Finite element analysis of a floating rectangular plate

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 03 | Mar-2014, Available @ http://www.ijret.org 237
FINITE ELEMENT ANALYSIS OF A FLOATING RECTANGULAR
PLATE
Bhagirathi Tripathy1
, Pratap Kumar Pani2
1
Lecturer, Department of Civil Engineering, IGIT, Sarang, Odisha, India
2
Professor, Department of Civil Engineering, IGIT, Sarang, Odisha, India
Abstract
The free vibration characteristics of a floating rectangular plate considering the fluid-structure interaction effects between the plate
and the reservoir are studied. The fluid is assumed to be inviscid, incompressible and the reservoir bottom is assumed to be horizontal
and rigid. Finite element technique is used to solve this interaction problem. The eigen frequencies of the plate are computed and the
effects of the flexibility of the plate, depth of the reservoir are investigated.
Keywords -inviscid,irrotational,eigenfrequency,FEM,VLFS, hydroelasticity.
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I. INTRODUCTION
With a growing population and the corresponding demand for
settlement and need for basic and recreational facilities in land,
countries with long coastlines, have resorted to land
reclamation. But, huge cost, impact on coast lines, marine eco
system and sometimes opposition from neighboring countries,
reclamation is not feasible all the time. Hence, the concept of
floating platforms came and has been successfully tried in many
places. Today floating platforms are used for various purposes
like exploration platforms, naval base, air strips, rescue bases,
amusement parks and settlement colonies etc. A lot of research
has been carried out. The earlier works like that of John [1]
were based on the motion of rigid plates due to wave action.
Later investigations were focused on the flexibility of the large
floating plates. Thus, hydroelastic analysis took centre stage in
the analysis of mat-like floating structures. Breakthrough works
by Bishop and Price [2] and Price and Wu [3] led to the full 3-
D hydroelasticity theory, where the Green function method is
used to model the fluid. The plate is modelled as an elastic thin
plate with free edges. The fluid is incompressible, inviscid and
its motion is irrotational so that a velocity potential exists. The
amplitude of the incident wave and the motions of the VLFS
are both small and only the vertical motion of the structure is
considered.
Eigen frequencies of a plate either immersed or in contact with
a fluid decrease significantly compared to those in vacuum,
especially for the fundamental one. This is because the
vibration of the plate is transferred to the fluid causing an
increase in the kinetic energy of the surrounding fluid. Though
exact solution for this type of interaction problem is difficult to
obtain, there are many analytical approaches for plates of
different shapes with some rational boundary conditions and
approximations of the fluid domain. In the early phase of work
on this area, Lamb (1920) studied the change in natural
frequencies of a thin clamped circular plate in an aperture of an
infinitely long plane rigid wall in contact with water. Later,
Powell and Roberts (1923) [4]experimentally verified the work
of Lamb and Mclachlan (1932)[5] extended Lamb’s work to
circular plates without any supports. Kwak and Kim (1991) [6]
and later Kwak (1991) [7] obtained the non-dimensional added
virtual mass incremental (NAVMI) factors for circular plates
placed on a free liquid surface using Hankel’s transformation
for axisymmetric modes and all other modes. All these works
are based on the assumption that mode shapes of the plate
remain same both in contact with the fluid and in vacuum.
Kwak (1996) [8] investigated the effect of water on mode
shapes and observed that except the fundamental mode, other
modes are influenced by the presence of water. The distortion
of mode shapes increases with the increase in mode numbers.
The plate was considered to be thin and made of isotropic,
homogeneous and linearly elastic material. Kirchhoff’s theory
for plate vibration and analytical-Ritz method for fluid-structure
interaction were adopted for the analysis.
For the solution of interaction problems, though analytical
methods provide better accuracy, their use is limited to either
very special or simple cases because of the mathematical
complexities involved. However, due to the availability of high
speed computational facilities, several numerical techniques
may be adopted to obtain a meaningful solution of such
complex problems. Among different numerical techniques used,
finite element method (FEM) is mostly preferred due to its easy
implementation in a wide range of problems.
The focus of the present work is to investigate the effect of the
flexibility of the plate and fluid depth on eigen frequencies of
the plate floating over the fluid.

__________________________________________________________________________________________
2. APPLICATION OF RECTANGULAR PLATE AS
VLFS
There are basically two types of very large floating structures
(VLFSs), namely the semisubmersible- type and the pontoon-
type. Semi-submersible type floating structures are raised above
the sea level using column tubes or ballast structural elements
to minimize the effects of waves while maintaining a constant
buoyancy force. Thus they can reduce the wave induced
motions and are therefore suitably deployed in high seas with
large waves. Floating oil drilling platforms used for drilling for
and production of oil and gas are typical examples of semi-
submersible-type VLFSs. When these semi-submersibles are
attached to the seabed, using vertical tethers with high
pretension as provided by additional buoyancy of the structure,
they are referred to as tension-leg platforms. In contrast,
pontoon-type floating structures lie on the sea level like a giant
plate floating on water. Pontoon-type floating structures are
suitable for use in only calm waters, often inside a cove or a
lagoon and near the shoreline. Large pontoon-type floating
structures have been termed Mega-Floats by Japanese
engineers. As a general rule of thumb, Mega-Floats are floating
structures with at least one of its length dimensions greater than
60 m. Referring to Fig. 1, a Mega-Float system consists of a (a)
very large pontoon floating structure, (b) mooring facility to
keep the floating structure in place, (c) an access bridge or
floating road to get to the floating structure from shore, and (d)
a breakwater (usually needed if the significant wave height is
greater than 4 m) for reducing wave forces impacting the
floating structure.
Fig-1 Components of Mega-Float System
3. THE MODEL OF INVESTIGATION
(RECTANGULAR VLFS)
A rectangular plate of width 2a and length 2b is floating on a
reservoir of infinite extent as shown in Fig. 1. The fluid is
considered to be incompressible and inviscid with small
amplitude motion. The effect of the static pressure is not
considered in the analysis. It is assumed that the reservoir floor
is horizontal and rigid. The plate is uniformly thick and its
material is homogeneous, isotropic and linearly elastic in
nature. In the analysis, the 3-D fluid domain is discretized
considering eight noded brick elements and the plate using four
noded quadrilateral plate elements. Mindlin’s plate bending
theory is used to analyze the plate.
Fig-2 Pontoon type of VLFS under wave action
The fluid-structure system and the coordinate system are shown
in Fig. 2. The origin of thecoordinate system is on the
undisturbed free surface. The z -axis is pointing upwards,
andthe sea-bed is assumed to be flat at z = - h. The VLFS has a
maximum length of 2a in the x direction, a maximum width of
2b in the y -direction, and a draft d in the z -direction.
Theproblem at hand is to determine the response of the VLFS
under the action of wave forces.
3.1. Basic Assumptions for Hydroelastic Analysis of
VLFS
In a basic hydroelastic analysis of pontoon-type VLFSs, the
following assumptions are usually made:
• The VLFS is modeled as an elastic (isotropic/orthotropic) thin
plate with free edges.
• The fluid is incompressible, inviscid and its motion is
irrotational so that a velocity potential exists.
• The amplitude of the incident wave and the motions of the
VLFS are both small and only the vertical motion of the
structure is considered (i.e. we constrained the plate from
moving horizontally in the analysis).
• There are no gaps between the VLFS and the free fluid
surface.
The analysis may be carried out in the frequency domain or in
the time domain. Most hydroelastic analyses are carried out in
the frequency-domain, being the simpler of the two. However,
for transient responses and for nonlinear equations of motion
due to the effects of a mooring system or nonlinear wave (as in
a severe wave condition), it is necessary to perform the analysis
in the time-domain. Below, we present the governing equations,
boundary conditions and briefly describe the commonly used
methods for the analysis in the frequency-domain and in the
time-domain.

__________________________________________________________________________________________
4. GOVERNING EQUATIONS FOR THE PLATE
STRUCTURE
The un-damped free vibration of the gate structure is given as
       0M W K W 
(1)
Where,
 M and  K are the mass and stiffness matrices of the
gate.
W   and
W 
 

are the nodal displacement and
acceleration vectors. In Mindlin’s plate theory where shear
deformations are taken into account, the nodal displacement is
represented as
z x yW W , ,       (2)
zW
is the transverse deflection along Z direction and x y, 
are
the rotations about Y and Z axes. The element stiffness matrix
 ek is split into flexure and shear parts to avoid shear locking.
The corresponding split matrices are represented as
 
3
12
Tf f fh
e A
t
k B D B dA           
(3)
Ts s s s
h
e A
k t B D B dA               
(4)
Where,
  s
D & D 
  are the constitutive matrices,
&f s
B B   
    are the strain-displacement matrices for the
flexure and the shear part respectively. The factor  depends
on the shape of the cross section of the element and is 5/6 for
rectangular cross section as considered here.
Governing equation for the fluid domain considering non
viscous incompressible fluid having irrotational motion is given
as
∇2
P = 0 (5)
5. BOUNDARY CONDITIONS
For the problem under consideration of the floating plate, the
following boundary conditions are adopted (Fig 2)
1. ( , ,0) 0p x y  on fS , where h is the height of the
free water surface above the reservoir bed.
2. ( , , ) 0
p
x y h
n

 

on bS , as the reservoir bed is
horizontal and rigid.
3.
𝜕𝑝
𝜕𝑛
𝑥, 0, 𝑧 = 0,
𝜕𝑝
𝜕𝑛
𝑥, 𝑎, 𝑧 = 0,
𝜕𝑝
𝜕𝑛
0, 𝑦, 𝑧 =
0,
𝜕𝑝
𝜕𝑛
(𝑏, 𝑦, 𝑧0 = 0 on sS , b is the width of the fluid
domain considered for analysis.
6. TIME –DOMAIN ANALYSIS
The commonly-used approaches for the time-domain analysis
of VLFS are the direct time integration method [9, 10] and the
method that uses Fourier transform [11–15]. In the direct time
integration method, the equations of motion are discretized for
both the structure and the fluid domain. In the Fourier transform
method, we first obtain the frequency domain solutions for the
fluid domain and then Fourier transform the results for
substitution into the differential equations for elastic motions.
The equations are then solved directly in the time domain
analysis by using the finite element method or other suitable
computational methods.
7. FINITE ELEMENT FORMULATION FOR THE
FLUID DOMAIN
The weighted integral form of eqn. 5 over an element is given
as
 2
e
e
p d

 
(6)
Where,
 is the weight function and e is the elemental
volume. The pressure at any point inside the element is
interpolated as
1
n
i i
i
p N p

 
(7)
ip (i = 1, 2,----n) is the nodal value of pressure at node i , iN
is the interpolation function corresponding to the node and the
summation is over all the n number of nodes. Using the
Galerkin’s weighted residual method, the ith equation may be
written as
     2 2 2
1 1 1
2 2 2
0
e
n n n
j j j j j j
j j j
i e
N p N p N p
N d
x y z
  

 
   
    
   
 
 
  

(8)

__________________________________________________________________________________________
Since, nodal pressure values are constant, taking them out of
integration and integrating the rest with integration by parts,
eqn. 8 may be represented as
Where, in is the normal to the surface and e indicates
integration over the surface only. Eqn. 9 may be expressed for
the whole domain in matrix form as
    G p Q
(10)
 
           
e
T T T
e
N N N N N N
G d
x x y y z z
      
    
       

(11)
   
e
T
e
S
p
Q N d
n

 


(12)
   nNNNN  ,, 21 (13)
 Q may be separated into different parts for different
boundaries.
         f fs b sQ Q Q Q Q   
(14)
At the free surface
  0fQ 
At the fluid structure interface
   fs f fsQ R a      (15)
   
fs e
T
fs e
S
R N N d

     
(16)
At reservoir bottom surface
  0bQ 
(17)
On either side of the plate, the assumed boundary condition
leads to
  0sQ (18)
Combining all these
    f fsG p R a      (19)
8. COUPLED MOTION OF THE PLATE AND THE
FLUID DOMAIN
The boundary motion is prescribed by the movement of the
plate. Replacing the acceleration
 a
in eqn. 15 with the nodal
acceleration of the gate
  fs eQ S w 
(20)
Where
   
fs e
T
f e
S
S N N d

   
(21)
 N is the interpolation function for pressure in the fluid
domain and
N   is the interpolation function for nodal
displacement of the plate. When the effect of dynamic pressure
on the plate structure is considered, eqn 1 may be rewritten as
       sM W K W f 
(22)
Where,
 sf
is the forcing term due to pressure from the fluid
at the interface.
     
1T T
s
f
f N pd S p

     
(23)
As
  pNp
Now, two coupled equations are obtained as
       G P Q S W  
(24)

__________________________________________________________________________________________
           
1
0
T
f
M W K W S P

  
(25)
      1
P G S W

 
(26)
So, eqn. 26 may be represented as
      0M W K W    

(27)
where,
         SGSMM
T
f
11 

 (28)
The eigenfrequencies are evaluated from eqn. 28 using Jacobi’s
method. The non dimensional frequencies are represented as
2 2 4
/p h pt a D  
(29)
Table-1: Eigenfrequencies of a plate of 1.0 m x 1.0 m x 0.02 m
floating over water of depth 1.0 m
Non
dimensional
frequency
1 2 3 4 5
In vacuum 3.70 9.12 24.60 28.83 35.86
On floating
condition
3.65 7.04 16.95 20.65 26.46
Fig-3 Non Dimensional Frequency
Table-2: Eigenfrequencies of a plate of 1.0 m x 1.0 m for
different thicknesses floating over water of depth 1.0 m
b/th
1 2 3 4 5
0.01 3.57 5.95 14.13 16.96 21.99
0.02 3.65 7.04 16.95 20.65 26.46
0.04 3.67 7.77 19.05 23.55 29.34
0.06 3.67 7.93 19.71 24.50 29.56
0.08 3.65 7.90 19.77 24.68 28.82
0.10 3.60 7.78 19.49 24.46 28.82
Fig-4 Eigenfrequencies for Different Thickness
Table-3: Eigenfrequencies of a plate of 1.0 m x 1.0 m x 0.02m
floating over water for different depth of the fluid
Depth of
fluid
In metre
1 2 3 4 5
1.0 3.65 7.04 16.95 20.65 26.46
1.5 1.75 4.44 10.04 16.79 20.24
2.0 0.99 3.05 5.89 11.67 15.44
2.5 0.62 2.28 3.76 8.47 10.386
3.0 0.44 1.77 2.77 6.30 8.02
Fig-5 Eigenfrequencies for Different Depth
0
5
10
15
20
25
30
35
Ω
Non Dimensional Freequency
Vaccum
Floating
0
10
20
30
40
0 0.05 0.1 0.15
Ω b/th
Eigenfrequencies for Different Thickness
Series1
Series2
Series3
Series4
Series5
0
10
20
30
0 2 4
Ω
Depth (m)
Eigenfrequencies for Different Depth
Series1
Series2
Series3
Series4
Series5

__________________________________________________________________________________________
9. RESULTS AND DISCUSSION & CONCLUSIONS
The changes in eigenfrequencies due to the presence of the fluid
is studied both for the flexibility of the plate and for the depth
of the fluid. As shown in Figure-3, the eigenfrequencies
decrease because of the presence of the fluid. The effect of the
flexural rigidity of the plate is investigated by observing the
reduction for different thicknesses of the plate. As observed
from Figure-4, reduction of the eigenfrequencies increases with
increase in flexibility of the plate. Depth of the fluid domain
plays an important role with eigenfrequencies decreasing with
the depth of the fluid as shown in Figure-5. The advanced
research in this field will definitely will bring a revolution
regarding the acute land problem and other disaster
management problem.
REFERENCES
[1]. John F. On the motion of floating bodies. ommunications
on Pure and Applied Mathematics, Part I: 1949;2:13–57; Part
II: 1950;3:45–100.
[2]. Bishop RED, Price WG” An introduction to ship
hydroelasticity” , Journal of Sound and Vibration, Volume87,
Issue 3, 8 April 1983, Pages 391-407.
[3]. Price WG, Wu Y. In: Niordson FI, Olhoff N, editors.
Hydroelasticity of marine structures, theoretical and applied
mechanics Elsevier Science Publishers; 1985, pp. 311–37
[4]. Powell, J. H., and Roberts, J. H. T., “On the frequency of
Vibration of CircularDiaphragms”, Proceedings of the Physical
Society, London, Vol. 35, pp. 170-182, 1923
[5]. McLachlan N. W., “The accession to inertia of flexible
discs vibrating in a fluid”,Proceedings of the Physical Society
of London, vol. 44, pp. 546-555, 1932
[6]. Kwak, M. K. and Kim, K. C., “Axisymmetric vibration of
circular plates in contact with fluid”,Journal of Sound and
Vibration, vol. 146(3), pp.381-389, 1991
[7]. Kwak, M. K., “Vibration of circular plates in contact with
water”, Journal of AppliedMechanics, ASME, vol. 58, pp. 480-
483, 1991
[8]. Kwak, M. K., “Hydroelastic vibration of rectangular
plates”, Journal of Applied Mechanics,ASME, vol. 63, pp. 110-
115, 1996
[9]. Watanabe E, Utsunomiya T. Transient response analysis of
a VLFS at airplane landing. In: Watanabe E, editor. Proc
International Workshop on Very Large Floating Strutures,
Hayama, Kanagawa, Japan, November 25–28. 1996, p. 24–7.
[10]. Watanabe E, Utsunomiya T, Tanigaki S. A transient
response analysis of a very large floating structure by finite
element method Structural Engrg/Earthquake Engrg, JSCE
1998;15(2): 155s–163s.
[11]. Miao Q, Du S, Dong S, Wu Y. Hydrodynamic analysis of
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ProcIntWorkshopo n Very Large Floating Structures, Hayama,
Kanagawa, Japan, November 25–28.1996, p. 201–8.
[12]. Endo H, Yago K, Chiaki S. Elastic responses of a floating
platform stimulated by dynamic load. Proc 14th Ocean
EngSymp, SocNav Arch Japan. 1998, p. 411–6.
[13]. Ohmatsu S. Numerical calculation of hydroelastic
behaviour of VLFS in time domain. Proc 2nd Int Conf.
Hydroelastic MarineTech, Fukuoka. 1998, p. 89–97.
[14]. Endo H. The behaviour of a VLFS and an airplane during
takeoff/landing run in wave condition. Marine Structures 2000;
13: 477–91.
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JMarine SciTechnol 2000; 5: 89–100.

Finite element analysis of a floating rectangular plate

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