)
2.1 Functionally graded material
Functionally graded material in this paper, is assumed to be
made from a mixture of ceramic and metal with the
volume-frac-tions given by a power law
Vmþ Vc¼ 1;
Vc¼ VcðzÞ ¼ 2z þ h
2h
;
where h is the thickness of panel; k P 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
;
qðzÞ ¼qmVmþqcVc¼qmþ ðqcqmÞ 2z þ h
2h
;
ð1Þ
the Poissons’s ratiomis assumed to be constant
2.2 Constitutive relations and governing equations
Consider a functionally graded cylindrical thin panel in-plane
edges a and b The panel is reinforced by eccentrically longitudinal
and transversal stiffeners The cylindrical panel is assumed to have
a relative small rise as compared with its span Let the (x1, x2) plane
of the Cartesian coordinates overlaps the rectangular plane area of
the panel Note that the middle surface of the panel generally is
de-fined in terms of curvilinear coordinates, but for the cylindrical
pa-nel, so the Cartesian coordinates can replace the curvilinear
coordinates on the middle surface (seeFig 1)
According to the classical shell theory and geometrical nonlin-earity in von Karman–Donnell sense, the strains at the middle sur-face and curvatures are related to the displacement components u,
v, w in the x1, x2, z coordinate directions as[28]
e0
¼@u
@x1þ1 2
@w
@x1
; v1¼@
2w
@x2;
e0¼@v
@x2
1
Rw þ
1 2
@w
@x2
; v2¼@
2w
@x2;
c0
12¼@u
@x2þ@v
@x1þ@w
@x1
@w
@x2
2
w
@x1@x2
;
ð2Þ
where R is radius of the cylindrical shell
The strains across the shell thickness at a distance z from the mid-surface are
e1¼e0 zv1; e2¼e0 zv2; c12¼c0
From Eq.(3) the strains must be relative in the deformation compatibility equation
@2e0
@x2 þ@
2e0
@x2 @
2c0 12
@x1@x2
2w
@x1@x2
!2
@
2w
@x2
@2w
@x21 R
@2w
Hook’s stress–strain relation is applied for the shell
rsh
1 ¼ EðzÞ
1 m2ðe1þme2Þ;
rsh
2 ¼ EðzÞ
1 m2ðe2þme1Þ;
ssh
12¼ EðzÞ 2ð1 þmÞc12;
ð5aÞ
and for stiffeners
rst
1 ¼ E0e1;
rst
where E0is Young’s modulus of ring and stringer stiffeners 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 panel, we obtain the expressions for force and moment resultants of an ES-FGM cylindrical panel
N1¼ A11þE0A1
s1
e0þ A12e0 ðB11þ C1Þv1 B12v2;
N2¼ A12e0
þ A22þE0A2
s2
e0
B12v1 ðB22þ C2Þv2;
N12¼ A66c 0
12 2B66v12;
ð6Þ
M1¼ ðB11þ C1Þe0þ B12e0 D11þE0I1
s1
v1 D12v2;
M2¼ B12e0
þ ðB22þ C2Þe0
D12v1 D22þE0I2
s2
v2;
M12¼ B66c 0
12 2D66v12;
ð7Þ
0
z
1
z
1
z 0
2
Trang 3where Aij, Bij, Dij(i, j = 1, 2, 6) are extensional, coupling and bending
stiffnesses of the panel without stiffeners
A11¼ A22¼
Zh=2
h=2
EðzÞ
1 m2dz ¼ E1
1 m2; A12¼
Z h=2
h=2
EðzÞm
1 m2dz ¼ E1m
1 m2;
A66¼
Z h=2
h=2
EðzÞ
2ð1 þmÞdz ¼
E1
2ð1 þmÞ;
B11¼ B22¼
Z h=2
h=2
zEðzÞ
1 m2dz ¼ E2
1 m2; B12¼
Z h=2
h=2
zEðzÞm
1 m2dz ¼ E2m
1 m2;
B66¼
Z h=2
h=2
zEðzÞ
2ð1 þmÞdz ¼
E2
2ð1 þmÞ;
D11¼ D22¼
Z h=2
h=2
z2EðzÞ
1 m2dz ¼ E3
1 m2; D12¼
Z h=2
h=2
z2EðzÞm
1 m2 dz ¼ E3m
1 m2;
D66¼
Z h=2
h=2
z2EðzÞ
2ð1 þmÞdz ¼
E3
2ð1 þmÞ;
ð8Þ
with
E1¼ EmþEc Em
k þ 1
h; E2¼ ðEc EmÞkh
2
2ðk þ 1Þðk þ 2Þ;
E3¼ Em
12þ ðEc EmÞ
1
k þ 3
1
k þ 2þ
1 4k þ 4
h3;
I1¼d1h
3
1
12 þ A1z
2; I2¼d2h
3 2
12 þ A2z
2:
ð9Þ
and
C1¼E0A1z1
s1
; C2¼E0A2z2
s2
;
z1¼h1þ h
2 ; z2¼
h2þ h
In above relations(6), (7) and (9)the quantityE0is the Young
modulus in the axial direction of the corresponding stiffener which
is assumed identical for both types of stiffeners, it takes the value
E0= Emif the full metal stiffeners are put at the metal-rich side of
the panel and conversely E0= Ecif the full ceramic ones at the
cera-mic-rich side Such FGM stiffened cylindrical panels provide
conti-nuity within panel and stiffeners and can be easier manufactured
The spacings of the longitudinal and transversal stiffeners are
de-noted by s1 and s2 respectively The quantities A1, A2 are the
cross-section areas of stiffeners and I1, I2, z1, z2are the second
mo-ments of cross section areas and the eccentricities of stiffeners
with respect to the middle surface of panel respectively
The strain-force resultant relations reversely are obtained from
Eq.(6)
e0
¼ A22N1 A12N2þ B11v1þ B12v2;
e0¼ A11N2 A12N1þ B 21v1þ B 22v2;
c0
12¼ A66þ 2B66v12;
ð10Þ
where
A11¼1
D A11þE0A1
s1
; A22¼1
D A22þE0A2
s2
;
A12¼A12
D ; A66¼ 1
A66
;
D¼ A11þE0A1
s1
A22þE0A2
s2
A212;
B
11¼ A22ðB11þ C1Þ A12B12;
B
22¼ A11ðB22þ C2Þ A12B12;
B
12¼ A22B12 A12ðB22þ C2Þ;
B
21¼ A11B12 A12ðB11þ C1Þ;
B
66¼B66
A :
ð11Þ
Substituting Eq.(10)into Eq.(7)yields
M1¼ B11N1þ B21N2 D11v1 D12v2;
M2¼ B12N1þ B22N2 D21v1 D22v2;
M12¼ B66N12 2D66v12;
ð12Þ
where
D
11¼ D11þE0I1
s1 ðB11þ C1ÞB11 B12B
21;
D22¼ D22þE0I2
s2
B12B12 ðB22þ C2ÞB22;
D
12¼ D12 ðB11þ C1ÞB
12 B12B
22;
D
21¼ D12 B12B
11 ðB22þ C2ÞB
21;
D
66¼ D66 B66B
66:
ð13Þ
The nonlinear equations of motion of a cylindrical thin panel based on the classical shell theory and the assumption (Refs [4,8,27]) u w andv w,q1@2u
@t 2! 0,q1@2v
@t 2! 0 are given by
@N1
@x1
þ@N12
@x2
¼ 0;
@N12
@x1 þ@N2
@x2¼ 0;
@2M1
@x2 þ 2@
2M12
@x1@x2þ@
2M2
@x2 þ N1
@2w
@x2þ 2N12
@2w
@x1@x2
þ N2
@2w
@x2 þ1
RN2þ q0¼q1@
2w
where
q1¼
Zh=2
h=2
qðzÞdz þq0 A1
s1þA2
s2
¼ qmþqcqm
k þ 1
h
þq0 A1
s1
þA2
s2
;
with
q0=qmfor metal stiffener,
q0=qcfor ceramic stiffener
The first two of Eq.(14)are satisfied automatically by choosing
a stress functionuas
N1¼@
2 u
@x2; N2¼@
2 u
@x2; N12¼ @
2 u
@x1@x2
The substitution of Eq.(10)into the compatibility Eqs.(4) and (12)into the third of Eq.(14), taking into account expressions(2) and (15), yields a system of equations
A 11
@4u
@x4 þ A66 2A12
/
@x2@x2þ A22
@4u
@x4þ B21
@4w
@x4
þ B 11þ B22 2B66 @4w
@x2@x2þ B12@
4w
@x4þ1 R
@2w
@x2
2w
@x1@x2
!2
@
2w
@x2
@2w
q1@
2w
@t2 þ D11@
4w
@x4þ D 12þ D21þ 4D66 @4w
@x2@x2þ D22@
4w
@x4
B 21
@4u
@x4 B
11þ B
22 2B 66
@x2@x2 B
12
@4u
@x4 1 R
@2u
@x2
@
2 u
@x2
@2w
@x2þ 2 @
2 u
@x1@x2
@2w
@x1@x2@
2 u
@x2
@2w
Trang 4For initial imperfection ES-FGM panels: The initial imperfection of the
panel considered here can be seen as a small deviation of the panel
middle surface from the perfect shape, also seen as an initial
deflec-tion which is very small compared with the panel dimensions, but
w0= w0(x1, x2) denote a known small imperfection, proceeding from
the motion Eqs.(16) and (17)of a perfect FGM cylindrical panel and
following to the Volmir’s approach[27]for an imperfection panel
we can formulate the system of motion equations for an imperfect
eccentrically stiffened functionally graded cylindrical panel
(imper-fect ES-FGM cylindrical panel) as
A11@
4u
@x4þ A 66 2A12 @4u
@x2@x2þ A22@
4u
@x4þ B21@
4ðw w0Þ
@x4
þ B
12
@4ðw w0Þ
@x4 þ B
11þ B
22 2B 66
ðw w0Þ
@x2@x2
2w
@x1@x2
!2
@
2w
@x2
@2w
@x2
2
4
3
5 þ @2w0
@x1@x2
!2
@
2w0
@x2
@2w0
@x2
2 4
3 5
þ1
R
@2ðw w0Þ
q1@
2w
@t2þ D11@
4
ðw w0Þ
@x4 þ D 12þ D21þ 4D66@4
ðw w0Þ
@x2@x2
þ D
22
@4ðw w0Þ
@x4 B
21
@4u
@x4 B
11þ B
22 2B 66
@x2@x2
B12
@4u
@x41
R
@2u
@x2@
2 u
@x2
@2w
@x2þ 2 @
2 u
@x1@x2
@2w
@x1@x2@
2 u
@x2
@2w
@x2¼ q0; ð19Þ
where w is a total deflection of panel
Hereafter, the couple of Eqs.(16) and (17)or of Eqs.(18) and
(19)are used to investigate the nonlinear vibration and dynamic
stability of panels They are nonlinear equations in terms of two
dependent unknowns w andu
3 Nonlinear dynamic analysis
3.1 Solution of the problem
Suppose that an imperfect ES-FGM cylindrical panel is simply
supported and subjected to uniformly distributed pressure of
intensity q0and in plane compressive load of intensities r0at its
cross-section (in Pa) Thus the boundary conditions considered in
the current study are
w ¼ 0; M1¼ 0; N1¼ r0h; N12¼ 0; at x1¼ 0; a;
w ¼ 0; M2¼ 0; N2¼ 0; N12¼ 0; at x2¼ 0; b: ð20Þ
where a and b are the lengths of in-plane edges of the panel
The mentioned conditions(20)can be satisfied identically if the
buckling mode shape is represented by
w ¼ f ðtÞ sinmpx1
npx2
where f(t) is time dependent total amplitude and m, n are numbers
of haft wave in axial and circumferential directions, respectively
The initial-imperfection w0is assumed to have similar form of
the panel deflection w, i.e
w0¼ f0sinmpx1
npx2
where f0is the known initial amplitude
Substituting Eqs.(21) and (22) into Eq (18) and solving
ob-tained equation for unknownulead to
u¼u1cos2mpx1
a þu2cos2npx2
b u3sinmpx1
npx2
b
r0hx
2
2;ð23Þ
where denote
u1¼ n
2k2f2
32m2A 11
;
2f2
32n2k2A 22
;
u3¼
B
21m4þ B11þ B22 2B66
m2n2k2þ B12n4k4a 2
p2 1m2
A11m4þ A 66 2A12
m2n2k2þ A22n4k4 :
ð24Þ
Substituting the expressions(21)–(23)into Eq.(19)and apply-ing Galerkin method to the resultapply-ing equation yield
M€f þ D þB
2
A
!
ðf f0Þ þ8mnk
2
3p2
B
Ad1d2ðf f0Þf þ H f
2
f2
þ K f 2 f2
f a
2h
p2r0m2f 4q0a4
where denote
M ¼a
4
p4q1;
A ¼ A
11m4
þ A66 2A12
m2n2
k2þ A22n4
k4;
B ¼ B
21m4þ B11þ B22 2B66
m2n2k2þ B12n4k4a
2
p2
1
Rm
2;
D ¼ D
11m4þ D
12þ D
21þ 4D 66
m2n2
k2þ D
22n4
k4;
2
3p2
B 21
A11þ
B 12
A22
2
6p4mn
n2k2
A11
1 R
d1d2;
K ¼ 1 16
m4
A 22
þn
4k4
A 11
!
; k¼a
b;
d1¼ ð1Þm 1;
d2¼ ð1Þn 1:
ð26Þ
The obtained Eq (25) is the governing equation for dynamic analysis of ES-FGM cylindrical panels in general Based on this equation the non-linear vibration of perfect and imperfect FGM cylindrical panels can be investigated and the dynamic buckling analysis of panels under various loading cases can be performed Particularly for a plate, R = 1 is taken in Eqs.(25) and (26) 3.2 Vibration analysis
Consider an imperfect ES-FGM cylindrical panel acted on by an uniformly distributed excited transverse load q0= Q sinXt and
r0= 0, the non-linear Eq.(25)has of the form
M€f þ D þB
2
A
!
ðf f0Þ þ8mnk
2
3p2
B
Ad1d2ðf f0Þf þ H f
2
f2
þ K f 2 f2
f ¼ 4a
4
By using Eq.(27), three aspects are taken into consideration: fundamental frequencies of natural vibration of ES-FGM panel and FGM panel without stiffeners, frequency–amplitude relation
of non-linear free vibration and non-linear response of ES-FGM pa-nel The non-linear dynamical responses of ES-FGM panels can be obtained by solving this equation combined with initial conditions
to be assumed as f ð0Þ ¼ 0; _fð0Þ ¼ 0 by using the Runge–Kutta iter-ation schema
Trang 5If the vibration is free and linear, Eq.(27)leads to
M€f þ D þB
2
A
!
from which the fundamental frequencies of natural vibration of
imperfect ES-FGM cylindrical panels can be determined by
xL¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
B2
A
! v
u
The equation of non-linear free vibration of a perfect panel can
be obtained from(27)
where denoting
H1¼x2
L¼1
B2
A
!
;
H2¼1
M
8mnk2
3p2
B
Ad1d2þ H
!
; H3¼K
Seeking solution as f(t) =gcosxt and applying procedure like
Galerkin method to Eq.(30), the frequency–amplitude relation of
non-linear free vibration is obtained
xNL¼xL 1 þ 8H2
3px2
L
gþ3H3
4x2 L
g2
wherexNLis the non-linear vibration frequency andgis the
ampli-tude of non-linear vibration
3.3 Nonlinear dynamic buckling analysis
Investigate the non-linear dynamic buckling of imperfect
ES-FGM cylindrical panels in some cases of active loads varying as
lin-ear function of time The aim of considered problems is to seek the
critical dynamic buckling loads They can be evaluated based in the
displacement responses obtained from the motion Eqs.(25) and
(26)
The criterion suggested by Budiansky and Roth[12]is employed
here as it is widely accepted This criterion is based on that, for
large value of loading speed, the amplitude-time curve of obtained
displacement response increases sharply depending on time and
this curve obtains a maximum by passing from the slope point
and at the corresponding time t = tcrthe stability loss occurs Here
t = tcris called critical time and the load corresponding to this
crit-ical time is called dynamic critcrit-ical buckling load
Consider an ES-FGM cylindrical panel subjected to axial load
r0(t) In this case q0= 0, Eq.(25)gives
M€f þ D þB
2
A
!
ðf f0Þ þ8mnk
2
3p2
B
Ad1d2ðf f0Þf þ H f
2 f2
þ K f2 f2
f m
2a2h
Omitting the term of inertia and putting f0= 0 in Eq.(33), yields
an equation for determining the static critical load of ES-FGM
cylindrical panels as
m2a2h
p2 r0f ¼ D þB
2
A
!
2
3p2
B
Ad1d2þ H
!
f2þ Kf3: ð34Þ
Taking f – 0, i.e considering the panel after the lost of stability
we obtain
m2a2h
p2 r0¼ D þB
2
A
!
2
3p2
B
Ad1d2þ H
!
From Eq.(35), the upper static buckling load can be determined
by putting f = 0
rupper¼ p2
m2a2h D þ
B2
A
!
and the lower static buckling load is found using the condition
dr 0
df ¼ 0, it follows
rlower¼ p2
m2a2h D þ
B2
A
^
H2
4K
!
where
b
H ¼8mnk
2
3p2
B
Ad1d2þ H:
Suppose axial load varying linearly on time r0= ct(c (in Pa/s) is a loading speed) and introduce parameters:
D ¼ D
h3; B ¼
B
h; A ¼ Ah; H ¼
H
h2; K ¼
K
h;
n¼f
h; n0¼
f0
h; s¼ r0
rscr¼ ct
rscr
;
ð38Þ
where rscr= minruppervs (m, n)
The non-dimension form of Eq.(33)is written as
1
S1
d2n
ds2þ D þB
2
A
!
ðn n0Þ þ8mnk
2
3p2
B
Ad1d1ðn n0Þnþ
"
þH n 2 n2
þk n 2 n2
ni p2
k4 b h2
rscr
k
2
where
S1¼p2r3 scrh
Solving Eq.(39)by Runge–Kutta method and applying Budian-sky–Roth criterion, the critical valuesdcr, the dynamic critical time
tdcr¼sdcr r scr
c and dynamical buckling load rdcr= ctdcrrespectively are obtained
4 Numerical results and discussions 4.1 Validation of the present formulation
In this section, first of all, the comparison on the fundamental frequency parameter ~x¼xLh ffiffiffiffiq
c
E c
q (xLis calculated from Eq.(29)) given by the present analysis with the results of Alijani et al.[20] based on the Donnell’s nonlinear shallow-shell theory, Chorfi and Houmat [21] based on the first-order shear deformation theory and Matsunaga[22]based on the two-dimensional (2D) higher-or-der theory for the perfect unreinforced FGM cylindrical panel
a¼ 1;h¼ 0:1
with simply supported movable edges is suggested The material properties in Refs [20–22]are aluminium and alu-mina, i.e Em= 70.109N/m2, qm= 2702 kg/m3 and Ec= 380.109
N/m2,qc= 3800 kg/m3 respectively The Poisson’s ratio is chosen
to be 0.3 As can be observed inTable 1, a very good agreement
is obtained in this comparison study
Next, the present frequencyxL(in Eq.(29)) is compared with the result of Szilard[29]and Troitsky[30]based on the classical assumptions of small deformations and thin plates Consider a sim-ply supported homogeneous plate that is biaxial stiffened with multiple stiffeners (seeFig 2) As shown inTable 2, a good agree-ment can be witnessed
Trang 64.2 Vibration results
To illustrate the proposed approach to eccentrically stiffened
FGM cylindrical panels, the panels considered here are cylindrical
panels and plates with in-plane edges a = b = 1.5 m; h = 0.008 m;
f0= 0 The panels are simply supported at all its edges The
combi-nation of materials consists of aluminum Em= 70 109N/m2;
qm= 2702 kg/m3 and alumina Ec= 380 109N/m2, qc= 3800 kg/
m3 The Poisson’s ratio is chosen to be 0.3 for simplicity Material
of reinforced stiffeners has elastic modulus E = 380.109N/m2;
q= 3800 kg/m3 The height of stiffeners is equal to 30 mm, its
width 3 mm, the spacing of stiffeners s1= s2= 0.15 m, the
eccen-tricities of stiffeners with respect to the middle surface of panel
z1= z2= 0.019 m
4.2.1 Results of fundamental frequencies of natural vibration
The obtained results inTable 3show that the effect of stiffeners
on fundamental frequencies of natural vibration x (x is
calculated from Eq.(29)) is considerable Obviously the natural fre-quencies of unreinforced and reinforced FGM cylindrical panels ob-served to be dependent on the constituent volume fractions, they decrease when increasing the power index k, furthermore with greater value k the effect of stiffeners is observed to be stronger This is completely reasonable because the lower value is the elas-ticity modulus of the metal constituent
4.2.2 Results of frequency–amplitude of non-linear free vibration Fig 3 shows the relation frequency–amplitude of non-linear free vibration of reinforced and unreinforced panel (calculated from Eq.(32)) with m = 1, n = 1 As expected the non-linear vibra-tion frequencies of reinforced panels are greater than ones of unre-inforced panels
4.2.3 Non-linear response results For obtaining the non-linear dynamical responses of FGM cylin-drical panel acted on by the harmonic uniformly load
q0(t) = Qsin(Xt) with Q = 5 103N/m2, X= 975 rad/s and
X= 950 rad/s, the Eq.(27)is solved using Runge–Kutta method Fig 4shows non-linear responses of ES-FGM cylindrical panel
In this case, exited frequencies are near to fundamental frequen-cies of natural vibrationx= 1011.97 rad/s (seeTable 3) From ob-tained results, the interesting phenomenon is observed like the harmonic beat phenomenon of a linear vibration, in which the amplitude of beats of reinforced panels increased rapidly when the exited frequency approached the natural frequency
When the exited frequenciesX= 500 rad/s andX= 600 rad/s are away from the natural frequencies of ES-FGM cylindrical panel The obtained non-linear dynamical responses are shown inFig 5
It shows that, the harmonic beat phenomenon does not appear
as in the previous case The amplitude of beats of reinforced panels increased slowly when the exited frequency is close to the natural frequency
Fig 6shows the Influence of initial imperfection with ampli-tudes f0= 0, f0= 105 and f0= 5 105m on the non-linear re-sponses of ES-FGM cylindrical panel The initial imperfection f0
has a slight influence to the nonlinear response of panel
Table 1
Comparison of ~xwith results reported by Alijani et al [20] , Chorfi and Houmat [21]
and Matsunaga [22]
a/R k Present Ref [20] Ref [21] Ref [22]
FGM plate
FGM cylindrical panel
0.6m
0.02222 m
0.0127 m
0.00633 m
E=211GPa
3
=0.3
=7830 kg/m
0.02222 m
Fig 2 Configuration of an eccentrically stiffened plate.
Table 2
Comparison of present frequency (Hz) with results reported by Szilard [29] and
Troitsky [30]
Table 3 The fundamental frequencies of natural vibration (rad/s) of FGM cylindrical panels.
R (m) k Unreinforced (m, n) Reinforced (m, n) 1.5
0.2 1172.51 (1, 3) 1571.27 (1, 2)
1 982.14 (1, 3) 1435.02 (1, 2)
5 822.19 (1, 3) 1266.54 (1, 2)
10 783.56 (1, 3) 1224.47 (1, 2) 3
0.2 803.92 (1, 2) 1192.51 (1, 2)
1 686.91 (1, 2) 1128.40 (1, 2)
5 556.39 (1, 2) 1011.97 (1, 1)
10 519.90 (1, 2) 924.63 (1, 1) 5
0.2 622.96 (1, 2) 930.82 (1, 1)
1 524.39 (1, 2) 812.67 (1, 1)
5 435.45 (1, 2) 647.97 (1, 1)
10 413.06 (1, 2) 599.93 (1, 1) 10
0.2 515.55 (1, 1) 551.26 (1, 1)
1 438.11 (1, 2) 494.97 (1, 1)
5 353.51 (1, 1) 427.52 (1, 1)
10 325.48 (1, 1) 411.30 (1, 1)
1 (plates)
0.2 197.11 (1, 1) 376.11 (1, 1)
1 162.11 (1, 1) 364.17 (1, 1)
5 139.79 (1, 1) 361.77 (1, 1)
10 135.38 (1, 1) 364.92 (1, 1)
Trang 74.3 Nonlinear dynamic buckling results
To evaluate the effectiveness of the reinforcement of stiffener in
the nonlinear dynamic buckling problem, we consider the case of
imperfect ES-FGM cylindrical panel subjected to an axial
compres-sive load The critical dynamic buckling loads is determined by
solving Eq.(39)and applying Budiansky–Roth criterion
Materials and structures used in this section are the same in the
previous section
Figs 7 and 8show the effect of buckling mode shapes on load –
deflection curve of reinforced and unreinforced FGM cylindrical
panel subjected to an axial compressive load with the power law
in-dex k = 1, R = 3 m and compressive load r0= 1.5 109t Clearly, the
smallest critical dynamic buckling load corresponds to the buckling
mode shape m = 5, n = 2 in the case of unreinforced panel and m = 2,
n = 2 in the case of reinforced panel This figure also shows that
there is no definite point of instability as in static analysis Rather,
there is a region of instability where the slope of n vs s curve increases rapidly In this paper, the critical parameterscris taken
as an intermediate value satisfying the conditiond2n
d s 2
s
¼ s cr
¼ 0 Table 4shows the critical loads of two cases of reinforcement and unreinforcement cylindrical panel The results show that the reinforcement by stiffeners has large effect in the dynamic stability problems of cylindrical panels under axial compressive load With the same input parameters, effectiveness of reinforcement in-creases as the curvature radius or the power index inin-creases.Table
4also considers the effect of loading speed to the dynamic buckling load; the results show that the dynamic buckling loads increases when the loading speed increases
Fig 9shows the influence of initial imperfection amplitude f0on the non-linear buckling of ES–FGM Cylindrical panel Clearly, the initial imperfection strongly influences on the critical dynamic buckling loads of ES-FGM cylindrical panel subjected to an axial compressive load
0.0E+0
3.0E+2
6.0E+2
9.0E+2
1.2E+3
1.5E+3
0.0E+0 3.0E-2 6.0E-2 9.0E-2 1.2E-1 1.5E-1
Reinforced Unreinforced
R=10m, k=0.2
R=10m, k=5
η (m)
ω (rad/s)NL
Fig 3 Frequency–amplitude relation.
-1.2E-2
-8.0E-3
-4.0E-3
0.0E+0
4.0E-3
8.0E-3
1.2E-2
0.4 0.2
0
Ω=975(rad/s) Ω=950(rad/s)
t (s)
f(m)
R=3m, k=5, q0=5000sin(Ωt)
Fig 4 Nonlinear response of ES-FGM cylindrical panel.
-9.0E-4
-6.0E-4
-3.0E-4
0.0E+0
3.0E-4
6.0E-4
9.0E-4
Ω=500 (rad/s) Ω=600 (rad/s)
t (s) f(m)
R=3m, k=5, q0=5000sin(Ωt)
Fig 5 Nonlinear response of FGM cylindrical panel.
-6.0E-4 -4.0E-4 -2.0E-4 0.0E+0 2.0E-4 4.0E-4 6.0E-4 8.0E-4
Perfect f =1e-5 f =5e-5 t(s) f(m) R=3m, k=5, q0=5000sin(500t)
Fig 6 Influence of initial imperfection on non-linear responses.
0 0.1 0.2 0.3 0.4 0.5 0.6
9.00E-01 9.50E-01 1.00E+00 1.05E+00 1.10E+00
m=5, n=2 m=6, n=1 m=4, n=3 m=5, n=1 ξ
τ
R=3m, k=1
Unreinforced Panel
Fig 7 Effect of buckling mode shapes on load–deflection curve of unreinforced panel.
0 0.2 0.4 0.6 0.8 1 1.2
9.00E-01 1.10E+00 1.30E+00 1.50E+00 1.70E+00
m=2, n=2 m=3, n=2 m=1, n=2 m=2, n=1 ξ
τ
R=3m, k=1 Reinforced Panel
Fig 8 Effect of buckling mode shapes on load–deflection curve of reinforced panel.
Trang 85 Conclusions
A formulation of the governing equations of eccentrically
rein-forced functionally graded cylindrical panels based upon the
clas-sical shell theory and the smeared stiffeners technique with von
Karman–Donnell nonlinear terms has been presented
By use of Galerkin method a nonlinear dynamic equation for
analysis of dynamic and stability characteristics of ES-FGM
cylin-drical panels is obtained
Fundamental frequencies of unreinforced and reinforced FGM
panels are considered Some results were compared with the ones
of other authors
Nonlinear dynamic responses and critical dynamic loads of
ES-FGM cylindrical panels are investigated according to the criterion
Budiansky–Roth They are significantly influenced by material
parameters, stiffeners and initial geometrical imperfection Clearly,
stiffeners enhance the stability and load-carrying capacity of FGM
cylindrical panels
Acknowledgements
This paper was supported by the National Foundation for
Science and Technology Development of Vietnam – NAFOSTED
The authors are grateful for this financial support
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Table 4
Nonlinear critical buckling loads of the cylindrical panels subjected to an axial compressive load (10 8
N/m 2 ).
c = 1.5 10 9
c = 2 10 9
c = 1.5 10 9
c = 2 10 9 3
0.2 5.1667 (5, 2) 5.1945 (5, 2) 5.2125 (5, 2) 9.5082 (2, 2) 9.6285 (2, 2) 9.6778 (2, 2)
1 3.3323 (5, 2) 3.3795 (5, 2) 3.4016 (5, 2) 7.1505 (2, 2) 7.2975 (2, 2) 7.3496 (2, 2)
5 1.9971 (5, 2) 2.0700 (5, 2) 2.0960 (5, 2) 5.0807 (2, 2) 5.2560 (2, 2) 5.3082 (2, 2)
10 1.7111 (5, 1) 1.7895 (5, 1) 1.8160 (5, 1) 4.6866 (2, 2) 4.8690 (2, 2) 4.9308 (2, 2) 5
0.2 3.0985 (3, 2) 3.1800 (4, 1) 3.2097 (4, 1) 6.5923 (2, 2) 6.7395 (2, 2) 6.7942 (2, 2)
1 2.0070 (3, 2) 2.1120 (4, 1) 2.1419 (4, 1) 5.2586 (2, 2) 5.4255 (2, 2) 5.4763 (2, 2)
5 1.1942 (3, 2) 1.3109 (4, 1) 1.3497 (4, 1) 3.3071 (1, 1) 3.6690 (1, 1) 3.7804 (1, 1)
10 1.0243 (4, 1) 1.1483 (4, 1) 1.1857 (4, 1) 2.7666 (1, 1) 3.1575 (1, 1) 3.2803 (1, 1) 10
0.2 1.5636 (3, 1) 1.7100 (3, 1) 1.7615 (3, 1) 2.9007 (1, 1) 3.2760 (1, 1) 3.3957 (1, 1)
1 1.0027 (3, 1) 1.1723 (3, 1) 1.2218 (3, 1) 2.1341 (1, 1) 2.5485 (1, 1) 2.6758 (1, 1)
5 0.6067 (3, 1) 0.7968 (3, 1) 0.8488 (3, 1) 1.4396 (1, 1) 1.9020 (1, 1) 2.0288 (1, 1)
10 0.5266 (3, 1) 0.7176 (3, 1) 0.7672 (3, 1) 1.3004 (1, 1) 1.7865 (1, 1) 1.9180 (1, 1)
1 (plates)
0.2 0.3204 (1, 1) 0.7958 (2, 1) 0.8613 (2, 1) 1.3503 (1, 1) 1.8405 (1, 1) 1.9772 (1, 1)
1 0.1948 (1, 1) 0.6194 (2, 1) 0.6906 (2, 1) 1.1552 (1, 1) 1.6575 (1, 1) 1.7740 (1, 1)
5 0.1285 (1, 1) 0.5138 (2, 1) 0.5905 (2, 1) 1.0309 (1, 1) 1.5315 (1, 1) 1.6686 (1, 1)
10 0.1171 (1, 1) 0.4980 (2, 1) 0.5773 (2, 1) 1.0236 (1, 1) 1.5255 (1, 1) 1.6607 (1, 1)
0
0.2
0.4
0.6
0.8
1
ξ =1e-5/h
ξ =2e-5/h
ξ =3e-5/h ξ
τ
Reinforced Panel
R=3m, k=1, m=2, n=2
0 0 0
Fig 9 Influence of initial imperfection on critical dynamic buckling load of
reinforced panel.
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