Accepted ManuscriptAn Analytical Approach: Nonlinear Vibration of Imperfect Stiffened Fgm Sandwich Toroidal Shell Segments Containing Fluid Under External To appear in: Composite Structu
Trang 1Accepted Manuscript
An Analytical Approach: Nonlinear Vibration of Imperfect Stiffened Fgm
Sandwich Toroidal Shell Segments Containing Fluid Under External
To appear in: Composite Structures
Received Date: 11 October 2016
Revised Date: 20 November 2016
Accepted Date: 23 November 2016
Please cite this article as: Huy Bich, D., Gia Ninh, D., An Analytical Approach: Nonlinear Vibration of ImperfectStiffened Fgm Sandwich Toroidal Shell Segments Containing Fluid Under External Thermo-Mechanical Loads,
Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.11.065
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AN ANALYTICAL APPROACH: NONLINEAR VIBRATION OF IMPERFECT STIFFENED FGM SANDWICH TOROIDAL SHELL SEGMENTS CONTAINING FLUID UNDER EXTERNAL
bLecturer, Ph D, School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam *Corresponding author Tel: +84 988 287 789 Email address:
ninhdinhgia@gmail.com and ninh.dinhgia@hust.edu.vn
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Highlights
• A new analytical approach to the nonlinear dynamical buckling of imperfect stiffened three-layered toroidal shell segment containing fluid under external thermal environment is studied
• The lowest natural frequencies corresponding to particular modes of both convex and concave are found in the considered case
• The fluid remarkably influenced on nonlinear vibration response of FGM sandwich toroidal shell segment Definitely, it makes the amplitudes of vibration of shell and frequencies decreased considerably
• The change of external temperature makes the deflection as well as the amplitudes of shell rocketed
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Abstract:
An analytical study to the nonlinear vibration of imperfect stiffened FGM sandwich toroidal shell segment containing fluid in external thermal environment is approached in this present The toroidal shell segments consist of two types convex shell and concave shell which are reinforced by ring and stringer stiffeners system Material properties of shell are assumed to
be continuously graded in the thickness direction Based on the classical thin shell theory with geometrical nonlinearity in von Karman-Donnell sense, Stein and McElman assumption, and the smeared stiffeners technique theoretical formulations are established In addition, the dynamical pressure of fluid is taken into account The fluid is assumed to be non-viscous and ideal incompressible The nonlinear vibration analyses of full-filled fluid toroidal shell segment are solved by using numerical method fourth-order Runge-Kutta Furthermore, effects of geometrical and material parameters, imperfection, fluid and change of temperature field on the nonlinear vibration responses of shells are shown in obtained results It is hoped that the obtained results will be used as benchmark solutions for an analytical approach of fluid-structures vibration in further research
Key words: Toroidal shell segment; thermal vibration; fluid-structures; imperfection; filled fluid
The sandwich structures have become pay attention in structural applications The smooth and continuous changes in material properties makes sandwich FGMs to avoid interface problems and unexpected thermal stress concentrations On the other hand, the sandwich structures also have the mentionable properties, especially thermal and sound insulation
Sofiyev et al [1, 2] investigated the parametric instability of simply-supported sandwich
cylindrical shell with a FGM core under static and time dependent periodic axial compressive loads; the influences of shear stresses and rotary inertia on the vibration of FG coated sandwich cylindrical shells resting on Pasternak elastic foundation The free vibration of sandwich plates with FGM face sheets in various thermal environments to improve high-order sandwich plate theory using Hamilton's principle was studied by Khalili and Mohammadi [3] A new approach was used to reduce the equations of motion and then solved them for both un-symmetric and symmetric sandwich plates Xia and Shen [4] anlyzed the small and large-amplitude vibration of compressively and thermally post-buckled sandwich plates with FGM face sheets in thermal environment using a higher-order shear deformation plate theory The formulations were based on a general von-Karman-type equation that included a thermal effect and the equations of motion were solved by an improved perturbation technique Wang and Shen [5] studied the nonlinear dynamic response
of sandwich plates with FGM face sheets assumed to be graded in the thickness direction according to the Mori–Tanaka scheme resting on elastic foundations in thermal environments Sburlati [6] gave an analytical solution in the framework of the elasticity theory to indicate the elastic bending response of axisymmetric circular sandwich panels with functionally graded material cores and homogeneous face-sheets The elastic solution was
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functions satisfying linear fourth-order partial differential equations Furthermore, Taibi et al
[7] proposed the deformation behavior of shear deformable FG sandwich plates resting on Pasternak foundation under thermo-mechanical loads Ninh and Bich [8] investigated the nonlinear torsional buckling and post buckling of eccentrically stiffened ceramic FGM metal layer cylindrical shell under thermo-mechanical load A layerwise shear deformation theory proposed by Ferreira for FGM sandwich shells and laminated composite shells using a
differential quadrature finite element method (DQFEM) was analyzed by Liu et al [9] The
combination of the DQFEM with Ferreira’s layerwise theory allows a very accurate prediction of the field variables
On the other hand, the fluid-structures problems have been attracted the special interest of
many authors in the world Chen et al [10] researched the vibration of fluid-filled orthotropic
FGM cylindrical shells based on the three-dimensional fundamental equations of anisotropic elasticity The frequency equation was deduced for an FGM cylindrical shell filled with a compressible, non-viscous fluid medium and the concept “added mass effect” appeared to show the effect of fluid to structures An analytical approach to research the elastic dynamic responses of FG plates to underwater shock with applications in deep sea exploration and
naval and coastal engineering was proposed by Liang et al [11] Taylor׳s one dimensional
fluid solid interaction (FSI) model was extended to fit a three dimensional model appropriate
to FG plates The extended FSI model and Laplace transform were integrated into the state space method, with the transient solution in the time domain being obtained by using the numerical inversion of the Laplace transform Sheng and Wang [12] investigated into the vibration of FGM cylindrical shells with flowing fluid in an elastic medium under mechanical and thermal loads using a modal expansion method Based on the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of
functionally graded cylindrical shells with flowing fluid was derived Amabili et al [13, 14]
presented circular cylindrical shells conveying incompressible flow using Donnell's linear theory retaining in-plane displacements and the Sanders–Koiter non-linear theory; geometrically nonlinear vibrations of thin infinitely long rectangular plates under axial flow and concentrated harmonic excitation using von Karman non-linear plate theory and geometric imperfections by employing Lagrangian approach The fluid was modelled by potential flow theory but the effect of steady viscous forces was investigated; moreover the flow perturbation potential was derived by applying the Galerkin technique A thin-walled beam made of functionally graded material (FGM) which was used as rotating blades in
non-turbomachinery under aerothermoelastic loadingwas presented Fazelzadeh and Hosseini
[15] Based on first-order shear deformation theory, the governing equations included the effects of the presetting angle, the secondary warping, temperature gradient through the wall thickness of the beam and also the rotational speed In addition, quasi-steady aerodynamic pressure loadings were determined using first-order piston theory, and steady beam surface
temperature was obtained from gas dynamics theory Eghtesad et al [16] researched the
Smoothed Particle Hydrodynamics (SPH) method to investigate elastic–plastic deformation
of AL and ceramic–metal FGM plates under the impact of water in a fluid–solid interface A new scheme called corrected smooth particle method (CSPM) was applied to both fluid and solid particles to improve the free surface behavior Khorshidi and Bakhsheshy [17]
Trang 6
investigated the vibration analysis of a functionally graded (FG) rectangular plate partially in contact with a bounded fluid The fluid velocity potential satisfying fluid boundary conditions was derived, and wet dynamic modal functions of the plate were expanded in terms of finite Fourier series for compatibility requirement along the contacting surface between the plate and the fluid
The vibration of FGM structures also is studied by many researchers Loy et al [18]
investigated vibration of FGM cylindrical shells based on Love’s shell theory and the eigenvalue governing equation was obtained using Rayleigh-Ritz method The nonlinear dynamic buckling of FGM cylindrical shells under time-dependent axial load and radial load using an energy method and Budiansky-Roth criterion were presented by Huang and Han [19, 20] Sofiyev [21] researched the dynamic buckling of truncated conical shells with FGM coatings subjected to a time dependent axial load in the large deformation The method of solution utilizes superposition principle and Galerkin procedure The nonlinear vibration of FGM cylindrical shell improved Donnell equations using Galerkin procedure was studied by Bich and Nguyen [22] The dynamic instability of simply supported, functionally graded (FG) truncated conical shells under static and time dependent periodic axial loads using Galerkin method was analyzed by Sofiyev and Kuruoglu [23] The domains of principal instability were determined by using Bolotin’s method Shariyat [24] presented dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment, with temperature-dependent properties Free vibration of simply supported FGM sandwich spherical and cylindrical shell geometries with a three-dimensional exact shell model and
different two-dimensional computational models was investigated by Fantuzzi et al [25]
Ferreira et al [26-27] analyzed nonlinear vibration of microstructure-dependent FGM piezoelectric material beams in pre/post buckling regimes based on Timoshenko beam theory with various inplane and out-of-plane boundary conditions in thermo-mechanical loads; the geometric nonlinear analysis of FGM plates and shells using the Marguerre shell element, modified to incorporate the graded properties across the thickness
The toroidal shell segment structures containing fluid have been applied in mechanical engineering, aerospace engineering, bio-mechanical engineering such as fusion reactor vessels, underwater toroidal pressure hull and nuclear reactor In the past, the initial post-buckling behavior of toroidal shell segments under several loading conditions using the basic
of Koiter’s general theory was studied by Hutchinson [28] Stein and McElman [29] considered the buckling of homogenous and isotropic toroidal shell segments Recently, Bich
et al. [30-34] have investigated the stability buckling of functionally graded toroidal shell segment under thermomechanical load based on the classical thin shell theory and the
smeared stiffeners technique Bich et al [35] have just studied the nonlinear dynamical
investigation of eccentrically stiffened FGM toroidal shell segments surrounded by elastic foundation in thermal environment
To the best of the authors’ knowledge, there are no any publications for nonlinear
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displacement relation of large deflection, the Galerkin method, Volmir’s assumption and the numerical method using fourth-order Runge-Kutta are implemented for nonlinear vibration responses of fluid-shells The fluid is assumed to be non-viscous and ideal incompressible The temperature impacts on from external environment to shells In addition, effects of geometrical and material parameters, imperfection and fluid on the nonlinear vibration responses of shells are shown in figures and discussion
The sandwich toroidal shell segment of thickness h, length L, which is formed by rotation
of a plane circular arc of radius R about an axis in the plane of the curve is shown in Figure 1
The coordinate system (x, y, z) is located on the middle surface of the shell, x and y is the axial and circumferential directions, respectively and z is the normal to the shell surface The thickness of the shell is defined in a coordinate system (y, z) in Fig 2 In this paper, FGM core and ceramic core structures are investigated
For FGM core, the inner layer (z = h/2) and the outer layer (z = -h/2) are isotropic homogenous with ceramic and metal, respectively Suppose that the material composition of the shell varies smoothly along the thickness in such a way that the inner surface is ceramic, the outer surface is metal and the core is FGM
For ceramic core, the inner layer (z = h/2) and the outer layer (z = -h/2) are metal when supposing that the material composition of the shell varies smoothly along the thickness from ceramic core to outside metal
The thickness of the shell is h, ceramic-rich and metal rich are h c , h m, respectively for FGM core and FGM coatings are hz1 and hz2 as in Fig.2 The subscripts m and c are refered to the metal and ceramic constituents respectively Denote V m and V c as volume - fractions of
metal and ceramic phases respectively, where V m + V c = 1 According to the mentioned law, the volume fraction is expressed as
For FGM core:
0,
22
,1)(
22
,2
/)
(
22
,0)(
z h
h z
V
h
h z h h h
h h
h h z z V
h
h z
h z
V
c c
c m
k
m c
m c
m c
(1)
For ceramic core:
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,
22
,2/)
(
22
,1)(
0,22
,2/)
(
2 2
1 2
1 1
z c
z z
c
z k
z c
h h z h h
h z z V
h
h z h
h z
V
k
h z h h h
h z z V
According to the mentioned law, the Young modulus, the mass density and the thermal expression coefficient of FGM core shell are expressed of the form respectively
)()(
)()
)()
)()
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Fig.2 The material properties of FGM sandwich:a) FGM core; b) ceramic core
Fig 3 Geometry and coordinate system of a stiffened FGM sandwich toroidal shell segment
containing fluid (a) stringer stiffeners; (b) ring stiffeners
The von Karman type nonlinear kinematic relation for the strain component across the shell thickness at a distance z from the middle surface are of the form [36]:
According to the classical shell theory the strains at the middle surface and curvatures are
related to the displacement components u, v, w in the x, y, z coordinate directions as [36]:
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂+
w y
w x
w y
w x
w x
v y
u
y
w y
w y
w a
w y v
x
w x
w x
w R
w x u
o o
o o
xy y x
2 2 2 2 2
χχχ
Trang 10
where w o (x, y) is a known function representing initial imperfection of the shell
From Eqs (4) the strains must be satisfied in the deformation compatibility equation
2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 0
2 2
∂
∂
∂+
w x
w x
w y
x
w y x
w x
a
w y
R
w y
x x
y
o o
o xy
)()()(
1
)
(
,,
1
)()()(
xy sh
xy
x y sh
y
y x sh
x
z
E
T z z E z
E
T T T T z z E z
,2
,)
(
,)
(
66 0
66
* 2
22 12
0 2
2 22
0
12
* 12
1 11 0 12 0 1
1 11
xy xy
xy
a a y x
y m x
y
a a y x
y x
m
x
B A
N
C B B
s
A E A A
N
B C
B A
s
A E
A
N
χγ
χχ
εε
χχ
εε
−
=
Φ+Φ++
=
Φ+Φ+
−+
−+
,)
(
,)
(
66 0
66
* 2
2 22 12
0 2 22 0
12
* 12
1
1 11 0 12 0 1
11
xy xy
xy
m m y m x
y x
y
m m y x
m y
x x
D B
M
s
I E D D
C B B
M
D s
I E D B
C
B
M
χγ
χχ
εε
χχ
εε
−
=
Φ+Φ+
−
−+
+
=
Φ+Φ+
−+
+
=
where A ij,B ij,D ij (i, j = 1, 2, 6) are extensional, coupling and bending stiffenesses of the
shell without stiffeners
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,)1(2
,1 ,1
,)1(2 ,1 ,1
,)1(2 ,1 ,1
3 66
2 3 12 2
3 22
11
2 66
2 2 12 2
2 22
11
1 66
2 1 12 2
1 22
11
νν
νν
νν
νν
νν
νν
E D
E
D
D
E B
E B
E
B
B
E A
E A
+ + +
=
=
+ +
h
hh
E
h E ) -h (h-h ) )(k )(k (k
E )
-h (h-h h h ) )(k (k
E )
-h (h-h h h k
E dz
E )- -h (h-h h h k
E h E - h h E zdz
z
E
E
, k
) h h (h E h E h E dz
z
E
E
m c c m m
c m m m m m c
c
c
c c m c cm
m c c cm
m c c cm h
h
m c cm
m c c cm c cm c cm h
h
m c cm c cm m h
h
2 2
3 3
2 2
3 3 2
2
3 3
2 1
2 2
2 1
2 2
1 )
(
2 1 2
1 2
2 )
(
1 )
(
3 3
3 3 2
2 2
1 (
2 2
1 3
2 4 )
( 2 ) (
4
) 3 )( 2 )(
1 (
2 2
) 2 )(
1 (
2 1
2 3
2 4 )
(
0 )
(
1 ) (
) (
1 1
3 1 2 1 2 1 2 3
1 3 2 2 1 2 2 1
2
2
3 2 2
2 2 2
2 2
3 2 2 2 2 2 2 2
1 2
+ +
k
h
E
h h k k
h E
h h k
h E h h h h h E h h h h h h
h
h
E
k k k
h E
h h k k
h E
k
h h h E h h h h h E dz
z
E
E
, k
h h E h h E h E dz
cm z
z cm z z z m z z z z z
z
c
z cm
z z
cm z z cm z z z m h
h
h
h
z z cm z z cm c h
h
in which E cm = E c - E m
,,
2
2 2 2
3 2 2
2 / 2 / 1
2 / 2 / 2
m m
s d
Trang 122 / 1
1
*
1
h h m m
h h m m
m m
1
1 1
*
2
2 2
s
h d
m m
where for FGM core:
.12
)(
1
)(
1
)(
)(
+
−
−+
+
−
−+
+
−
−+
−+
+
=
k
h h h E
k
h h h E k
h h h E h h E h E
h
E
c m m c c c m
m
α α
α α
α α
and for ceramic core:
.12
)(
1
)(
1
)(
)(
)
2 1 2
1
+
++
+
++
+
++
−
−+
+
=
k
h h E
k
h h E k
h h E h h h E h h
E
z z c c z z
m
m
α α
α α
α
The spacings of the stringer and ring stiffeners are denoted by s 1 and s 2 respectively The
quantities A 1 , A 2 are the cross section areas of the stiffeners and I 1 , I 2 , z 1 , z 2 are the second moments of cross section areas and eccentricities of the stiffeners with respect to the middle surface of the shell respectively
The reverse relations are obtained from Eqs (7)
.2
,)
(
,)
(
* 66
*
* 11
* 11
* 12
* 22
* 21
* 12
*
* 22
* 22
* 12
* 12
* 11
* 12
*
22
0
xy xy
xy
a a
a y
x x
y
y
a a
a y
x y
x
x
B N
A
A A
A A B
B N A
N
A
A A
A A B
B N A
N
A
χ γ
χ χ
ε
χ χ
ε
+
=
Φ+Φ
−Φ
−++
+
−
=
Φ+Φ
−Φ
−++
,)
(
)(
,)
(
)(
* 66
*
* 11
* 11
* 12 2
22
*
* 12
*
* 22
* 22
* 12 12
* 22
* 21
* 22
*
* 11
*
* 22
* 22
* 12 1 11
* 12
* 11
* 21
*
11
xy xy
xy
m m a a
a
a a
a y
x y
x
y
m m a a
a
a a
a y
x y
x
x
D N
B
M
A A
A A
C
B
A A
A A B D
D N B N
B
M
A A
A
A
B
A A
A A C B D
D N B
χχ
−
=
Φ+Φ+Φ+Φ
−Φ
−+
+
+Φ+Φ
−Φ
−+
−
−+
=
Φ+Φ+Φ+Φ
−Φ
−
+
Φ+Φ
−Φ
−+
+
−
−+
=
(11)
where
Trang 13
),(
),(
,)
(
,)
(
;
,
1
,
,
1 ,1
66
66
* 66 1
11
* 12 12
* 11
* 21 2
22
* 12 12
* 11
* 22 12
* 12 1 11
2 22
1
1 11
66
* 66 12
* 12 2
2 22
* 22 1
1 11
*
11
A
B B C
B A B A B C
B A B
A
B
B A C B A B B
A C B
A
B
A s
A E A s
A
E
A
A A
A A s
A E A A
s
A E A
A
m m
m m
=+
−
=+
−
=
−+
=
−+
,)
(
,)(
,)
(
* 66 66
66
*
66
* 21 2 22
* 11 12
12
*
21
* 22 12
* 12 1 11
12
*
12
* 22 2 22
* 21 12 2
2 22
*
22
* 21 12
* 11 1 11 1
1 11
*
11
B B
D
D
B C B B B
D
D
B B B C B
D
D
B C B B B s
I E
D
D
B B B C B s
I E
=
−+
−+
=
Consider a toroidal shell segment filled inside by an incompressible fluid and subjected to a
lateral pressure q(t) varying on time Under this mechanical load the shell and fluid will
vibrate simultaneously, moreover the fluid reacts on the shell by a dynamic fluid pressurep L
The nonlinear motion equations of the considered shell based on the classical shell theory are given by [36]:
, - 2 2
2
, ,
L 1 2 2 1 2
2 2 2 2
2 2
2 2 2 2
2 2
2
2
2 2 1
2 2 1
p t
w t
w q
a
N R
N y
w y
w N y x
w y x
w N x
w x
w N y
M y
N
x
N
y x o y
o xy
o x
y xy
∂
∂
= + + +
∂
∂ +
∂
∂
∂ +
∂
∂ +
∂
∂ +
ρ
ρ
(12) where ε is damping coefficient, and
+
−
−+
+
=
2 2 1
1 1
A s
A k
) h h (h h
c cm
−
−++
++
+
=
2 2 1
1 2
1 2
1 2
1
1
)(
)(
s
A s
A h
h h k
h h h
z z
ρ
The dynamic pressure of fluid p acting on the shell is expressed as follows: L
Trang 14
t
L L
where ρLis the mass density of fluid and ϕLis the fluid velocity potential
According to the Stein-McElman assumption [29], for a shallow toroidal shell segment the equation of the fluid velocity potential can be written approximately in the cylindrical
coordinate system (x, θ, r)
01
1
2 2 2 2 2
2
=
∂
∂+
∂
∂++
∂
∂+
∂
∂
x r
r r r
L L
L
θ
ϕϕ
)
L
x m t f
Then the solution of Eq (16) can be expressed as following:
θ π π
L
x m L
r m I t
in which I n is Bessel’s function of the first kind of order n
Certainly substitution Eq (19) into the left side of Eq (16), leads to the identity Satisfying boundary condition (17) with the use of Eqs (18) (19), we obtain
dt df L
a m I m
L t
aI m n m
m n L
)(
' λλ
λ
L
a m
Trang 15
2 2
'
)(
)(
t
w m t
w I
aI t
m n m
m n L L L L
λρ
ϕ
in which
)(
)(
'
m n m
m n L L
I
aI m
λλ
λρ
= ( wherem Lis the mass of fluid corresponding to the vibration of shell; so-called “added mass effect”)
Putting the expression of dynamic fluid pressure (21) into Eqs (12) and using the Volmir’s assumption [37] ρ1(d2U/dt2)→0, ρ1(d2V/dt2)→0 because of u << w,v << w, we can rewrite the system of motion equations (12) as follows:
2 ) ( 2
2 2 2 2
2 2
2 2 2 2
2 2
2
2
t
w t
w m q
a
N R
N y
w y
w N y x
w y x
w N x
w x
w N y
M y
x o y
o xy
o x
y xy
∂
∂ +
= + + +
∂
∂ +
∂
∂
∂ +
∂
∂ +
∂
∂ +
(22) Two first equations of Eqs (22) are satisfied identically by introducing the stress function as:
y x
F N
x
F N
2 2
2
,
Substituting Eqs (21) into deformation compatibility equation (5) and substituting Eqs (11) into the third equation of motion (22), taking into account expressions (4) and (23), yields a system of equations:
,02
1
1)
2(
)2(
2 2 2 2 2 2 2
2 2
2 2 2 2 2 2 2
2
2
2 2 4
4
* 12 2 2
4
* 66
* 22
* 11 4
4
* 21 4
4
* 22 2 2
4
* 12
∂
∂
∂
∂+
∂
∂+
∂
∂
∂
−++
∂
∂+
∂
∂+
∂
∂
∂
−+
∂
∂
x
w y
w y
w x
w y x
w y x
w y
w x
w y
x
w x
w
a
y
w R y
w B y x
w B
B B x
w B y
F A y x
F A
o
.2
11
)2(
)4(
2
2 2 2 2 2
2 2
2 2 2
2 2 2 2 2 2 2 2 2 4
4
*
12
2 2
4
* 66
* 22
* 11 4
4
* 21 4
4
* 22 2 2
4
* 66
* 21
* 12 4
4
* 11 1
2
2
1
q y
w y
w x
F y x
w y x
w y x
F x
w x
w y
F x
F a y
F R
y
F
B
y x
F B
B B x
F B y
w D y x
w D
D D x
w D t
w t
w
m
o o
o L
∂
∂
∂+
++
∂
∂+
∂
∂+
shell segments
Trang 16
Suppose that the shell is simply supported at its edges and acted on by a lateral pressure q and
a pre-axial compression p The boundary conditions can be expressed as:
w = 0, M x = 0, N x = -ph, N xy = 0 at x = 0; L
Satisfying these conditions the deflection and the imperfection of the shell can be chosen in the form
y x
t
f
w= ( )sinγm sinβn ,
y x
numbers along y-axis, respectively,
o
f is constant, f o can be put as: f o =µh(0≤µ<1), h is the thickness of shell
Substituting Eq (26) into the left side of Eq (24), the solution for stress function F of this
equation can be expressed as:
22
sinsin2
cos2
cos
2 2 2 01 3
2 1
x N
y N y x F
y F
x F
where N 01 = -p o h is the pre-axial compression load, N o2 is the negative average circumferential load and
)2
2
* 12
* 66 4
* 11
2 2
4
* 12 2
2
* 66
* 22
* 11 4
2(2
//
416
)2(
2
n n
m m
m n
n n
m m
A A
A A
a R
B B
B B B
F
β β
γ γ
γ β
β β
γ γ
+
−+
−
−+
−++
Furthermore, the toroidal shell segments have to satisfy the circumferential closed condition [20] as:
Trang 17
02
w y
w a
w dxdy
4
22
*
* 12
*
* 11
* 11
* 12
2 2
2
2 1 2
1
* 22 2
1
* 21
* 12 1 2
1
* 3
* 12
* 11 2 2
+Φ
−Φ
−+
a L A A
A A f f a L f
a
L
f a f
B f
B a L A N f F A a L A N f
F
A
a a
a o
n n
n m m
n n
m o
m
n o
n
m
ππ
βπ
β
βγ
δδγ
δδββ
δδγπ
γ
δδβπ
Ω
−+
+
11
* 12 1 2
++
−
n m m
n n
m m
n n
a
B B
F A F
A a
δ δ γ
δ δ β β
δ δ γ γ
δ δ β β
δ δ γ π
*
* 11
2sin
sin2
cos2
cos
2
* 11
* 12 1 2
* 11
2 2 1 3
2 1
x A
A N f f A
y N y x F
y F
x F
o n m n
++
+
−+
(31) Substituting Eq (26) and (31) into Eq (25) and then applying Galerkin’s procedure in the range 0≤x≤L; 0≤y≤2πa, we obtain the following equation:
0 2 2
2 1
2
2 2 1 2
2
1
* 11
* 12 2
1 1 2
* 11
* 12 2 1 2 2
* 11
* 12 2 1 3 2 2 3 1 1 2
2
1
= Ω
− Ω
Ω +
+ +
+ + +
∂
∂ +
∂
∂
+
a L
f a
L
q
a L aA
A R N
f A
A N
f f A
A N
f U f U f U t
f t
f
m
n m o n n
m
n m o o n m o n n
m o L
π β γ δ δ β
γ δ δ β
γ β
β γ
ε ρ
ρ
(31)
in which
Trang 18
11 2
1
1 2 (F F ) /8A
a L a A
f A
a L
F R
F a
F B
F B
F a L F
F f
U
m
n o n
n
n m m
m m
m n
m o
n
m
πγ
δδββ
ψβ
π
δδβγβ
γβ
γπ
βγ
δδβ
γ
2 1
* 11
* 2 2
* 1 2
* 12 4
* 2
* 21 4
* 1 2
1
* 2
* 1 2
2
2
48
9
204
416
163
2)(
6
−+
+
++
++
=
ψπβγ
δδψβ
π
δδβγβ
γβ
γπ
βγ
δδβ
γ
γβββ
γγ
ββ
γγ
a L a
f a
L F f R
F a
F B
F B F a L
f F
F
f
F a
F R F B F B
B B F B D D
D D
D
U
n m o n o
n m m
m m
m n
m
o o
n
m
m n
n n
m m
n n
m m
2 1 2
* 3 2 1
* 2 2
* 1 2
* 12 4
* 2
* 21 4
* 1 2
1
* 2
2
* 3
2
* 3 4
* 12
* 3 2 2
* 66
* 22
* 11
* 3 4
* 21 4
* 22 2 2
* 66
* 21
* 12 4
*
11
3
2 9
20 4
4 16
16 3
4 ) (
4
) 2 (
) 4 (
− + +
+ + +
−
− +
− + + +
+ +
+ +
=
The fundamental frequencies of the shell can be drawn from Eq (31) as follows:
L
n n
m o mn
m A
A N
U
+
Ω+
=
1
2 2
* 11
* 12 2 1 3
ρ
β β
γ
Equation (31) is an equation of motion to investigate the nonlinear vibration of imperfect
stiffened FGM sandwich toroidal shell segments containing fluid Using the fourth-order
Runge-Kutta method into Eq (31) combined with initial conditions, the nonlinear dynamic
responses of full-filled fluid shells can be observed
To the best of the authors’ knowledge, there are no any publications about thermal
vibration of imperfect three-layered toroidal shell segments containing full-filled fluid Thus,
the results in this paper are compared with FGM cylindrical shell (i.e a toroidal shell
segment with R → ∞) and hc = hm = 0
Firstly, comparison of natural frequencies of perfect un-stiffened FGM cylindrical
shells without containing fluid are illustrated in Table 1a Those were also compared with
these of Ref [18] using the energy method and Love’s shell theory It is observed in Table 1
that very good agreements are obtained in this comparison
shells
1 Ref [18] 12.917 13.126 13.177 13.234 13.344 13.457 13.528 13.549 13.572
Trang 19
2 Ref [18] 31.603 32.151 32.277 32.418 32.683 32.959 33.157 33.221 33.296 Present 31.687 32.231 32.331 32.467 32.731 33.007 33.186 33.239 33.298
3 Ref [18] 88.267 89.818 90.173 90.569 91.309 92.075 92.617 92.795 93.001 Present 88.566 90.000 90.347 90.742 91.499 92.274 92.766 92.911 93.064
4 Ref [18] 168.99 171.97 172.65 173.41 174.83 176.29 177.33 177.66 178.06 Present 169.48 172.31 172.94 173.66 175.06 176.54 177.50 177.79 178.09
5 Ref [18] 273.14 277.95 279.05 280.28 282.57 284.93 286.61 287.15 287.79 Present 273.90 278.34 279.41 280.61 282.94 285.34 286.88 287.33 287.82
Secondly, the natural frequencies of isotropic full-filled fluid cylindrical shells in Table 1b are computed and compared with the results of Pellicano and Amabili [38] using continuation techniques and direct simulations based on geometric nonlinearities with the Donnell’s nonlinear shallow-shell theory As can be seen, a very good agreement is obtained
in the comparison with the results of Ref [38]
Table 1b Comparisons of natural frequencies (rad/s) of ful-filled fluid cylindrical shells
In next sections, thermal vibration of imperfect three-layered toroidal shell segments containing fluid under mechanical loads and effects of geometrical parameters, imperfections, fluid and temperature are considered The Aluminum and Alumina materials are used with
2 9/10
E m = × ;ρm =2702kg/m3; αm =23×10−6 0C−1; E c =380×109N/m2;3800
=
c
ρ kg/m3; αc =5.4×10−6 0C−1 and Poisson’s ratio is chosen to be 0.3 The density of fluid is chosen ρL =1000kg/m3 The parameters n 1 = 50 and n 2 = 50 are the number of stringer and ring stiffeners, respectively and the some parameters are confirm as: h1=h/2;
h2=h/2; d2=h/2;d1=h/2
Fundamental frequencies
The geometrical parameters of problem are given as follows:
h = 0.01m; hc = 0.2h; hm = 0.1h; h1=h/2; h2=h/2; d2=h/2;d1=h/2; a = 200h; R = 400h; L = 2a