Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation’’ [Compos.. 99 2013 88–96] Nguyen Dinh Duc⇑ Univer
Trang 1Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentrically
stiffened FGM double curved shallow shells on elastic foundation’’
[Compos Struct 99 (2013) 88–96]
Nguyen Dinh Duc⇑
University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Available online 10 April 2013
Keywords:
Nonlinear dynamic
Eccentrically stiffened FGM double curved
shallow shells
Imperfection
Elastic foundation
a b s t r a c t
This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiff-ened functionally graded double curved shallow shells resting on elastic foundations and being subjected
to axial compressive load and transverse load The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation The non-linear equations are solved
by the Runge–Kutta and Bubnov–Galerkin methods Obtained results show effects of material and geo-metrical properties, elastic foundation and imperfection on the dynamical response of reinforced FGM shallow shells Some numerical results are given and compared with ones of other authors
Ó 2013 Elsevier Ltd All rights reserved
1 Introduction
Functionally Graded Materials (FGMs), which are
microscopi-cally composites and made from mixture of metal and ceramic
constituents, have received considerable attention in recent years
due to their high performance heat resistance capacity and
excel-lent characteristics in comparison with conventional composites
By continuously and gradually varying the volume fraction of
con-stituent materials through a specific direction, FGMs are capable of
withstanding ultrahigh temperature environments and extremely
large thermal gradients Therefore, these novel materials are
cho-sen to use in temperature shielding structure components of
air-craft, aerospace vehicles, nuclear plants and engineering
structures in various industries As a result, in recent years
impor-tant studies have been researched about the stability and vibration
of FGM plates and shells
The research on FGM shells and plates under dynamic load is
attractive to many researchers in the world
Firstly we have to mention the research group of Reddy et al
The vibration of functionally graded cylindrical shells has been
investigated by Loy, Lam and Reddy[1] Lam and Hua has taken
into account the influence of boundary conditions on the frequency
characteristics of a rotating truncated circular conical shell[2] In
[3]Pradhan et al studied vibration characteristics of FGM
cylindri-cal shells under various boundary conditions Ng et al studied the
dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading[4] The group of Ng and Lam also pub-lished results on generalized differential quadrate for free vibration
of rotating composite laminated conical shell with various bound-ary conditions in 2003[5] In the same year, Yang and Shen[6]
published the nonlinear analysis of FGM plates under transverse and in-plane loads
In 2004, Zhao et al studied the free vibration of two-side sim-ply-supported laminated cylindrical panel via the mesh-free kp-Ritz method[7] About vibration of FGM plates Vel and Batra[8]
gave three dimensional exact solution for the vibration of FGM rectangular plates Also in this year, Sofiyev and Schnack investi-gated the stability of functionally graded cylindrical shells under linearly increasing dynamic tensional loading in[9]and obtained the result for the stability of functionally graded truncated conical shells subjected to a periodic impulsive loading[10], and they pub-lished the result of the stability of functionally graded ceramic– metal cylindrical shells under a periodic axial impulsive loading
in 2005[11] Ferreira et al received natural frequencies of FGM plates by meshless method[12], 2006 In [13], Zhao et al used the element-free kp-Ritz method for free vibration analysis of con-ical shell panels Liew et al studied the nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a tempera-ture gradient [14] and dynamic stability of rotating cylindrical shells subjected to periodic axial loads[15] Woo et al investigated the non linear free vibration behavior of functionally graded plates
[16] Kadoli and Ganesan studied the buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition[17] Also in this year,
0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved.
DOI of original article: http://dx.doi.org/10.1016/j.compstruct.2012.11.017
⇑Tel.: +84 4 37547989; fax: + 84 4 37547724.
E-mail address: ducnd@vnu.edu.vn
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 2Wu et al published their results of nonlinear static and dynamic
analysis of functionally graded plates[18] Sofiyev has considered
the buckling of functionally graded truncated conical shells under
dynamic axial loading[19] Prakash et al studied the nonlinear
axi-symmetric dynamic buckling behavior of clamped functionally
graded spherical caps[20] In[21], Darabi et al obtained the
non-linear analysis of dynamic stability for functionally graded
cylin-drical shells under periodic axial loading Natural frequencies and
buckling stresses of FGM plates were analyzed by Matsunaga using
2-D higher-order deformation theory[22] In 2008, Shariyat also
obtained the dynamic thermal buckling of suddenly heated
tem-perature-dependent FGM cylindrical shells under combined axial
compression[23]and external pressure and dynamic buckling of
suddenly loaded imperfect hybrid FGM cylindrical with
tempera-ture-dependent material properties under
thermo-electro-mechanical loads[24] Allahverdizadeh et al studied nonlinear free
and forced vibration analysis of thin circular functionally graded
plates[25] Sofiyev investigated the vibration and stability
behav-ior of freely supported FGM conical shells subjected to external
pressure[26], 2009 Shen published a valuable book ‘‘Functionally
Graded materials, Non linear Analysis of plates and shells’’, in which
the results about nonlinear vibration of shear deformable FGM
plates are presented[27] Last years, Zhang and Li published the
dynamic buckling of FGM truncated conical shells subjected to
non-uniform normal impact load[28], Bich and Long studied the
non-linear dynamical analysis of functionally graded material
shal-low shells subjected to some dynamic loads[29], Dung and Nam
investigated the nonlinear dynamic analysis of imperfect FGM
shallow shells with simply supported and clamped boundary
con-ditions[30] Bich et al has also considered the nonlinear vibration
of functionally graded shallow spherical shells[31]
In fact, the FGM plates and shells, as other composite
struc-tures, ussually reinforced by stiffening member to provide the
benefit of added load-carrying static and dynamic capability with
a relatively small additional weight penalty Thus study on static
and dynamic problems of reinforced FGM plates and shells with
geometrical nonlinearity are of significant practical interest
How-ever, up to date, the investigation on static and dynamic of
eccen-trically stiffened FGM structures has received comparatively little
attention Recently, Bich et al studied nonlinear dynamical
analy-sis of eccentrically stiffened functionally graded cylindrical panels
[32]
This paper presents an dynamic nonlinear response of double
curved shallow eccentrically stiffened shells FGM resting on elastic
foundations and being subjected to axial compressive load and
transverse load The formulations are based on the classical shell
theory taking into account geometrical nonlinearity, initial
geo-metrical imperfection and the Lekhnitsky smeared stiffeners
tech-nique with Pasternak type elastic foundation The nonlinear
transients response of doubly curved shallow shells subjected to
excited external forces obtained the dynamic critical buckling
loads are evaluated based on the displacement response using
the criterion suggested by Budiansky–Roth Obtained results show
effects of material, geometrical properties, eccentrically stiffened,
elastic foundation and imperfection on the dynamical response of
FGM shallow shells
2 Eccentrically stiffened double curved FGM shallow shell on
elastic foundations
Consider a ceramic–metal stiffened FGM double curved shallow
shell of radii of curvature Rx, Rylength of edges a, b and uniform
thickness h resting on an elastic foundation
A coordinate system (x, y, z) is established in which (x, y) plane
on the middle surface of the panel and z is thickness direction (h/
2 6 z 6 h/2) as shown inFig 1
The volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution
VmðzÞ ¼ 2z þ h
2h
where N is volume fraction index (0 6 N < 1) Effective properties
Preffof FGM panel are determined by linear rule of mixture as
where Pr denotes a material property, and subscripts m and c stand for the metal and ceramic constituents, respectively Specific expressions of modulus of elasticity E(z) andq(z) are obtained by substituting Eq.(1)into Eq.(2)as
½EðzÞ;qðzÞ ¼ ðEm;qmÞ þ ðEcm;qcmÞ 2z þ h
2h
ð3Þ
where
Ecm¼ Ec Em;qcm¼qcqm; mðzÞ ¼ const ¼m ð4Þ
It is evident from Eqs.(3), (4)that the upper surface of the panel (z = h/2) is ceramic-rich, while the lower surface (z = h/2) is me-tal-rich, and the percentage of ceramic constituent in the panel is enhanced when N increases
The panel–foundation interaction is represented by Pasternak model as
qe¼ k1w k2r2
where r2= @2/@x2+ @2/@ y2,w is the deflection of the panel, k1 is Winkler foundation modulus and k2is the shear layer foundation stiffness of Pasternak model
3 Theoretical formulation
In this study, the classical shell theory and the Lekhnitsky smeared stiffeners technique are used to obtain governing equa-tions and determine the nonlinear dynamical response of FGM curved panels The strain across the shell thickness at a distance
z from the mid-surface are
ex
ey
cxy
0 B
1 C
A ¼
e0 x
e0 y
c0 xy
0 B
1 C
A z
kx
ky
2kxy
0 B
1
wheree0
x; e0
xandc0
xyare normal and shear strain at the middle sur-face of the shell, and kx, ky,kxy are the curvatures The nonlinear strain–displacement relationship based upon the von Karman the-ory for moderately large deflection and small strain are:
h
Rx
z
y
x
Fig 1 Geometry and coordinate system of an eccentrically stiffened double curved shallow FGM shell on elastic foundation.
Trang 3x
e0
y
c0
xy
0
B
1
C
A ¼
u;x w=Rxþ w2
;x=2
v;y w=Ryþ w2
;y=2
u;yþv;xþ w;xw;y
0
B
1 C A;
kx
ky
kxy
0 B
1 C
A ¼
wx;x
wy;y
w;xy
0 B
1
In which u,vare the displacement components along the x, y
directions, respectively
The force and moment resultants of the FGM panel are
deter-mined by
ðNi;MiÞ ¼
Z h=2
h=2
The constitutive stress–strain equations by Hooke law for the
shell material are omitted here for brevity The shell reinforced
by eccentrically longitudinal and transversal stiffeners is shown
inFig 1 The shallow shell is assumed to have a relative small rise
as compared with its span The contribution of stiffeners can be
ac-counted for using the Lekhnitsky smeared stiffeners technique
Then intergrading 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 shallow shell
are obtained[32]:
1 m2þEA1
s1
e0
xþ E1m
1 m2e0
y E2
1 m2þ C1
kx E2m
1 m2ky
Ny¼ E1m
1 m2e0
xþ E1
1 m2þEA2
s2
e0
y E2m
1 m2kx E2
1 m2þ C2
ky
1 m2þ C1
e0
xþ E2m
1 m2e0
y E3
1 m2þEI1
s1
kx E3m
1 m2ky
My¼ E2m
1 m2e0
xþ E2
1 m2þ C2
e0
y E3m
1 m2kx E3
1 m2þEI2
s2
ky
2ð1 þmÞ E1c0
xy 2E2kxy
2ð1 þmÞ E2c0
xy 2E3kxy
ð9Þ
where
E1¼ Emþ Ecm
N þ 1
h
E2¼ EcmNh
2
2ðN þ 1ÞðN þ 2Þ
E3¼ Em
12þ Ecm
1
N þ 3
1
N þ 2þ
1 4N þ 4
h3
C1¼EA1z1
s1
; C2¼EA2z2
s2
ð10Þ
In above relations(9) and (10), the quantities A1, A2are the cross
section areas of the stiffeners and I1, I2, z1, z2are the second
mo-ments of cross section areas and eccentricities of the stiffeners with
respect to the middle surface of the shell respectively, E is elasticity
modulus in the axial direction of the corresponding stiffener witch
is assumed identical for both types of stiffeners (Fig 2) In order to
provide continuity between the shell and stiffeners, suppose that
stiffeners are made of full metal (E = Em) if putting them at the
me-tal-rich side of the shell, and conversely full ceramic stiffeners
(E = Ec) at the ceramic-rich side of the shell[32]
The nonlinear dynamic equations of a FGM shallow shells based
on the classical shell theory are[33]
Nx;xþ Nxy;y¼q@
2u
@t2
Nxy;xþ Ny;y¼q@
2v
@t2
Mx;xxþ 2Mxy;xyþ My;yyþNx
Rx
þNy
Ry
þ Nxw;xxþ 2Nxyw;xyþ Nyw;yyþ q
k1w þ k2r2w ¼q@
2
w
where
q¼
Zh 2
h 2
qðzÞ dz þq0
A1
s1
þA2
s2
¼ qmþ qcm
N þ 1
h þq0
A1
s1
þA2
s2
ð12Þ
in whichq0is the mass density of stiffeners;q@ 2 u
@t 2! 0 andq@ 2v
@t 2! 0 into consideration because of u w,v w the Eq (11) can be rewritten as:
Mx;xxþ 2Mxy;xyþ My;yyþNx
RxþNy
Ryþ Nxw;xxþ 2Nxyw;xyþ Nyw;yyþ q
k1w þ k2r2w ¼q@
2w
Calculating from Eq.(9), obtained:
e0
x¼ A22Nx A12Nyþ B11kxþ B12ky
e0
y¼ A11Ny A12Nxþ B21kxþ B22ky
c0
xy¼ A66Nxyþ 2A66kxy
ð14Þ
where
A11¼1 D
EA1
s1 þ E1
1 m2
; A22¼1 D
EA2
s2 þ E1
1 m2
A12¼1 D
E1m
1 m2; A66¼2ð1 þmÞ
E1
D¼ EA1
s1
þ E1
1 m2
2
s2
þ E1
1 m2
E1m
1 m2
2
B11¼ A22 C1þ E2
1 m2
A12
E2m
1 m2; B22¼ A11 C2þ E2
1 m2
A12
E2m
1 m2
B12¼ A22
E2m
1 m2 A12
E2
1 m2þ C2
; B21¼ A11
E2m
1 m2 A12
E2
1 m2þ C1
B66¼E2
E1
ð15Þ Substituting once again Eq.(14)into the expression of Mijin(9)
leads to:
Mx¼ B11Nþ
xB21N
yD11kx D12ky
Mx¼ B12Nþ
xB22N
yD21kx D22ky
Mxy¼ B66N2D66kxy
ð16Þ
x1
x2
a
b
s2
h
z2
z1
1
2
s2 s2 s2
s1
s1
s1
s1
b O
z
Fig 2 Configuration of an eccentrically stiffened shallow shells
Trang 4D11¼EI1
s1
1 m2 C1þ E2
1 m2
B11 E2m
1 m2B21
D22¼EI2
s2
1 m2 C2þ E2
1 m2
B22 E2m
1 m2B12
D12¼ E3m
1 m2 C1þ E2
1 m2
B12 E2m
1 m2B22
D21¼ E3m
1 m2 C2þ E2
1 m2
B21 E2m
1 m2B11
D66¼ E3
2ð1 þmÞ
E2
2ð1 þmÞB66
ð17Þ
Then Mijinto Eq.(13)and f(x, y) is stress function defined by
For an imperfect FGM curved panel, Eq.(13)are modified into
form
B21f;xxxxþ B12f;yyyyþ ðB11þ B22 2B66Þf;xxyy D11w;xxxx
D22w;yyyy ðD12þ D21þ 4D66Þw;xxyyþ D11w
;xxxx
þ D22w
;yyyyþ ðD12þ D21þ 4D66Þw
;xxyyþ f;yyw;xx 2f;xyw;xy
þ f;xxw;yyþf;yy
Rx
þf;xx
Ry
þ q k1w þ k2r2w ¼q@
2w
in which w⁄(x, y) is a known function representing initial small
imperfection of the eccentrically stiffened shallow shells The
geo-metrical compatibility equation for an imperfect shallow shells is
written
e0
x;yyþe0
y;xxc0
xy;xy¼ w2;xy w;xxw;yy w2
;xyþ w
;xxw
;yy
w;yy w
;yy
Rx w;xx w
;xx
Ry
From the constitutive relations(18)in conjunction with Eq.(14)
one can write
e0
x¼ A22f;yy A12f;xxþ B11w;xxþ B12w;yy
e0
y¼ A11f;xx A12f;yyþ B21w;xxþ B22w;yy
c0
xy¼ A66f;xyþ 2A66w;xyÞ
ð21Þ
Setting Eq.(21)into Eq.(20)gives the compatibility equation of
an imperfect eccentrically stiffened shallow FGM shells as
A11f;xxxxþ ðA66 2A12Þf;xxyyþ A22f;yyyyþ B21w;xxxx
þ ðB11þ B22 2B66Þw;xxyyþ B12w;yyyy¼ w2
;xy w;xxw;yy
w2
;xyþ w
;xxw
;yyw;yy w
;yy
Rx w;xx w
;xx
Eqs.(19) and (22)are nonlinear equations in terms of variables
w and f and used to investigate the nonlinear dynamic and
nonlin-ear stability of thick imperfect stiffened FGM double curved panels
on elastic foundations subjected to mechanical, thermal and
ther-mo mechanical loads
4 Nonlinear dynamic analysis
In the present study, suppose that the stiffened FGM shallow
shell is simply supported at its all edges and subjected to a
trans-verse load q(t), compressive edge loads r0(t) and p0(t) The
bound-ary conditions are
w ¼ Nxy¼ Mx¼ 0; Nx¼ r0h at x ¼ 0; a
where a and b are the lengths of in-plane edges of the shallow shell
The approximate solutions of w, w⁄and f satisfying boundary conditions(23)are assumed to be[27–31]
where km= mp/a, dn= np/b and W is the maximum deflection; W0is
a constant; h(x) andx(y) chosen such that:
Subsequently, substitution of Eqs.(24a and b)into Eqs.(22) and (24c) into Eq (19) and applying the Galerkin procedure for the resulting equation yield leads to:
gA11m4
þ ðA66 2A12Þm2n2
k2þ A22n4
k4
a
2
p2
n2k2
Rx
2
Ry
!
ðW W0Þ
þ W B21m4
þ ðB11þ B22 2B66Þm2n2
k2þ B12n4
k4
þ16 3
mnk2
p2 W2 W20
g p 4
a4 B21m4þ ðB11þ B22 B66Þn2m2
k2þ B12n4
k4
ðW
W0Þp4
a4 D11m4þ ðD12þ D21þ 4D66Þn2m2
k2þ D22n4
k4
þ32
3 Wgmnp2k2
a4þp2hW
a2 ðm2r0þ n2p0k2Þ
p2
a2g m
2
Ry
þn
2k2
Rx
!
mnp2
r0
Rx
þp0
Ry
mnp2 k1W
k2Wp2
a2ðm2þ k2n2Þ ¼q@
2W
where m,n are odd numbers, and k ¼a Eliminatinggfrom two obtained equations leads to non-linear second-order ordinary differential equation for f(t):
W p2h
a2 ðm2r0þ n2p0k2Þ k1 k2p2
a2ðm2þ k2n2Þ p4
a4
P2
P1
þp2
a2
m2
Ry
þn
2k2
Rx
!
P2
P1
þ ðW W0Þ p2
a2
m2
Ryþn
2k2
Rx
!
P2
P1p4
a4P3 m
2
Ryþn
2k2
Rx
!2 1
P1
2 4
3 5
þ ðW2 W2Þ 1
a2
m2
Ryþn
2k2
Rx
! 16mnk2 3P1 16mnp2k2
a4
P2
P1
þ W232mnp2k2
3a4
P2
P1þ WðW W0Þ32mnk
2 3a2
m2
Ryþn
2k2
Rx
! 1
P1
WðW2 W2Þ512m
2n2k 9a4
1
P1 16h
mnp2
r0
Rxþp0
Ry
þ 16q
mnp2¼q@
where
P1¼ A11m4þ ðA66 2A12Þm2n2k2þ A22n4k4
P2¼ B21m4þ ðB11þ B22 2B66Þm2n2
k2þ B12n4
k4
P3¼ D11m4
þ ðD12þ D21þ 4D66Þm2n2
k2þ D22n4
k4
ð29Þ
The obtained Eq (28) is a governing equation for dynamic imperfect stiffened FGM doubly-curved shallow shells in general The initial conditions are assumed as Wð0Þ ¼ W0; Wð0Þ ¼ 0 The_ nonlinear Eq.(28)can be solved by the Newmark’s numerical inte-gration method or Runge–Kutta method
4.1 Nonlinear vibration of eccentrically stiffened FGM shallow shell Consider an imperfect stiffened FGM shallow shell acted on by uniformly distributed excited transverse q(t) = QsinXt, i.e
p = r = 0, from(28)we have
Trang 5Q1W þ Q2ðW W0Þ þ Q3 W2
W2
Q4W2
þ Q5WðW
W0Þ Q6W W 2 W20
þ Q7sinXt ¼q@
2W
where
Q1¼ k1þ k2
p2
a2ðm2þ k2n2Þ þp4
a4
P2
P1p2
a2
m2
Ry þn
2k2
Rx
!
P2
P1
Q2¼ p2
a2
m2
Ry
þn
2k2
Rx
!
P2
P1
þp4
a4P3þ m
2
Ry
þn
2k2
Rx
!2
1
P1
Q3¼1
a2
m2
Ry
þn
2k2
Rx
! 16mnk2
3P1
16mnp2k2
a4
P2
P1
Q4¼32mnp2k2
3a4
P2
P1
Q5¼32mnk
2
3a2
m2
Ry
þn
2k2
Rx
! 1
P1
Q6¼512m
2n2k
9a4
1
P1
Q7¼16Q0
mnp2
ð31Þ
From Eq.(30)the fundamental frequencies of natural vibration
of the imperfect stiffened FGM shell can be determined by the
relation:
xL¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
qðQ1þ Q2Þ
s
ð32Þ
The equation of nonlinear free vibration of a perfect FGM
shal-low panel can be obtained from:
€
where denoting
H1¼x2
L¼1
qðQ1þ Q2Þ
H2¼Q4 Q3 Q5
q
H3¼Q6
q
ð34Þ
Seeking solution as W(t) =scosxt and applying procedure like
Garlerkin method to Eq (33), the frequency-amplitude relation of
nonlinear free vibration is obtained
xNL¼xL 1 þ 8H2
3px2
L
sþ3H3
4x2 L
s2
ð35Þ
wherexNLis the nonlinear vibration frequency andsis the
ampli-tude of nonlinear vibration
4.2 Nonlinear dynamic buckling analysis of imperfect eccentrically
stiffened FGM shallow shell
The aim of considered problems is to search the critical dynamic
buckling loads They can be evaluated based on the displacement
responses obtained from the motion Eq.(28) This criterion
sug-gested by Budiansky and Roth is employed here as it is widely
ac-cepted This criterion is based on that, for large values of loading
speed the amplitude-time curve of obtained displacement
re-sponse increases sharply depending on time and this curve
ob-tained a maximum by passing from the slope point, and at the
time t = tcra stability loss occurs, and here t = tcris called critical
time and the load corresponding to this critical time is called
dy-namic critical buckling load
4.2.1 Imperfect eccentrically stiffened FGM cylindrical panel acted on
by axial compressive load
Eq (28) in this case Rx?1, Ry= R, p0= q = 0;r0–0 can be rewritten as:
W p2h
a2 m2r0 k1 k2
p2
a2ðm2þ k2n2Þ p4P2
a4P1
þp2m2P2
a2RP1
þ ðW W0Þ p2m2P2
a2RP1 p4
a4P3 m
4
R2P1
þ W 2 W20 1
a2
16m3nk2
3P1R
16mnp2k2
a4
P2
P1
þ W2
32mnp2k2 3a4
P2
P1
þ WðW W0Þ32mnk
2
3a2
m2
R
1
P1
W W 2 W20 512m2n2k
9a4
1
P1
¼q@
2W
The static critical load can be determined by the equation to be reduced from Eq.(36)by putting €W ¼ 0; W0¼ 0
Wp2h
a2 m2r0¼ W k1þ k2
p2
a2ðm2þ k2n2
Þ þp4P2
a4P1
p2m2P2
a2RP1 p2m2P2
a2RP1 þp4
a4P3þ m
4
R2P1
a2
16m3nk2
3P1R
16mnp2k2
a4
P2
P1
"
32mnp2k2 3a4
P2
P1
2
3a2
m2
R
1
P1
#
þ W3512m
2n2k 9a4
1
P1
ð37Þ
Taking of W – 0, i.e considering the shell after the loss of stabil-ity we obtain
p2h
a2 m2r0¼ k1þ k2
p2
a2ðm2þ k2n2Þ þp4P2
a4P1
p2m2P2
a2RP1
p2m2P2
a2RP1
þp4
a4P3þ m
4
R2P1
a2
16m3nk2
3P1R
16mnp2k2
a4
P2
P1
"
32mnp2k2 3a4
P2
P1þ32mnk
2
3a2
m2
R
1
P1
#
þ W2512m
2n2k 9a4
1
P1
ð38Þ
From Eq.(38)the upper buckling load can be determined by
W = 0
2
m2hp2 k1þ k2
p2
a2ðm2þ k2n2Þ þp4P2
a4P1
p2m2P2
a2RP1
p2m2P2
a2RP1
þp4
a4P3þ m 4
R2P1
ð39Þ
And the lower buckling load is found using the conditiondr 0
dW¼ 0,
it follows:
rlower¼ a
2
p2hm2 k1þ k2
p2
a2ðm2þ k2n2Þ þp4P2
a4P1
p2m2P2
a2RP1
p2m2P2
a2RP1
þp4
a4P3þ m
4
R2P1
4P1 1024m2n2k 1
a2
16m3nk2 3P1R
16mnp2k2
a4
P2
P1
32mnp2k2 3a4
P2
P1
2 3a2
m2 R
1
P1
2n2k 9a4
1
P1
1
a2
16m3nk2 3P1R
16mnp2k2
a4
P2
P1
"
32mnp2k2 3a4
P2
P1þ
32mnk2 3a2
m2 R
1
P1
##
ð40Þ
Trang 64.2.2 Imperfect eccentrically stiffened shallow FGM cylindrical panel
subjected to transverse load
Eq.(28)in this case Rx?1,Ry= R, p0= r0= 0 can be rewritten as:
W k1 k2
p2
a2ðm2þ k2n2Þ p4
a4
P2
P1
þp2n2k2P2
a2RP1
þ ðW W0Þ p2m2P2
a2RP1
p4
a4P3m
4
RP1
þ W 2 W20 1
a2
16m3nk2
3RP1
16mnp2k2
a4
P2
P1
þ W232mnp2k2
3a4
P2
P1
þ WðW W0Þ32m
3nk2
3Ra2
1
P1
W W 2 W2 512m2n2k
9a4
1
P1þ 16q
mnp2¼q@
2W
The static critical load can be determined by the equation to be
re-duced from Eq.(41)by putting €W ¼ 0; W0¼ 0 and using condition
dq
dW¼ 0
4.2.3 Imperfect eccentrically stiffened FGM shallow spherical panel
under transverse load
Eq.(28)in this case Rx= Ry= R, p0= r0= 0 can be rewritten as:
W k1 k2
p2
a2ðm2þ k2n2Þ p4
a4
P2
P1
þp2
a2
m2þ n2k2 R
!
P2
P1
þ ðW W0Þ p2
a2
m2þ n2k2 R
!
P2
P1
p4
a4P3 m
2þ n2k2 R
!2
1
P1
2
4
3 5
þ W 2 W20 1
a2
m2þ n2k2 R
! 16mnk2
3P1
16mnp2k2
a4
P2
P1
þ W232mnp2k2
3a4
P2
P1
þ WðW W0Þ32mnk
2
3a2
m2þ n2k2 R
! 1
P1
W W 2 W20 512m2n2k
9a4
1
P1
mnp2¼q@
2W
The static critical load can be determined by the equation to be
reduced from Eq.(42)by putting €W ¼ 0; W0¼ 0 and using
condi-tiondq
dW¼ 0
5 Numerical results and discussion
The eccentrically stiffened FGM shells considered here are
shal-low shell with in-plane edges:
a ¼ b ¼ 2 m; h ¼ 0:01 m;
Em¼ 70 109N=m2; Ec¼ 380 109N=m2;
qm¼ 2702 kg=m3; qc¼ 3800 kg=m3;
s1¼ s2¼ 0:4; z1¼ z2¼ 0:0225 ðmÞ; m¼ 0:3
ð43Þ
Table 1presents the dependence of the fundamental frequencies
of nature vibration of spherical FGM shallow shell on volume ratio N
in which m ¼ n ¼ 1; a ¼ b ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ; K1¼ 200; K2¼
10, R¼
xRy¼ 3 ðmÞ; W0¼ 1e 5
From the results ofTable 1, it can be seen that the increase of
volume ration N will lead to the decrease of frequencies of nature
vibration of spherical FGM shallow shell
Table 2presents the frequencies of nature vibration of spherical
double curved FGM shallow shell depending on elastic
founda-tions These results show that the increase of the coefficients of
elastic foundations will lead to the increase of the frequencies of
nature vibration Moreover, the Pasternak type elastic foundation
has the greater influence on the frequencies of nature vibration
of FGM shell than Winkler model does
Based on(28)the nonlinear vibration of imperfect eccentrically
stiffened shells under various loading cases can be performed
Table 1 The dependence of the fundamental frequencies of nature vibration of spherical FGM double curved shallow shell on volume ratio N.
55.667 10 5
23.477 10 5
Table 2 The dependence of the fundamental frequencies of nature vibration of spherical FGM double curved shallow shell on elastic foundations.
K 1 = 200, K 2 = 0 33.574 10 5
32.865 10 5
K 1 = 200, K 2 = 10 39.034 10 5
38.515 10 5
K 1 = 200, K 2 = 20 44.079 10 5
43.273 10 5
K 1 = 200, K 2 = 30 48.535 10 5
46.371 10 5
K 1 = 0, K 2 = 10 26.734 10 5 25.646 10 5
K 1 = 100, K 2 = 10 31.534 10 5 30.078 10 5
K 1 = 150, K 2 = 10 35.585 10 5
35.033 10 5
K 1 = 200, K 2 = 10 39.034 10 5
38.515 10 5
Table 3 Comparison of-with result reported by Bich et al [32] , Alijani et al [34] , Chorfi and Houmat [35] and Matsunaga [36]
(a/R x , b/R y ) N Present Ref [32] Ref [34] Ref [35] Ref [36] FGM plate
(0, 0) 0 0.0562 0.0597 0.0597 0.0577 0.0588
0.5 0.0502 0.0506 0.0506 0.0490 0.0492
1 0.0449 0.0456 0.0456 0.0442 0.0403
4 0.0385 0.0396 0.0396 0.0383 0.0381
10 0.0304 0.0381 0.0380 0.0366 0.0364 FGM cylindrical panel
(0, 0.5) 0 0.0624 0.0648 0.0648 0.0629 0.0622
0.5 0.0528 0.0553 0.0553 0.0540 0.0535
1 0.0494 0.0501 0.0501 0.0490 0.0485
4 0.0407 0.0430 0.0430 0.0419 0.0413
10 0.0379 0.0409 0.0408 0.0395 0.0390
5.7413 5.7414 5.7415 5.7416 5.7417 5.7418
5.7419x 10
4
τ
ω NL
Reinforced, Rx=Ry=3(m), N=5 Reinforced, R
x =R(y)=3(m), N=0 Unreinforced, Rx=Ry=3(m), N=5 Unreinforced, Rx=Ry=3(m), N=0
Trang 7Particularly for spherical panel we put1
x¼ 1
yin(28), for cylindrical shell 1
x¼ 0 and for a plate1
x¼ 1
y¼ 0
Table 3presents the comparison on the fundamental frequency
parameter-¼xLh ffiffiffiffiq
c
E c
q (In theTables 1–3,xLis calculated from
Eq.(32)) given by the present analysis with the result of Alijani
et al.[34]based on the Donnell’s nonlinear shallow shell theory,
Chorfi and Haumat[35]based on the first–order shear deformation
theory and Matsunaga [36] based on the two-dimensional (2D)
higher order theory for the perfect unreinforced FGM cylindrical
panel The results in Table 3 were obtained with m = n = 1,
a = b = 2 (m), h = 0.02 (m), K1= 0, K2= 0; W⁄= 0 and with the
cho-sen material properties in(43) As inTable 3, we can observe a very
good agreement in this comparison study
Fig 3shows the relation frequency-amplitude of nonlinear free vibration of reinforced and unreinforced spherical shallow FGM shell on elastic foundation (calculated from Eq (35)) with m ¼
n ¼ 1; a ¼ b ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ; K1¼ 200, K2¼ 10; R¼
xRy¼ 3 ðmÞ; W0¼ 1e 5 As expected the nonlinear vibration frequencies
of reinforced spherical shallow FGM shells are greater than ones of unreinforced shells
The nonlinear Eq.(28) is solved by Runge–Kutta method The below figures, except Fig 6, are calculated basing on k1= 100;
k2= 10
Fig 4shows the effect of eccentrically stiffeners on nonlinear respond of the FGM shallow shell on elastic foundation The FGM
Fig 4 Effect of eccentrically stiffeners on nonlinear dynamic response of the
shallow spherical FGM shell.
Fig 5 Deflection-velocity relation of the eccentrically stiffened shallow spherical
FGM shell.
Fig 6 Influence of elastic foundations on nonlinear dynamic response of the eccentrically stiffened shallow spherical FGM shell.
Fig 7 Effect of volume metal-ceramic on nonlinear response of the eccentrically stiffened shallow spherical FGM shell.
Trang 8shell considered here is spherical panel Rx= Ry= 5 m Clearly, the
stiffeners played positive role in reducing amplitude of maximum
deflection Relation of maximum deflection and velocity for
spher-ical shallow shell is expressed inFig 5
Fig 6shows influence of elastic foundations on nonlinear
dy-namic response of spherical panel Obviously, elastic foundations
played negative role on dynamic response of the shell: the larger
k1and k2coefficients are, the larger amplitude of deflections is
Fig 7shows effect of volume metal-ceramic on nonlinear
dy-namic response of the eccentrically stiffened shallow spherical
FGM shell
Figs 8 and 9show effect of dynamic loads and Rxon nonlinear
dynamic response of the eccentrically stiffened shallow spherical
FGM shell
Fig 10shows influence of initial imperfection on nonlinear dy-namic response of the eccentrically stiffened spherical panel The increase in imperfection will lead to the increase of the amplitude
of maximum deflection
Fig 11shows nonlinear dynamic response of shallow eccentri-cally stiffened spherical and eccentrieccentri-cally stiffened cylindrical FGM panels For eccentrically stiffened cylindrical FGM panel, in this case, the obtained results is identical to the result of Bich in[32]
6 Concluding remarks This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundations and
Fig 8 Effect of dynamic loads on nonlinear response.
Fig 9 Effect of R x on nonlinear dynamic response.
Fig 10 Influence of initial imperfection on nonlinear dynamic response of the eccentrically stiffened spherical panel.
Fig 11 Nonlinear dynamic response of eccentrically stiffened spherical and cylindrical FGM panel.
Trang 9being subjected to axial compressive load and transverse load The
formulations are based on the classical shell theory taking into
ac-count geometrical nonlinearity, initial geometrical imperfection
and the Lekhnitsky smeared stiffeners technique with Pasternak
type elastic foundation The nonlinear equations are solved by
the Runge–Kutta and Bubnov-Galerkin methods Some results
were compared with the ones of the other authors
Obtained results show effects of material, geometrical
proper-ties, eccentrically stiffened, elastic foundation and imperfection
on the dynamical response of reinforced FGM double curved
shal-low shells Hence, when we change these parameters, we can
con-trol the dynamic response and vibration of the FGM shallow shells
actively
Acknowledgments
This work was supported by Project in Mechanics of the National
Foundation for Science and Technology Development of
Vietnam-NAFOSTED The author is grateful for this financial support
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