Hydroelastic analysis of circular very large floating structures

HYDROELASTIC ANALYSIS OF CIRCULAR VERY
LARGE FLOATING STRUCTURES

LE THI THU HANG

NATIONAL UNIVERSITY OF SINGAPORE
2005


HYDROELASTIC ANALYSIS OF CIRCULAR
VERY LARGE FLOATING STRUCTURES

BY

LE THI THU HANG
B.E. (Hanoi University of Civil Engineering)

A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2005


ACKNOWLEDGEMENTS

I wish to convey my sincere gratitude to my supervisor Professor Wang Chien
Ming for his encouragements, critical comments and suggestions throughout the
research work. His invaluable guidance and advice have greatly shaped my thinking
over the past two years and what I have learnt and experience will undoubtedly be
useful for my future studies.

I am indeed grateful to Professor Tomoaki Utsunomiya from Kyoto University for
his advice and useful discussions on this research study.
I would like to express my thanks to the National University of Singapore for
providing the financial support in the form of the NUS scholarship and facilities to
carry out the research. Thanks are also extended to my colleagues in Civil Engineering
Department for their kind assistance.
Finally, special thanks to my family and my friends for their encouragements and
love in many respects.

Le Thi Thu Hang

i


TABLE OF CONTENTS

ACKNOWLEGEMENTS ..................................................................................................... i
TABLE OF CONTENTS…………………………………………………………………...ii
SUMMARY……………………………………………………………………………...........v
NOTATIONS………………………………………………………………………………..vii
LIST OF FIGURES………………………………………………………………………....x
LIST OF TABLES……………………………………………………………………..….xiv

CHAPTER 1 INTRODUCTION ............................................................................... 1
1.1 BACKGROUND INFORMATION ON VLFS ........................................................1
1.2 LITERATURE REVIEW ........................................................................................5
1.3 OBJECTIVE OF RESEARCH .................................................................................8
1.4 LAYOUT OF THESIS..............................................................................................9

CHAPTER 2 VIBRATION ANALYSIS OF UNIFORM

CIRCULAR PLATES....................................................................... 11
2.1 PROBLEM DEFINITION ......................................................................................11
2.2 GOVERNING EQUATIONS AND METHOD OF SOLUTION ..........................12
2.3 RESULTS AND DISCUSSIONS ...........................................................................16
2.4 CONCLUDING REMARKS..................................................................................31

CHAPTER 3 VIBRATION ANALYSIS OF STEPPED
CIRCULAR PLATES....................................................................... 32
3.1 PROBLEM DEFINITON .......................................................................................32
3.2 METHOD OF SOLUTION AND MATHEMATICAL MODELLING.................32

ii


3.3 RESULTS AND DISCUSSIONS ...........................................................................38
3.4 CONCLUDING REMARKS..................................................................................61

CHAPTER 4 HYDROELASTIC ANALYSIS OF UNIFORM
CIRCULAR VLFS ............................................................................ 62
4.1 BASIC ASSUMPTIONS AND PROBLEM DEFINITION ...................................62
4.2 BOUNDARY VALUE PROBLEMS AND GOVERNING EQUATIONS ...........63
4.3 MODAL EXPANSION OF MOTION ...................................................................66
4.4 SOLUTIONS FOR RADIATION POTENTIALS .................................................67
4.5 SOLUTIONS FOR DIFFRACTION POTENTIALS .............................................69
4.6 EQUATION OF MOTION IN MODAL COORDINATES ...................................70
4.7 NUMERICAL RESULTS.......................................................................................72
4.8 CONCLUDING REMARKS..................................................................................76

CHAPTER 5 HYDROELASTIC ANALYSIS OF STEPPED
CIRCULAR VLFS ............................................................................ 77

5.1 PROBLEM DEFINITION ......................................................................................77
5.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS ......................77
5.3 EQUATIONS OF MOTION IN MODAL COORDINATES.................................80
5.4 RESULTS AND DISCUSSIONS ...........................................................................83
5.5 CONCLUDING REMARKS..................................................................................89

CHAPTER 6

CONCLUSIONS............................................................................. 90

6.1 CONCLUSIONS.....................................................................................................90
6.2 RECOMMENDATIONS ........................................................................................91

iii


REFERENCES ..................................................................................................................... 92

APPENDICES ...................................................................................................................... 97
APPENDIX 1 ELEMENTS OF MATRIX [K]9x9 FOR NON-AXISYMETRIC
VIBRATION OF STEPPED CIRCULAR PLATE ..............................97
APPENDIX 2 ELEMENTS OF MATRIX [K]6x6 FOR AXISYMETRIC
VIBRATION OF STEPPED CIRCULAR PLATE ............................102

iv


SUMMARY

This thesis presents a hydroelastic analysis of pontoon-type, circular, very large

floating structure (VLFS) under action of waves. The coupled fluid-structure
interaction problem may be solved by firstly decomposing the unknown deflection of
the plate into modal functions associated with a freely vibrating plate in air. The
second step involves substituting the modal functions into the hydrodynamic equations
and solving the boundary value problem using the boundary element method. The
modal amplitudes of the set of equations of motion obtained are then back substituted
into the modal functions and the stress-resultants functions for the deflections and
stress-resultants of the VLFS under the action of waves.
Although one may use any form of admissible functions for the vibration
modes, it is essential that the final stress-resultants satisfy the natural boundary
conditions along the free edges of the plate. Recently, Wang et al. (2001) and Xiang et
al. (2001) showed that the use of the classical thin plate theory for modeling the
pontoon-type VLFS and well-known energy methods (such as the Ritz method and the
finite element method) for vibration analysis yield modal stress resultants that (a) do
not satisfy the natural boundary conditions and (b) contain oscillations/ripples in their
distributions, affecting the accuracy of the peak values and their locations. When these
modal solutions are used in the hydrodynamic analysis, the final stress-resultants will
also contain these aforementioned inaccuracies. The use of the more refined plate
theory of Mindlin (1951) that incorporates the effects of transverse shear deformation
and rotary inertia, the accuracy of the stress-resultants, especially the transverse shear
forces and twisting moments maybe improved. In order to develop accurate numerical

v


solution for detecting the hydroelastic response of VLFS, there is a need to obtain
benchmark solutions, especially the vibration modes and modal stress-resultants of
freely vibrating plates. As circular plate with free edge is the one can be obtained the
exact vibration results, this study focuses on VLFS with a circular planform. By
obtaining exact mode shapes and modal stress-resultants of circular Mindlin plate, the

hydroelastic results are expected to be accurate.
More specifically, we consider circular VLFS with constant thickness as well as
thickness variation. A comparative study on deflection and stress-resultants between
two kinds of circular plates (by keeping constant volume of material) is conducted.
Numerical results show that the stepped circular VLFS gives much better performance
than uniform circular plate because the final deflection and modal stress-resultants
maybe reduced. Therefore, it would be beneficial to design stepped circular VLFS
instead of uniform thickness one.
The formulations for vibration analysis and hydroelastic analysis for uniform and
stepped circular VLFS are given in explicit forms and the solutions obtained maybe
regarded as almost exact. Such exact solutions should be extremely useful for the
preliminary design of a circular VLFS.

vi


NOTATIONS

A

amplitude of incident wave (m)

b

step location for step thickness junction

D

plate rigidity of uniform circular plate (kNm)

D0

plate rigidity of reference uniform circular plate (kNm)

D1

plate rigidity of annular sub-plate (kNm)

D2

plate rigidity of core circular sub-plate (kNm)

E

Young modulus (kN/m2)

G

shear modulus (kN/m2)

g

gravitational acceleration (m/sec2)

h

thickness of uniform circular plate (m)

h0

thickness of reference uniform circular plate (m)

h1

thickness of annular sub-plate (m)

h2

thickness of core circular sub-plate (m)

k

wave number

M rr

bending moment per unit length the radial direction (kNm)

M rr

= M rr R / D0 = non-dimensional bending moment

M rθ

twisting moment (kNm)

M rθ

= M rθ R / D0 = non-dimensional twisting moment

n

number of nodal diameter

Qr

shear forces (kN)

Qr

=Q r R 2 / D0 = non-dimensional shear force

vii


R

radius of circular plate (m)

r

radial coordinate (m)

S

non-dimensional plate rigidity of uniform circular plate

s

number of sequence for each mode

S0

non-dimensional plate rigidity of reference uniform circular plate

w

vertical displacement (m)

w

= w / R = non-dimensional vertical displacement

wmax

maximum deflection (m)

α

step ratio of thicknesses = h2 h1

χ

non-dimensional radial coordinate = r / R

φ

velocity potential

φ Dn

diffraction potential

φ ns

radiation potential

γ

density of plate material (kg/m3)

κ2

correction factor

λ

= ϖR 2 γh D non-dimensionalized frequency parameter of circular plate

λ0

= ϖ 0 R 2 γh0 D 0 non-dimensionalized frequency parameter of reference
circular plate

λs

= ϖ s R 2 γh0 D 0 non-dimensionalized frequency parameter of stepped circular
plate

ν

Poisson’s ratio

Θi

potential functions

θ

circumferential coordinate (radiants)

viii


ρ

density of the fluid (kg/m3)

τ

thickness ratio of uniform circular plate =

τ0

thickness ratio of reference uniform circular plate =

τ1

thickness ratios of stepped circular plate =

h1
R

τ2

thickness ratios of stepped circular plate =

h2
R

ω

natural frequency of uniform circular plate

ω0

natural frequency of reference uniform circular plate

ωs

natural frequency of stepped circular plate

ψr

rotary displacement along radial axis of circular plate

ψθ

rotary displacement along circumferential of circular plate

ζ ns

modal apmplitude

∇(•)

Laplacian operator

ix

h0
R
h0
R


LIST OF FIGURES
Figure 1.1

Mega Float in Tokyo Bay, Japan............................................................ 2

Figure 1.2

Floating Oil Storage at Kamigoto, Japan................................................ 2

Figure 1.3

Yumeshima-Maishima Floating Bridge in Osaka, Japan ....................... 3

Figure 1.4

Floating Rescue Emergency Base at Tokyo Bay, Japan ........................ 3

Figure 1.5

Floating island at Onomichi Hiroshima, Japan....................................... 3

Figure 1.6

Floating pier at Ujina Port Hiroshima, Japan ......................................... 3

Figure 1.7

Floating Restaurant in Yokohoma, Japan............................................... 3

Figure 1.8

Floating heliport in Vancouver, Canada................................................. 3

Figure 1.9

Nordhordland Brigde Floating Bridge, Norway.................................... 3

Figure 1.10

Hood Canal Floating Bridge, USA......................................................... 3

Figure 1.11

Types of Floating Structures ................................................................... 4

Figure 2.1

Geometry of a Circular Mindlin Plate ................................................... 12

Figure 2.2a SAP2000 modal stress resultants associated with the fundamental

frequency of a uniform circular plate with free edges .......................... 21
Figure 2.2b Exact modal stress resultants associated with the fundamental frequency

of a uniform circular plate with free edges ........................................... 21
Figure 2.2c Mode shapes and modal stress resultants for free circular plates based on

classical thin plate theory and Mindlin plate theory ............................ 21
Figure 2.3

3D-mode shape plots of uniform circular Mindlin plate.............. 22 to 24

Figure 2.4a

Mode shapes and modal stress resultants for free circular Mindlin plates
with thickness ratio h/R = 0.01. The number of nodal diameters n = 0
(axisymmetric modes) ........................................................................... 22

x


Figure 2.4b Mode shapes and modal stress resultants for free circular Mindlin plates

with thickness ratio h/R = 0.01. The number of nodal diameters is n = 1,

2, 3 and 4, respectively.......................................................................... 23
Figure 2.4c

Mode shapes and modal stress resultants for free circular Mindlin plates
with thickness ratio h/R = 0.01. The number of nodal diameters is n = 5,
6, 7 and 8, respectively.......................................................................... 24

Figure 2.5a

Mode shapes and modal stress resultants for free circular Mindlin plates
with thickness ratio h/R = 0.10. The number of nodal diameters is n = 0
(axisymmetric modes) ........................................................................... 25

Figure 2.5b

Mode shapes and modal stress resultants for free circular Mindlin plates
with thickness ratio h/R = 0.1. The number of nodal diameters is n = 1,
2, 3 and 4, respectively......................................................................... 26

Figure 2.5c

Mode shapes and modal stress resultants for free circular Mindlin plates
with thickness ratio h/R = 0.1. The number of nodal diameters is n = 5,
6, 7 and 8, respectively.......................................................................... 27

Figure 3.1

Geometry of a Stepped Circular Plate ................................................... 29

Figure 3.2

Frequency parameter λ s versus step location b for Mindlin plates with
reference thickness ratio τ 0 = 0.1, α = 0.5 to 2 .................................... 43

Figure 3.3

Frequency parameter λ s versus step location b for plates with τ 0 = 0.1,

α = 2 and n = 2 .................................................................................... 45
Figure 3.4a Frequency parameter λ s versus reference thickness ratios τ 0 for plates

with step location b = 0.5 and stepped thickness ratio α = 0.5 ............ 46
Figure 3.4b

Frequency parameter λ s versus reference stepped thickness ratio α for
plates with step location b=0.5 and reference thickness ratio τ 0 =0.1 .. 46

xi


Figure 3.5a

Mode shapes (with n = 2, s = 1) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 47

Figure 3.5b

Mode shapes (with n = 0, s = 1) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 48

Figure 3.5c

Mode shapes (with n = 3, s = 1) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 49

Figure 3.5d

Mode shapes (with n = 1, s = 1) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 50

Figure 3.5e

Mode shapes (with n = 4, s = 1) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 51

Figure 3.5f

Mode shapes (with n = 5, s = 1) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 52

Figure 3.5g

Mode shapes (with n = 2, s = 2) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 53

Figure 3.5h

Mode shapes (with n = 0, s = 2) and modal stress resultants for stepped
plates and their reference constant thickness plates ............................ 54

Figure 3.6

Three Dimensional Stress-resultant Plots of Uniform and Stepped
Circular Plates ............................................................................. 58 to 60

Figure 4.1

Geometry of an Uniform Circular VLFS ............................................. 57

Figure 4.2

Deflection for Problem 1, Real part and Imaginary part ...................... 68

Figure 4.3

Deflection Amplitude for Problem 1 .................................................... 68

Figure 4.4

Deflection for Problem 2, Real part and Imaginary part ...................... 68

Figure 4.5

Deflection Amplitude for Problem 2 .................................................... 68

Figure 4.6

Bending moment amplitude for Problem 2 .......................................... 69

Figure 4.7

Twisting moment amplitude for Problem 2.......................................... 69

xii


Figure 4.8

Shear force amplitude for Problem 2.................................................... 69

Figure 5.1

Geometry of a Stepped Circular VLFS ................................................ 72

Figure 5.2

Displacements and Bending Moments for stepped VLFSs and the
reference constant thickness VLFS ...................................................... 81

Figure 5.3

Twisting Moments and Shear Forces for stepped VLFSs and the
reference constant thickness VLFS ...................................................... 82

Figure 5.4

Stresses M rri R /(τ i2 D0 A) , M rθi R /(τ i2 D0 A) , Qri R 2 /(τ i D0 A) for stepped
VLFSs and the reference constant thickness VLFS ............................. 83

xiii

LIST OF TABLES
Table 2.1

Frequency parameters λ = ωR 2 γh / D for free circular Mindlin plates
(ν = 0.3 , κ 2 = 5 / 6 )................................................................................. 20

Table 3.1a Frequency parameter λs for stepped plates with step location at b = 1/2,

reference constant thicknesses τ o = 0.01 and 0.1 ................................. 39
Table 3.1b Frequency parameter λs for stepped plates with step location at b = 1/2 ,

reference constant thicknesses τ o = 0.125 and 0.15 .............................. 40
Table 3.2a Frequency parameter λs for stepped plates with step location at b = 1/3,

reference constant thicknesses τ o = 0.01 and 0.1 .................................. 41
Table 3.2b Frequency parameter λ s for stepped plates with step location at b = 1/3,

reference constant thicknesses τ o = 0.125 and 0.15 ............................. 42
Table 4.1

Parameters for Analyzed Circular VLFSs................................................ 66

Table 5.1

Parameters for Analyzed Stepped Circular VLFSs.................................. 79

xiv

Chapter 1

INTRODUCTION

This chapter introduces the very large floating structures (VLFSs)
and their applications. A literature review on hydroelastic analysis of
pontoon-type VLFS, the objective of research work and layout of the
thesis are presented.

1.1 BACKGROUND INFORMATION ON VLFS
With a growing of population and a corresponding expansion of urban development
in land-scarce island countries and countries with long coastlines, the governments of
these countries have resorted to land reclamation from the sea in order to ease the
pressure on existing heavily-used land space. There are, however, constraints on land
reclamation works, such as the negative environmental impact on the country’s and
neigbouring countries’ coastlines and marine eco-system as well as the huge economic
costs in reclaiming land from deep coastal waters, especially when the cost of sand for
reclamation is very high. In response to both the aforementioned needs and problems,
engineers have proposed the construction of very large floating structures (VLFS) for
industrial space, airports, storage, facilities and even habitation. Japan, for instance,
has constructed the Mega-Float in the Tokyo Bay (Fig. 1.1), the floating oil storage
base Shirashima and Kamigoto (Fig. 1.2), the Yumeshima-Maishima floating bridge in
Osaka (Fig. 1.3), the floating emergency rescue bases in Tokyo Bay, Osaka Bay (Fig.
1.4), the floating island at Onomichi Hiroshima (Fig. 1.5), the floating pier at Ujina
Port Hiroshima (Fig. 1.6), and the floating restaurant in Yokohoma (Fig.1.7). Canada
has constructed a floating heliport in Vancouver (Fig. 1.8) and the Kelowna floating
bridge on Lake On in British Columbia. Norway has the Bergsoysund floating bridge

1

Introduction
and Nordhordland Brigde (Fig. 1.9), while the United States has the Hood Canal
floating bridge (Fig. 1.10) and Korea has a floating hotel. These VLFSs have
advantages over the traditional land reclamation solution in the following aspects:
• They are cost effective when the water depth is large and sea bed is soft
• Environmentally friendly as they do not damage the marine eco-system or silt-up
deep harbours or disrupt the ocean currents
• They are easy and fast to construct and therefore sea-space can be speedily
exploited
• They can be easily removed or expanded
• The structures on VLFSs are protected from seismic shocks since the floating
structure is inherently base isolated
• They do not suffer from differential settlement due to reclaimed soil
consolidation
• Their positions with respect to the water surface are constant and thus facilitate
small boats and ship to come alongside when used as piers and berths
• Their location in coastal water provide scenic body of water all around, making
them suitable for developments associated with leisure and water sport activities.

Figure 1.1 Mega Float in Tokyo Bay, Figure
Japan

2

1.2 Floating Oil
Kamigoto, Japan

Storage

at


Introduction

Figure 1.3 Yumeshima-Maishima Floating
Bridge in Osaka, Japan

Figure 1.4 Floating Rescue Emergency
Base at Tokyo Bay, Japan

Figure 1.5 Floating island at Onomichi
Hiroshima, Japan

Figure 1.6 Floating pier at Ujina Port
Hiroshima, Japan

Figure 1.7 Floating Restaurant in
Yokohoma, Japan

Figure 1.8 Floating heliport in Vancouver,
Canada

Figure 1.9 Nordhordland Brigde Floating Figure 1.10 Hood Canal Floating Bridge,
Bridge, Norway
USA
_________________________
Figures courtesy of Prof E Watanabe, Kyoto University
3

Introduction
VLFS may be classified under two categories, namely the semi-submersible type
and the pontoon-type (see Fig. 1.11). The semi-submersibles type is designed to
minimize the effects of waves while maintaining a constant buoyant force. Thus it can
reduce the wave-induced movement of the structure, and therefore it is suitable for
areas where the water is very deep and very high waves. The semi-submersibles are
kept in position by either tethers or thrusters. In contrast, pontoon-type floating
structures lie on the sea level like a giant plate floating on water (see Fig. 1.1).
Pontoon-type floating structures are suitable for use in only calm waters, often near the
shoreline. The pontoon-type VLFS is very flexible when compared to other kinds of
offshore structures and so the elastic deformations are more important than their rigid
body motions.

Semi-submersible-type
Pontoon-type

Figure 1.11 Types of Floating Structures
This thesis deals with the hydroelastic analysis of pontoon-type circular VLFSs
under action of waves. Both uniform circular VLFS and stepped circular VLFS’s
solutions are considered. This study develops analytical approach for hydroelastic
analysis of these VLFS structures. Exact deflections and stress resultants are given and
should be useful as they served as benchmark solutions for verification of numerical
programs such as BEM or FEM for VLFS analysis.

4


Introduction

1.2 LITERATURE REVIEW
The hydroelastic analysis of very large floating structures has attracted the attention
of many researchers, especially with the construction of the Mega-Float in Tokyo Bay
in 1995. Many researchers analyzed pontoon-type VLFS of a rectangular planform
(Utsunomiya et al. 1998, Mamidipudi and Webster 1994, Endo 2000, Ohkusu and
Namba 1998, Namba and Ohkusu 1999), mainly because of practical reasons for this
shape and also it lends itself for the construction of semi-analytical methods for
solution. There are very few studies on non-rectangular VLFS. Hamamota and Fujita
(2002) treated L-shaped, T-shaped, C-shaped and X-shaped VLFSs. It was suggested
that hexagonal shaped VLFSs be constructed as Japanese Society of Steel Construction
(1994). Circular pontoon-type VLFSs are considered by Hamamoto (1994), Zilman
and Miloh (2000), Tsubogo (2001), Peter et al. (2003) and Watanabe et al. (2003). So
it so appears that more studies on VLFSs of circular shape should be carried out.
The hydroelastic analysis of VLFS may be conducted 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. The commonly-used approaches for the analysis of VLFS
in the frequency domain are the modal expansion method and the direct method. The
principal difference between the modal superposition method and the direct method
lies in the treatment of the radiation motion for determining the radiation pressure.
In the direct method, the deflection of the VLFS is determined by directly solving
the motion of equation without any help of eigenmodes. Mamidipudi and Webster
(1994) pioneered this direct method for a VLFS. In their solution procedure, the
potential of diffractions and radiation problems were established first, and the
deflection of VLFS was determined by solving the combined hydroelastic equation via
the finite difference scheme. Their method was modified by Yago and Endo (1996)

5


Introduction

who applied the pressure distribution method and the equation of motion was solved
using the finite element method. Ohkusu and Namba (1996) proposed a different type
of direct method which does away with the commonly-used two-step modal expansion
approach. Their approach is based on the ideal that the thin plate is part of the water
surface but with different physical characteristics than those of the free surface of the
water. The problem is considered as a boundary value problem in hydrodynamics
rather than the determination of action. In Kashiwagi’s direct method (1998), the
pressure distribution method was applied and the deflection was solved from the
vibration equation of the structure. In order to archive a high level of accuracy in a
very short wave length regime as well as short computational times and fewer
unknowns, he uses bi-cubic B-spline functions to present the unknown pressure and a
Galerkin’s method to satisfy the boundary conditions. His method for obtaining
accurate results in the short wave length regime is a significant improvement over the
numerical techniques proposed by other researchers (Maeda et al. 1995, Takaki and
Gu 1996, Yago 1995, Wang et al. 1997), who have also employed the pressure
distribution method.
The modal expansion method consists of separating the hydrodynamic analysis and
the dynamic response analysis of the plate. The deflection of the plate with free edges
is decomposed into vibration modes that can be arbitrarily chosen. In this regard,
researchers have adopted different modal functions such as products of free-free beam
modes (Maeda et al. 1995, Wu et al. 1995, 1996 and 1997, Kashiwagi 1998, Nagata et
al. 1998, Utsunomiya et. al. 1998, Ohmatsu 1998), B-spline function (Lin and Takaki
1998), Green function (Eatock Taylor and Ohkusu 2000), two-dimensional polynomial
functions (Wang et al. 2001) and finite element solutions of freely vibrating plates
(Takaki and Gu 1996). Also, it should be remarked that the modes may be that of the

6


Introduction

dry type or the wet type. While the most analyses used the dry-mode approach (Wu et
al. 1997) because of its simplicity and numerical efficiency, Hamamoto et al. (1995,
1996, 1997, 2002) have conducted studies using the wet-mode approach. Although the
dry-modes superposition and wet-modes superposition can lead to the same solution,
the wet-mode superposition approach is considered to be rather complex (for example,
an iterative procedure is needed to obtain a wet-mode).
In order to validate the numerical methods and to check the accuracy and
convergence of solutions, analytical solutions are important needed for hydrodynamic
response of VLFSs. Moreover it was shown that numerical techniques such as the
finite element method (FEM) and the Rayleigh-Ritz method actually do not furnish
accurate distributions of modal stress-resultants (Wang et al. 2001, Xiang et al. 2001).
In fact, the distributions of the numerically obtained modal stress-resultants contain
oscillations and they do not satisfy completely the natural boundary conditions at the
free edges. The reason for this shortcoming is that the FEM and the Rayleigh-Ritz
method do not directly enforce the natural boundary conditions as is done for the
essential boundary conditions. Therefore exact vibration solutions, especially exact
modal stress resultants, for free plates are important to have as benchmark solutions for
assessing the accuracy of numerical results.
A plate shape that admits the derivation of exact solutions for plates with free edges
is the circular shape. Probably, the first paper on hydroelastic analysis of circular
VLFS is the one written by Hamamoto and Tanaka (1992). They developed an
analytical approach to predict the dynamic response of a flexible circular floating
island subjected to stochastic wind-waves and seaquakes (see also Hamamoto, 1994).
Their approach was based on the superposition of wet-modes (free vibration modes in
still-water).

7


Introduction

Researchers have also been seeking analytical solutions. Zilman and Miloh (2000)
obtained a closed form solutions of the hydroelastic response of a circular floating
plate in shallow waters. Tsubogo (2000, 2001) solved the same floating circular plate
problem independently. However, the assumption of shallow water limits the
applicability range, and the extension of their method to finite-water depth has not yet
been made.

1.3 OBJECTIVE OF RESEARCH
Complementing the above studies, this study will develop analytical approach for
hydroelastic analysis of a circular VLFS. The analysis is carried out in the frequency
domain using the modal expansion method (dry-mode superposition). The aims of the
present study are
• to determine the exact mode shape and modal stress-resultants of freely vibrating
circular plate with uniform thickness as well as stepped thickness variation.
• to solve the hydroelastic problem of pontoon-type circular VLFS under action of
waves.
In the open literature, many analysts used the classical thin plate theory for
modeling the pontoon-type VLFS. For more accurate evaluation of modal stress
resultants, the more refined plate theory proposed by Mindlin (1951) should be
adopted instead. The Mindlin plate theory allows for the effect of transverse shear
deformation and rotary inertia which become significant in high vibration modes.
Moreover, the stress-resultants are evaluated from first order derivatives of deflection
and rotations. In contrast, the stress-resultants in the classical thin plate theory are
expressed in terms of second order and third order derivatives of deflection. Therefore,
more accurate stress-resultants can be obtained by using the Mindlin plate theory.

8


Introduction

VLFSs are usually designed as optimally as possible with properties sometimes
varying abruptly over their cross-sections for economic distribution of materials.
Owing to the variations in structural properties, the deflection pattern may have a very
different spatial character from a similar structure with uniform structural property
characteristics. Therefore, vibration problem of stepped circular plate is tackled with
and the hydroelastic analysis solution for stepped circular VLFS is presented in this
thesis. The influence of the stepped thickness design on the vibration frequencies,
mode shapes and modal stress resultants is explored by comparing with the
corresponding results of a reference circular plate of constant thickness and equal
volume. Comparison study of the deflections and stress-resultants of stepped circular
VLFS and its reference uniform thickness circular VLFS are also given. These exact
solutions and research findings should be useful in the hydroelastic analysis and costeffective design of circular VLFSs with a stepped thickness variation.

1.4 LAYOUT OF THESIS
This thesis comprises of two parts. Part 1, consisting of Chapters 2 and 3, deals
with the free vibration analysis of a uniform and non-uniform circular plates vibrating
in air, normally reformed to dry mode solution. Part 2, consisting of Chapters 4 and 5,
is concerned with the hydroelastic analysis of these circular VLFSs under actions of
waves.
More specifically, Chapter 2 deals with the free vibration analysis of circular plates
with uniform thickness. Adopting the Mindlin plate theory, the governing equations
and the boundary condition are presented. They are solved analytically and the natural
frequencies, mode shapes and modal stress-resultants are given.

9