Page 1
Vibration analysis of rectangular plates coupled with fluid
Y. Kerbouaa, A.A. Lakisa,*, M. Thomasb, L. Marcouillerc
aMechanical Engineering Department, E´cole Polytechnique of Montre ´al, Canada C.P. 6079, Succursale Centre-ville,
Montre ´al, Que ´bec, Canada H3C 3A7
bMechanical Engineering Department, E´cole de Technologie Supe ´rieure, Canada 1100, Notre Dame Ouest,
Montreal, Quebec, Canada H3C 1K3
cInstitut de Recherche d’Hydro Que ´bec, 1800 Lionel-Boulet Varennes, Quebec, Canada J3X 1S1
Received 15 January 2007; received in revised form 13 September 2007; accepted 17 September 2007
Available online 5 November 2007
Abstract
The approach developed in this paper applies to vibration analysis of rectangular plates coupled with fluid. This case is
representative of certain key components of complex structures used in industries such as aerospace, nuclear and naval.
The plates can be totally submerged in fluid or floating on its free surface. The mathematical model for the structure is
developed using a combination of the finite element method and Sanders’ shell theory. The in-plane and out-of-plane dis-
placement components are modelled using bilinear polynomials and exponential functions, respectively. The mass and stiff-
ness matrices are then determined by exact analytical integration. The velocity potential and Bernoulli’s equation are
adopted to express the fluid pressure acting on the structure. The product of the pressure expression and the developed
structural shape function is integrated over the structure-fluid interface to assess the virtual added mass due to the fluid.
Variation of fluid level is considered in the calculation of the natural frequencies. The results are in close agreement with
both experimental results and theoretical results using other analytical approaches.
? 2007 Elsevier Inc. All rights reserved.
Keywords: Vibration; Finite element; Added mass; Floating and submerged rectangular plates
1. Introduction
Plates have wide applications in areas such as; modern construction engineering, aerospace and aeronau-
tical industries, aircraft construction, shipbuilding, and the components of nuclear power plants. It is therefore
very important that the static and dynamic behaviour of plates when subjected to different loading conditions
be clearly understood so that they may be safely used in these industrial applications.
It is well known that the natural frequencies of structures in contact with fluid are different from those in
vacuo. Therefore, the prediction of natural frequency changes due to the presence of the fluid is important for
designing structures which are in contact with or immersed in fluid. In general, the effect of the fluid force on
0307-904X/$ - see front matter ? 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2007.09.004
*Corresponding author. Tel.: +1 514 340 4711x4906; fax: +1 514 340 4176.
E-mail address: aouni.lakis@meca.polymtl.ca (A.A. Lakis).
Available online at www.sciencedirect.com
Applied Mathematical Modelling 32 (2008) 2570–2586
www.elsevier.com/locate/apm
Page 2
the structure is represented as added mass, which lowers the natural frequency of the structure from that
which would be measured in a vacuum. This decrease in the natural frequency of the fluid-structure system
is caused by increasing the kinetic energy of the coupled system without a corresponding increase in strain
energy.
An extraordinary amount of effort has been carried out on problems involving dynamic interaction between
an elastic structure and a surrounding fluid medium. Lamb [1] calculated the first bending mode shape of a
circular plate fixed at its circumference, in contact with water. The developed method was based on a calcu-
lation of the kinetic energy of the liquid. The resonant frequency was determined using Rayleigh’s method.
Powell and Robert [2] experimentally verified Lamb’s results. They underlined that their frequencies were
slightly higher than those of Lamb [1]. Lindholm et al. [3] carried out an extensive experimental study of
the response of cantilever plates in air and in contact with water. The plates with different aspect ratios
and thickness were horizontally and vertically placed or inclined. The results were compared with theoretical
predictions. Volcy et al. [19] reported measured results of the fundamental natural frequencies of vertical can-
tilever plates partially and totally submerged in liquid. These experimental results were compared with results
obtained using the fluid finite element method. Values for the fundamental natural frequency obtained using
the finite element method were about 15% higher than those measured in the experiments. This discrepancy
was due to a difference in boundary conditions between experimental and theoretical approaches. In some
cases however, good agreement was reported between the measured and calculated values for the fundamental
natural frequencies. Fu and Price [4] studied the vibration responses of cantilevered vertical and horizontal
plates partially or totally immersed in water. It was assumed that the plates vibrated in a semi-infinite fluid
medium. The vibrational mode shapes of plates in vacuo and in contact with fluid were considered the same.
They used a combination of finite element method and a singularity distribution panel approach to analyze the
dynamic responses of plates in vacuo and also to determine the hydrodynamic coefficients for each element
that is in contact with fluid. They also examined the effect of the free surface. Kwak and Kim [5] studied
the free vibrations of a floating circular plate in contact with the free surface of fluid. They only considered
the axisymmetric vibrations. They calculated the non-dimensional added virtual mass incremental (NAVMI)
factors for clamped, simply supported and free plates. They compared their results with other results from the
literature. The discrepancy in results of Lamb [1] and Powell et al. [2] is explained later by Kwak [6] on the
basis of differences in the fluid boundary conditions between the two methods. An analytical solution for a
simply supported rectangular plate carrying liquid with reservoir conditions at its edges is presented by Soedel
[20]. The harmonic response of the plate-liquid system to a dynamic pressure distribution was expressed in
terms of orthogonal plate-liquid modes. Kwak [13] calculated the added virtual mass of uniform rectangular
plates with simply supported and clamped boundary conditions, vibrating in contact with water. The Green
function was employed to solve the boundary value problem of the water domain. The interfacing domain was
subdivided into a multitude of small panels. This method was combined with the Rayleigh–Ritz method, lead-
ing to the equation of motion for rectangular plates in contact with water. Two cases were considered for the
outside boundary condition, i.e., the case of a plate placed in an aperture of an infinite rigid plane wall and the
case of a plate independently resting on a free surface. The natural frequencies of annular plates in contact
with a fluid on one side were theoretically obtained by Amabili [21] using the added mass approach. The Han-
kel transform is applied to solve the coupled fluid-structure system. Meylan [28] studied the forced vibration of
an arbitrary thin plate floating on the surface of an infinite liquid. The linear potential problem for the liquid is
solved using the appropriate Green’s function. A variation equation is derived and a solution by the Rayleigh–
Ritz method is presented. Haddara and Cao [7] investigated dynamic responses of rectangular plates sub-
merged in fluid. An approximate expression for the evaluation of the modal added mass has been derived.
The effect of water depth on the dynamic behaviour of free edge circular plates was studied by Kwak and
Han [24]. They adopted the Hankel transformation method, which resulted in dual integral equations. The
solution of dual integral equations was calculated numerically using Fourier–Bessel series. The Kirchhoff the-
ory of plates is used to model the elastic thin plate. Amabili [8] studied the vibrations of circular plates resting
on a sloshing liquid free surface. The fully coupled problem between sloshing modes of the free surface and
bulging modes of the plate is solved using the Rayleigh–Ritz method. The case of a circular plate composing
the base of a rigid cylindrical tank filled with fluid was studied by Cheung and Zhou [23] including consider-
ation of the fluid free surface effect. The method of separation of variables was applied to obtain the solution
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
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of the velocity potential of the liquid and the Galerkin method was applied to derive the eigenfrequency equa-
tion of the liquid-plate system. Cheung and Zhou [26] also studied the case of a horizontal rectangular plate
composing the base of a rigid rectangular container. The dynamic characteristics of a vertical cantilever plate
partially in contact with fluid were investigated by Ergin and Ugurlu [27]. The in-vacuo dynamic properties of
the plate were obtained using standard finite-element software. The fluid-structure interaction effects were cal-
culated in terms of the generalized added-mass values independent of frequency using a boundary integral
equation method together with the method of images in order to impose a null potential at the free surface.
Liang et al. [25] adopted an empirical added-mass formulation to determine the frequencies and mode shapes
of submerged cantilevered plates. A numerical study was also performed of the vibrations of cantilever plates
in air and water. These results were then compared with experimental and numerical data.
One of the authors of this article developed, for the case of cylindrical shells, a finite element where the dis-
placement functions were derived from shells’ equations of motion instead of using polynomial displacement
functions [10,22,30–36]. The new model was generated using a more efficient approach combining finite ele-
ment method and classical shell’s theory. In this previous work however, because the element uses a cylindrical
frustum, the method could only be applied to cylindrical, conical shells or spherical shells. In this present arti-
cle, we are using the same approach with a plate finite-element with the intention of expanding the approach
enable modeling of any kind of curved structure subjected to turbulent flow. Because of its usage of classical
plate (shell) theory for the displacement function, the presented method may easily be adapted to take hydro-
dynamic effects into account. For the same reason, we can obtain the high as well as low frequencies with high
accuracy. This is normally of little interest for free vibration analysis, but is of considerable importance in the
determination of the response of such a structure to random pressure fields, such as those generated by upper
and lower turbulent flow.
The transverse displacement function of the plate finite-element is derived from the equation of motion.
Then, mass and stiffness matrices required by the finite element method are determined by exact analytical
integration. The velocity potential and Bernoulli’s equation are adopted to express the fluid pressure acting
on the structure. The product of the pressure expression and the developed structural shape function is inte-
grated over the structure-fluid interface to assess the virtual added mass due to the fluid.
2. Structural model
A typical four-node element and nodal degrees of freedom are shown in Fig. 1b. Each node has six degrees
of freedom that represent the in-plane and out-of-plane displacement components and their spatial
derivatives.
Sanders’ thin shell theory [9] is based on Love’s first approximation and gives zero strain for small rigid-
body motion; this is not the case with other theories (Reissner [16], Vlasov [17], Timoshenko [18]). Using
the general equations of shells, it is possible to derive the equations of motion corresponding to a sphere, cone,
cylinder, plate, etc. The equations of motion of plates may be derived from either the general equations of
shells or other geometries similar to those of cylindrical shells [9]. The outcome is the same. To develop the
equilibrium equations of the rectangular plates, the Sanders’ equations for cylindrical shells are used assuming
the radius to be infinite, h = y and rdh = dy. Both membrane and bending effects are taken into account in this
theory.
2.1. Equilibrium equations and displacement functions
The equilibrium equations of a rectangular plate as a function of displacement components of the reference
surface resulting from the Sanders’ theory can be written as follows:
?
P11o2U
ox2þ P12
oy2
P22o2V
oy2þ P21o2U
oxoyþ P33
o2U
oxoyþo2V
?
ox2
?
?
¼ 0;
ð1Þ
o2V
oxoyþ P33
o2V
oxoyþo2U
¼ 0;
ð2Þ
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Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
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P44o4W
ox4þ
o4W
ox2oy2ðP45þ P54þ 2P66Þ þ P55o4W
oy4¼ 0;
ð3Þ
where U and V represent the in-plane displacement components of the middle surface in X and Y-directions,
respectively. W is the transversal displacement of the middle surface (see Fig. 1). Pij(i = 1,6j = 1,6) are the
coefficients of elasticity matrix [P] given in Appendix.
It is important to note that the development of the circumferential and longitudinal elements used in
dynamic analysis of the closed and open cylindrical shells [10,11,22,30–36], were based on exact solution of
the equilibrium equations of cylindrical shells. This approach resulted in a very precise element that leads
to a fast convergence and less numerical difficulties from the computational point of view. It encouraged us
to develop an element using the same approach for dynamic analysis of rectangular plates. Generally, exact
solution of the equilibrium equations is difficult for the case of rectangular plates. To avoid this issue, we pres-
ent the membrane displacement components in terms of bilinear polynomials and the bending displacement
component by an exponential function. Hence, the displacement field may be defined as follows:
Uðx;y;tÞ ¼ C1þ C2x
V ðx;y;tÞ ¼ C5þ C6x
X
where A and B are the plate dimensions in X and Y directions, respectively (see Fig. 1a). ‘‘x’’ is the natural
frequency of the plate (rad/s). ‘‘i’’ is a complex number and Cjare unknown constants.
It is necessary that our model takes into account membrane effects because the fluid-plate finite element
developed in this paper will be used to model systems composed of parallel plates, cylindrical shells or rect-
angular reservoirs as well as turbine hubs. In this category of structures, the finite element defined in its local
Aþ C3y
Aþ C7y
Bþ C4
xy
AB;
xy
AB;
ð4Þ
Bþ C8
ð5Þ
W ðx;y;tÞ ¼
24
i¼9
Cjeip
x
Aþy
B
ðÞeixt;
ð6Þ
{ } {
i
δ
T
xy, i
y , i
x , iiii
W,WW,W,V,U
=
k { }
δ
k
Y
ye
xe
X
Z
U
V
W
i j
l
{ }
δ
j
{ }
lδ
{ }
iδ
ye
X
A
Y
xe
B
Z
Plate finite element
a
b
}
Fig. 1. (a) Finite element discretization of rectangular plate and (b) geometry and displacement field of a typical element.
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
2573
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co-ordinates does not coincide with the global axes of the structure. The local displacements, mass and stiff-
ness matrices must be transformed to the global system, and therefore membrane effects become extremely
important. Eq. (6) can be developed in Taylor’s series as [12]:
W ðx;y;tÞ ¼ C9þ C10x
Aþ C11y
Bþ C12
x3y
6A3Bþ C20
x2
2A2þ C13
x2y2
4A2B2þ C21
xy
ABþ C14
y2
2B2þ C15
x3
6A3þ C16
x3y2
12A3B2þ C23
x2y
2A2Bþ C17
x2y3
12A2B3þ C24
xy2
2AB2
þ C18
y3
6B3þ C19
xy3
6AB3þ C22
x3y3
36A3B3:
ð7Þ
The displacement field may be rewritten in the form of matrix relations as follows:
8
>
with
U
V
W
>
:
fCg ¼ fC1;C2;...;C24gT;
where [R] is a matrix of order (3 · 24) given in Appendix and {C} is the vector for the unknown constant. The
components of this last vector can be determined using twenty-four degrees of freedom presented for a plate
element as shown in Fig. 1b. The displacement vector of each element is given as:
n
Each node, i.e. ‘‘node i’’, possesses a nodal displacement vector composed of the following terms:
?
where Uiand Viare in-plane displacement components and Wirepresent the displacement components normal
to the middle surface as shown in Fig. 1b. By introducing Eqs. (4, 5 and 7) into relation (10), the elementary
displacement vector can be defined as
<
9
>
>
;
=
¼ ½R?fCg;
ð8Þ
ð9Þ
fdg ¼fdigT;fdjgT;fdkgT;fdlgT
oT:
ð10Þ
fdig ¼ Ui;Vi;Wi;oWi=ox;oWi=oy;o2Wi=oxoy
?T;
ð11Þ
fdg ¼ ½A?fCg;
where [A] is a (24, 24) matrix.
The vector {C} in Eq. (8) will be then replaced by the generalized displacement vector of a quadrilateral
finite element. The displacement field may be described by the following relation:
8
>
where matrix [N] of order (3 · 24) is the displacement shape function of the finite element and the terms of
matrix [A]?1are given in Appendix.
ð12Þ
U
V
W
>
:
<
9
>
>
;
=
¼ ½R?½A??1fdg ¼ ½N?fdg;
ð13Þ
2.2. Kinematics relations
Strain-displacement relations for the rectangular plates are given as [9]
8
>
>
:
ex
ey
2exy
jx
jy
jxy
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
9
>
>
;
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
¼
oU=ox
oV =oy
oV =ox þ oU=oy
?o2W =ox2
?o2W =oy2
?2o2W =oxoy
8
>
>
:
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
9
>
>
;
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
:
ð14Þ
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Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
Page 6
Substituting the displacement components defined in Eq. (13) into the strain–displacement relationship
(14), one obtains an expression for the strain vector as a function of nodal displacements.
feg ¼ ½Q?½A??1fdg ¼ ½B?fdg;
where matrix [Q], of order (6 · 24), is given in Appendix.
ð15Þ
2.3. Constitutive equations
The stress–strain relationship of an anisotropic rectangular plate is defined as follows
frg ¼ ½P?feg;
where [P] is the elasticity matrix (6 · 6) given in Appendix. The elements of [P] characterize the shell anisot-
ropy, which depends on the mechanical properties of the material of the structure. The element of [P] may be
obtained experimentally or analytically depending on the material anisotropy [10,22,29]. The development in
this section covers only the case of the isotropic plate. Substituting Eq. (15) into Eq. (16) results in the follow-
ing expression for the stress vector as a function of nodal displacements.
ð16Þ
frg ¼ ½P?½B?fdg:
The mass and stiffness matrices for one finite element can be expressed as
Z Z
½ms?e¼ qsh
A
ð17Þ
½ks?e¼
A
Z Z
½B?T½P?½B?dA;
ð18:aÞ
½N?T½N?dA;
ð18:bÞ
where dA is the element surface area, h is the plate thickness and qsis the material density and [P],[N] and [B]
are defined in Eqs. (16, 13 and 15), substituting them into Eqs. (18.a and 18.b) we obtain:
h
½ms?e¼ qsh ½A??1
00
½ks?e¼ ½A??1
iT
Zye
iT
0
Zxe
Zye
0
½Q?T½P?½Q?dxdy
Zxe
??
?
½A??1;
ð19:aÞ
h
½R?T½R?dxdy
?
½A??1;
ð19:bÞ
where xeand yeare dimensions of an element according to the X and Y coordinates, respectively. These inte-
grals are calculated using Maple mathematical software.
3. Fluid modeling
Generally, the fluid pressure acting upon the structure is expressed as a function of acceleration. The fluid
force matrices are superimposed onto the structural matrices to form the dynamic equations of a coupled
fluid-structure system. Linear potential flow is applied to describe the fluid effect that leads to the fluid
dynamic forces. The mathematical model is based on the following assumptions: (i) the fluid flow is potential,
(ii) vibration is linear, (iii) since the flow is inviscid, there is no shear and the fluid pressure is purely normal to
the plate wall, and (v) the fluid is incompressible. Based on the aforementioned hypothesis the potential func-
tion, which satisfies the Laplace equation, is expressed in the Cartesian coordinate system as:
r2/ ¼o2/
ox2þo2/
oy2þo2/
oz2¼ 0;
ð20Þ
where / is the potential flow function. Using Bernoulli’s equation the fluid pressure at the solid–fluid interface
may be expressed as:
????
Pjz¼0¼ ?qf
o/
ot
z¼0
:
ð21Þ
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The following separate variable relation is assumed for the potential velocity function:
/ðx;y;z;tÞ ¼ FðzÞSðx;y;tÞ;
where F(z) and S(x. y, z) are two separate functions to be determined.
The impermeability condition of the structure surface requires that the out-of-plane velocity component of
the fluid on the plate surface should match the instantaneous rate of change of the plate displacement in the
transversal direction. This condition implies a permanent contact between the plate surface and the peripheral
fluid layer, which should be:
????
The following expression may be defined by introducing Eq. (22) into Eq. (23)
1
dFð0Þ=dz
For X and Y in the finite element domain (see Figs. 1b and 2), the potential and pressure at the interface are
coupled by the transverse movement of the plate W(x,y,t) and its derivatives. Eq. (24) describes the function
S(x,y,t) in terms of this transverse movement of the plate which itself varies as a function of plate geometry
and time. Therefore, the movement of the fluid at any point on the interface (including the boundaries X and
Y) is intimately linked to the movement of the edges of the plate.
Then, substituting Eq. (24) into (22), results in the following expression for the potential function:
ð22Þ
o/
oz
z¼0
¼oW
ot:
ð23Þ
Sðx;y;tÞ ¼
oW
ot:
ð24Þ
/ðx;y;z;tÞ ¼
FðzÞ
dFð0Þ=dz
oW
ot:
ð25Þ
X
Z
i
j
k
l
{ }
iδ
{ }
δ
j
{ }
δ
k
Fluid
Free Surface
h1
Solid-Fluid Interface
Plate
Element
{ }
lδ
X
Z
j
k
l
i { }
iδ
{ }
δ
j
{ }
δ
Plate finite Element
k
{ }
lδ
Fluid
Rigid Wall
h1
Solid-Fluid Interface
b
a
Fig. 2. (a) Coupled fluid-structure element possessing a free surface of fluid at Z = h1and (b) plate element in contact with fluid bounded
by a rigid wall at Z = h1.
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Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
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Fluid boundary conditions are introduced separately for each element, which allows us to study partially or
totally submerged plates, i.e. vertical plates or inclined plates as well as floating plates. Substituting Eq. (25)
into relation (20) leads to the following differential equation of second order:
d2FðzÞ
dz2
? l2FðzÞ ¼ 0;
ffiffiffiffiffiffiffiffiffiffiffiffiffi
ð26Þ
where l ¼ p
The general solution of Eq. (26) is given as
FðzÞ ¼ A1elzþ A2e?lz:
Substituting Eq. (27) into (25), one gets the following expression for the potential function:
1
A2þ1
B2
q
ð27Þ
/ðx;y;z;tÞ ¼
A1elzþ A2e?lz
dFð0Þ=dZ
ðÞ
oW
ot;
ð28Þ
where A1and A2are two unknown constants. The potential function ‘/’ must be verified for given boundary
conditions at the fluid-structure interface and the fluid extremity surfaces (Z = h1or Z = h2) as well.
3.1. Plate-fluid model with free surface
Assuming that perturbations due to free surface motion of the fluid are insignificant, the following condi-
tion may be applied at the fluid free surface to the velocity potential, see Fig. 2a.
????
where ‘g’ is acceleration due to gravity. The introduction of Eq. (28) simultaneously into relation (29) and (23),
results in the following expression for the potential function
?
where
o/ðx;y;z;tÞ
oz
z¼h1
¼ ?1
g
o2/
ot2;
ð29Þ
/ðx;y;z;tÞ ¼1
l
elzþ Ce?lðz?2h1Þ
1 ? Ce2lh1
?oW
ot;
ð30Þ
C ¼ ðgl ? x2Þ=ðgl þ x2Þ:
The graphical presentation in Fig. A.1 of Appendix demonstrates that the value of C tends asymptotically
towards ?1. This approximation is made in order to avoid non-linear eigenvalues problem. The corresponding
dynamic pressure at the fluid-structure interface becomes
?
ð31Þ
P ¼ ?qf
l
1 þ Ce2lh1
1 ? Ce2lh1
?o2W
ot2¼ Zf1o2W
ot2:
ð32Þ
3.2. Plate-fluid model bounded by a rigid wall
The boundary condition at the upper surface of the fluid represented in Fig. 2b was studied by Lamb [1] and
referred to as the null-frequency condition. This rigid wall boundary condition is expressed as
????
Similarly, by introducing Eq. (28) into relations (33) and (23), we obtain the following expression for the veloc-
ity potential as follows:
?
o/
oz
z¼h1
¼ 0:
ð33Þ
/ðx;y;z;tÞ ¼1
l
elðz?2h1Þþ e?lz
e?2lh1? 1
?oW
ot:
ð34Þ
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2577
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The dynamic pressure for this case is determined as
P ¼ ?qf
l
e?2lh1þ 1
e?2lh1? 1
??o2W
ot2¼ Zf2o2W
ot2:
ð35Þ
In case of totally submerged plate (Fig. 3), the total dynamic pressure will be a combination of the pressures
corresponding to the fluid boundary conditions at both top and bottom surfaces of the plate. The total
dynamic pressure is therefore the sum of lower and upper pressures and can be calculated using Eqs. (32)
and (35), respectively. The resulting pressure is obtained as
P ¼ ?qf
l
1 þ Ce2lh1
1 ? Ce2lh1þe?2lh2þ 1
e?2lh2? 1
??o2W
ot2¼ Zf3o2W
ot2;
ð36Þ
where h1and h2are fluid level on top of the plate and fluid level below the plate surface, respectively. In the
case of floating plate on the fluid free surface (Fig. 4) the resulting pressure is calculated using Eq. (35) at h2
level.
P ¼ ?qf
l
e?2lh2þ 1
e?2lh2? 1
??o2W
ot2¼ Zf4o2W
ot2:
ð37Þ
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 10 203040
Frequency (Hz)
C
A(B)=0.05
A(B)=0.4
A(B)=0.1
A(B)=0.2
A(B)=3
A(B)=1
Fig. A.1. Variation of C with respect to frequency for different values of A and B.
Rigid Wall
Y
Fluid
Plate
Free Surface
Z
h1
h2
Rigid Wall
Y
Fluid
Plate
Free Surface
Z
h1
h2
a
b
Fig. 3. Boundary conditions of plates totally submerged in water: (a) Rectangular plate simply supported at its two short sides, and
(b) Cantilever square plate.
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Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
Page 10
3.3. Calculation of fluid-induced force
Using the procedure of the classical finite element, the fluid-induced force vector can be expressed for one
finite element by the following relation:
Z
where [N] is the shape function matrix of the finite element defined in Eq. (13) and {P} is a vector expressing
the pressure applied by the fluid on the plate (Eqs. 32, 35, 36, and 37). Substituting the transversal displace-
ment given in Eq. (13) into the appropriate pressure expressions which depend on the fluid-structure contact
model as well as the boundary conditions, one obtains the fluid pressure that is applied on the plate finite ele-
ment. The dynamic pressure is then defined as
fFge¼
A
½N?TfPgdA;
ð38Þ
fPg ¼ Zfi½Rf?½A??1f€dg:
The coefficients Zfi(i = 1–4) depend on the fluid-structure contact model (Eqs. 32, 35, 36, and 37). The matri-
ces [Rf] and [A]?1are given in Appendix. Placing Eq. (39) and shape function matrix [N] into relation (38)
leads to the vector of fluid forces as
Z
where dA is elementary surface of fluid-structure interface area.
ð39Þ
fFge¼ Zfi
A
½A??1
hiT½R?T½Rf?½A??1dAf€dg;
ð40Þ
4. Calculation of global mass and stiffness matrices
The studied rectangular plate is subdivided into a series of quadrilateral finite elements (see Fig. 1a) such
that each of them is a smaller rectangular plate. An in-house computer code has been developed to establish
the structural matrices of each element based on the equations developed using this theoretical approach. The
global matrices are obtained by superimposing the mass and stiffness matrices for each individual element.
After applying the boundary conditions, the matrices are reduced to square matrices of order 6*N–NC, where
N is the number of nodes in the structure and NC is the number of constraints applied. The global equations
system for motion of a rectangular plate interacting with a fluid can be represented as follows:
½Ms? ? ½Mf?ð Þf€dTg þ ½Ks?fdTg ¼ f0g;
ð41Þ
where subscripts ‘s’ and ‘f’ refer to the plate in vacuo and fluid filled respectively. [Ms] and [Ks] are the global
mass and stiffness matrices of the plate in vacuo. [Mf] represents the inertial force of the fluid; {dT} is the dis-
placement vector. The natural frequencies and mode shapes of the coupled fluid-structure system are obtained
by solving the typical eigenvalue problem expressed by
?
Det ½Ks? ? x2½Ms? Mf?
?¼ 0:
ð42Þ
Plate
Plane YZ
Rigid Wall
Y
Fluid
Free
Surface
Z
h2
Plate
Plane XZ
Rigid Wall
X
Fluid
Free
Surface
Z
h2
Fig. 4. Rectangular plate clamped at all sides and floated on the free surface of the fluid.
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
2579
Page 11
5. Results and discussions
The solid plate finite element used in this work had been previously employed to study the dynamic behav-
iour of a plate without fluid [14]. The computed natural frequencies were compared to those obtained by other
theories and from experiments and the results were in very good agreement. This approach resulted in a very
precise element that led to fast convergence and less numerical difficulties from the computational point of
view.
In this paper, the modal results obtained using the proposed theory of plates coupled with fluid are com-
pared with those of experiments and other research work to show the applicability, reliability and effectiveness
of the presented formulation. The first example calculates the natural frequencies of a steel rectangular plate
simply supported at its two short sides as shown in Fig. 3a. The plate is totally submerged in water. This case
was experimentally studied by Haddara and Cao [7]. The mechanical and geometrical properties of plate are:
Young’s modulus E = 207 Gpa, material density qs= 7850 kg/m3, Poisson’s ratio m = 0.3, A = 0.20165 m,
B = 0.655 m and thickness h = 0.963 mm. The experimental apparatus was set up by placing the plate in a
rectangular reservoir with dimensions 1.3 m · 0.55 m · 0.8 m. The total dynamic pressure is therefore the
sum of lower and upper pressures and can be calculated using Eqs. (32) and (35), respectively. The results
obtained using the present theory are listed in Table 1 along with those of using the experimental approach
described in Ref. [7]. As can be seen, there is good agreement between the numerical results and corresponding
experimental values.
The modal analysis of a cantilever square plate that is totally submerged in water (see Fig. 3b) is carried out
in the next example. The plate dimensions and mechanical properties are
A ¼ B ¼ 10m; h ¼ 0:238m; qs¼ 7830 kg=m3; m ¼ 0:3
This plate was studied experimentally by Lindholm et al. [3] and theoretically by Fu and Price [4]. Accord-
ing to the results listed in Table 2, we can conclude that the dynamic responses of the square plates are com-
puted with a good precision when compared with experimental and other analytical approaches.
The dynamic behaviour of structures may be influenced by changing the fluid height. The variation of the
natural frequencies as a function of fluid level is verified in the following example. Modal frequencies of the
previous cantilever square plate submerged in water (see Fig. 3b) at different fluid levels (h1) have been calcu-
lated in [4]. The same structure is investigated here by using our finite element. It is assumed that the water
height under the plate is sufficiently high (h2> plate length) so that it has no influence on the dynamic behav-
iour of the coupled system [7]. The natural frequencies of the first three modes calculated at different fluid lev-
els are listed in Table 3. We note that the frequencies decrease with increasing fluid level (h1) and they become
independent of h1(fluid level) when this reaches a level higher than 50% of the plate length, which is the same
and
E ¼ 206 Gpa:
Table 1
Natural frequencies of a plate, simply supported at two opposite sides, submerged in water
Mode number Experiment [7]Present theory
1
2
3
4
5
28.72
117.125
154.51
281.79
335.04
31.28
126.40
141.78
285.98
304.57
Table 2
Natural frequencies of a cantilever square plate totally submerged in water
Mode numberIn vacuoTotally submerged in water
Present theory Fu and price [4]Present theoryFu and price [4] Experiment [3]
1
2
3
12.93
31.69
79.37
12.94
31.93
79.8
7.0
17.16
42.98
7.35
20.19
50.11
6.56
19.66
45.32
2580
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
Page 12
percentage stated in Refs. [3,4]. The results obtained by the present method are in good agreement with those
of Fu and Price [4] and Lindholm et al. [3].
The last example in this series of horizontal plates deals with the dynamic analysis of a rectangular plate
clamped at all sides and floated on the free surface of the fluid as shown in Fig. 4. In this case the fluid pressure
is expressed by Eq. (35) at h2level. The eigenvalues calculated using the present theory are compared with
results obtained analytically by Kwak [13] and with those generated by experiments [15]. Kwak [13] also stud-
ied the same plate in contact with fluid and placed into the hole of an infinite rigid wall. Using the theoretical
and experimental factors of NAVMI (non-dimensional added virtual mass incremental), the variation of fre-
quency ratio (Frequency in vacuo/ Frequency in contact with water) of a rectangular plate is presented for
different ratios of plate dimensions in Fig. 5. The results coincide well with the experimental and analytic val-
ues. It is noted that the results calculated using this theory are much more closer to those of a floating plate on
the fluid free surface than a plate put in a reservoir bounded by a rigid wall. This confirms that the proposed
boundary conditions are similar to those of a plate in contact with fluid without an infinite rigid wall [13].
When a vertical plate is partially immersed in a container of fluid, its behaviour is completely different from
that of a horizontal submerged plate since the immersed finite elements are subjected to the fluid pressure
whereas the dry elements vibrate in vacuum. In order to validate our model we considered a plate belonging
to the series of cantilevered plates studied experimentally by Lindholm et al. [3] (see Fig. 6). The same plate
was calculated by Ergin and Ugurlu [27] using a boundary integral equation method. The plate is made of steel
with an area of 203.2 mm · 1016 mm and a thickness of 4.84 mm. The physical properties of the material are
as follows: Young’s modulus = 206 GPa, Poisson ratio = 0.3 and mass density = 7830 kg/m3. The fluid den-
sity is 1000 kg/m3. Vibration analysis was carried out at four levels of immersion (25%,50%,75% and 100%).
The fluid pressure applied on the submerged part of the plate is equal to twice the pressure expressed by Eq.
(35). Fig. 7 shows the natural frequencies corresponding to the first two modes calculated for different level of
Table 3
Natural frequencies of a cantilever square plate submerged in water as function of fluid level (h1variable and h2? A)
Mode
h1/A = 0.05
h1/A = 0.1
h1/A = 0.3
h1/A P 0.5
Present theoryRef. [4] Present theoryRef. [4] Present theory Ref. [4] Present theoryRef. [4] Ref. [3]
Flexion
Torsion
Flexion
8.6
21.09
52.92
8.95
23.1
55.7
8.17
20.05
50.32
8.04
21.54
53.23
7.41
18.16
45.59
7.51
20.34
50.83
7.002
17.16
42.98
7.35
20.19
50.11
6.56
19.66
45.32
1.6
1.8
2
2.2
2.4
2.6
2.8
0.51.0 1.5
Plate dimension ratio (A/B)
2.0 2.5 3.03.5
Frequency ratio ωvacuo/ωfluid
Kim et al.Experimental [15]
Present Theory
Kw ak's Theory [13]
Fig. 5. Fundamental frequency ratio (x(in vacuo)/x(floated)) of a plate fixed at its four sides, floating on a fluid free surface, as a function of
plate dimension ratio (A/B).
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
2581
Page 13
immersion. The results are in good agreement with those of other works [3,27]. It can be noted that the natural
frequency of a cantilevered plate decreases significantly when the immersed part is less than half the length of
the plate.
Fluid Free surface
Fluid
Cantilever
plate
B
Y
Z
S
Fig. 6. Vertical cantilever plate gradually submerged in a large fluid reservoir.
1
2
3
4
5
0 0.250.5 0.751
Immersed part/Plate length
Immersed part/Plate length
Natural frequency (Hz)
Present method
Lindholm et al. [3]
Ergin et al. [27]
5
10
15
20
25
30
0 0.25 0.50.751
Natural frequency (Hz)
Present method
Lindholm et al. [3]
Ergin et al. [27]
b
a
Fig. 7. Variation of natural frequencies of a vertical cantilever plate gradually immersed in fluid as a function of (S/B) ratio: (a) first mode,
(b) second mode. where S is the immersed section and B is the plate’s length.
2582
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
Page 14
By examining the results obtained above for different examples we can conclude that this numerical model
is able to predict the vibrational characteristics of plates in contact with or submerged in fluid.
6. Conclusions
The purpose of the investigation described in this paper is to determine natural frequencies of rectangular
plates in water, which constitute important structural components in various sectors of industry. The plates
may be completely submerged or floating on the free surface of fluid. Natural frequency is affected by the
added mass associated with the inertia of the fluid that is forced to oscillate when the structure vibrates. Thus
it is necessary to evaluate the virtual added mass due to fluid when predicting and analyzing the vibratory
responses and stresses of structures that are in contact with fluid. The model is developed using a Hybrid
approach; combining finite element frame work and classical shell theory. This theoretical approach is much
more precise than usual finite element methods. The in-plane and out-of-plane displacement components are
modelled using bilinear polynomials and exponential functions, respectively. The transverse displacement
function of the plate finite-element is derived from equilibrium equations of a rectangular plate. The mass
and stiffness matrices are determined by exact analytical integration. The velocity potential and Bernoulli’s
equation are adopted to express the fluid pressure acting on the structure. The product of the pressure expres-
sion and the developed structural shape function is integrated over the structure-fluid interface to assess the
virtual added mass due to the fluid. The effect of the fluid depth is explored in this work. We can conclude
from our work that the accuracy of the frequencies calculated using our model is either very good or at least
sufficient for practical purposes. Our results compare favourably with other analytical or experimental models
for the cases studied.
Moreover, it is worthy to note that the present method offers the following advantages compared to other
existing models: (i) it can be used for rectangular plates, either uniform or non-uniform (thickness or other
geometric discontinuities) subjected to any boundary conditions, (ii) the simplifying hypothesis that the mode
shapes of the plate in contact with the fluid are the same as in a vacuum is not assumed in this work, and (iii)
the pressure of the fluid is calculated and applied separately at each finite element. This provides the advantage
that our method can be used to study vertical, horizontal or angled plates which are totally or partially
submerged.
The presented approach can also be used to model any curved structure subjected to random pressure.
Examples of industrial applications include; (a) dynamic analysis of steam generator components; (b) a set
of parallel plates; and (c) turbine blades that may be subjected to random pressure induced by fully developed
turbulent flow.
Appendix.
This appendix contains some equations, which are referred to in this work. Matrix [R] (3 · 24)
½R? ¼
1
x
A
y
B
xy
AB
0000000000
00001
x
A
y
B
xy
AB
000000
000000001
x
A
y
B
x2
2A2
xy
AB
y2
2B2
2
66664
0000000000
0000000000
x3
6A3
x2y
2AB
xy2
2AB2
y3
6B3
x3y
6A3B
x2y2
4A2B2
xy3
6AB3
x3y2
12A3B2
x2y3
12A2B3
x3y3
36A3B3
3
77775
Matrix [Q] (6 · 24)
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
2583
Page 15
½Q? ¼
0
1
A
0
y
AB
000000000000
000000
1
B
x
AB
00000000
00
1
B
x
AB
0
1
A
0
y
AB
00000000
00000000000
?1
0
A2
00
x
A3
?
y
A2B
000000000000
?1
0
B2
00
000000000000
?2
AB
0
?2x
A2B
2
666666666666664
00000000
00000000
00000000
00
?
xy
A3B
?
y2
2A2B2
0
?
xy2
2A3B2
?
y3
6A2B3
?
xy3
6A3B3
?
x
AB2
?y
B3
0
?
x2
2A2B2
?
xy
AB3
?
x3
2A3B2
?
x2y
6A2B3
?
x3y
6A2B3B2
?
2y
A2B
0
?x2
A3B
?
2xy
A2B2
?
y2
AB3
?
x2y
A3B2
?
xy2
A2B3
?
x2y2
2A3B3
3
7777777777777775
Matrix [P] (6 · 6)
½P? ¼
p11
p12
0
p14
p15
0
p21
p22
0
p24
p25
0
00
p33
00
p36
p41
p42
0
p44
p45
0
p51
p52
0
p52
p55
0
00
p63
00
p66
2
666666666664
3
777777777775
:
The elements of the elasticity matrix for an isotropic plate are
P11¼ P22¼ D
¼ ð1 ? mÞK=2
with K ¼
thickness. Matrix [Rf] (3 · 24)
2
P44¼ P55¼ KP12¼ P21¼ mDP45¼ P54¼ mKP33¼ ð1 ? mÞD=2
P66
Eh3
12ð1?m2Þand D ¼
Eh
1?m2where E is the modulus of elasticity, m is the Poisson’s coefficient and h is the plate
½Rf? ¼
0000000000000000
0000000000000000
000000001
x
A
y
B
x2
2A2
xy
AB
y2
2B2
x3
6A3
x2y
2AB
6664
00000000
00000000
xy2
2AB2
y3
6B3
x3y
6A3B
x2y2
4A2B2
xy3
6AB3
x3y2
12A3B2
x2y3
12A2B3
x3y3
36A3B3
3
7775
Matrix [A]?1(24 · 24)
2584
Y. Kerboua et al. / Applied Mathematical Modelling 32 (2008) 2570–2586
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