SYSTEM PERFORMANCE OF A COMPOSITE STEPPED-SLOPE
FLOATING BREAKWATER
LIM CHAI HENG
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Engineering (Coastal and Maritime)
Faculty of Civil Engineering
Universiti Teknologi Malaysia
FEBRUARY 2006
iii
To my friends, teachers, lecturers and professors, who patiently share their ideas,
knowledge and skills with me all these years; and
To students, researchers, academicians and engineers, who have spent time to read
my thesis on the Stepped-Slope Floating Breakwater System; and
⤂㒭ϔⳈ䰾៥៤䭓ⱘᆊҎЁ᮹0DQ݁নঞᏆ䗱ⱘ໻ྥDŽ
iv
ACKNOWLEDGEMENTS
I would like to express my sincere appreciation and gratitude to my thesis
supervisor, Professor Hadibah Ismail, for her supervision, helpful encouragement,
knowledge, continued guidance and moral support throughout my studies as well as
freedom provided to work on this research. I would also like to thank Associate
Professor Ir. Faridah Jaffar Sidek who has given me a tremendous amount of
suggestions, advice, knowledge and guidance as well as for having many meaningful
conversations. I am also grateful to my colleagues, Eldina and Sabri, for sharing their
ideas and providing valuable suggestions; laboratory assistants, Pak Din and Helmy, for
their patience, for answering endless questions and making the time spent together
working at the coastal laboratory a valuable experience for me; and my former colleague
in COEI as well as former flatmate in Wangsa Maju, Teh Hee Min, for sharing his
knowledge and laboratory experience. Their boundless enthusiasm for coastal
engineering is contagious and served as my sources of inspiration when there was doubt.
Many thanks are also due to the staff of COEI, Halim, Asrol (former staff), Kak Ani,
Kak Zim and Azuan, for their support and assistance in my studies.
Associate Professor Ron Cox from the University of New South Wales helped
guide me in my study during his visit to COEI. His expertise in coastal engineering
proved instrumental in helping to efficiently solve problems that arose. I gratefully
acknowledge Dr. Nor Azizi from PPD, UTM, Dr. Colin Christian from the University of
Auckland, Dr. Torsten Schlurmann from Bergische Universität - Gesamthochschule
Wuppertal, Pei Fung and Yew Kim for their contributions.
Special thanks to Andreas Büttcher, for his valuable time and effort to produce
those beautiful drawings for me and making the time in university, K2 and B-1-53 a true
joy; and Edward Andrewes for his effort and contributions to this project. Their great
dedication towards assisting me was above and beyond what I expected. I truly
appreciated their true friendship. I would also like to thank Jeff, Mathieu Mirmont,
Makara Ty, Abdelghani El Mahrad, Wong Teck Soon, Inayati, Jörg Weigl, Chai Chuen
Loon, Lim Meng Hee, Wong Yi Kang, Nadia and those mates in K2 for their moral
support, encouragement and friendship.
I wish to extend my gratitude to COEI for providing necessary funding and
facilities for the study and to Seginiaga Rubber Industries Sdn. Bhd. for the manufacture
of the STEPFLOAT models. Last, but most importantly, I would like to thank Papa,
Mama, my sister and brother, Alan (Soh Chong Zeh), Man (Mati-ur Rehman), my uncle
and late auntie for their unswerving belief and support to me all these years.
v
ABSTRACT
With the increasing demand for multi-purpose use of coastal sea areas in recent
years, the composite stepped-slope floating breakwater system (STEPFLOAT) has been
designed and developed as an alternative engineering solution, mainly for shore
protection and coastal shelter to pioneer the floating breakwater technology in Malaysia.
The unique stepped-slope and multiple sharp-edge features of the STEPFLOAT serve to
intercept waves by dissipating (rather than reflecting) the wave energy through the
formation of wave breaking, turbulence and eddies around the polyhedron as the waves
impinge on the surface of the structure. Laboratory experiments were conducted to study
the performance of the STEPFLOAT as a wave attenuator under unidirectional
monochromatic wave only environment on various system arrangements, i.e. 2-row, 3row, G = b and G = 2b systems. A suggested mooring method using vertical piles as a
modification to the classical mooring system using chains or cables is applied to the
STEPFLOAT system to overcome the problem of roll and sway motions. Additional
tests on the 2-row chain-moored STEPFLOAT were also conducted to allow
comparisons with the fundamental design of the SSFBW system as well as the pilesupported STEPFLOAT. Experiments on restrained case for 2-row and 3-row systems
were performed to evaluate the effect of heave and limited roll motions of the floating
body on wave attenuation. For the present study, a simple conventional method is
applied to decompose the co-existing composite wave record in front of the model into
the incident and reflected waves. Transmitted wave heights were measured at the lee
side of the model. Measured transmission coefficient (Ct), reflection coefficient (Cr) and
loss coefficient (Cl) were related to the non-dimensional structural geometric parameters,
i.e. relative width (B/L), relative draft (D/L) and relative pontoon spacing (G/L), and
hydraulic parameters, i.e. wave steepness (H/L) and relative depth (d/L). Two new nondimensional composite parameters, i.e. BD number and BDG number were introduced
and examined. Experimental results for Ct are presented and compared to the results of
previous studies of various floating breakwater designs done by other researchers.
Empirical equations for predicting the transmission coefficient are developed for each
tested system using Multiple Linear Regression Analysis. The STEPFLOAT, with
relatively smaller structure width, generally has excellent wave attenuation ability over
most of the previous floating breakwaters. The experimental results showed that the
composite pile-supported STEPFLOAT with 3-row, G = b and G = 2b arrangements are
capable to attenuate waves up to 80% of the incident wave height for wave period of less
than 1.33 seconds.
vi
ABSTRAK
Berikutan dengan peningkatan permintaan terhadap penggunaan kawasan pantai
sejak kebelakangan ini, sistem pemecah ombak terapung komposit bercerun tingkat
(STEPFLOAT) telah direkabentuk dan dibangunkan sebagai satu penyelesaian
kejuruteraan alternatif, khasnya untuk kawalan dan perlindungan pantai bagi merintis
teknologi pemecah ombak terapung di Malaysia. Bentuk STEPFLOAT yang bercerun
tingkat dan berbucu tajam berfungsi untuk memintas ombak dengan mengurangkan
tenaganya melalui pembentukan pemecahan ombak, gelora dan eddi di sekitar struktur
polihedron tersebut apabila ombak bertindak pada permukaannya. Ujikaji makmal telah
dijalankan dalam keadaan ombak seragam sehala bagi pelbagai penyusunan sistem, iaitu
sistem 2-baris, 3-baris, G = b dan G = 2b bagi menilai prestasi STEPFLOAT sebagai
struktur pelemah ombak. Penggunaan cerucuk menegak sebagai pengubahsuaian kepada
sistem tambatan secara tradisional yang menggunakan rantai atau kabel telah
diaplikasikan dalam sistem STEPFLOAT bagi mengatasi masalah gerakan oleng dan
huyung. Ujikaji tambahan terhadap STEPFLOAT berbaris dua yang ditambat oleh rantai
juga dilakukan untuk perbandingan dengan sistem SSFBW dan STEPFLOAT yang
ditambat oleh cerucuk. Eksperimen untuk kes terhalang bagi sistem 2-baris dan 3-baris
telah dilaksanakan bagi menilai kesan gerakan lambung dan oleng yang terhad pada
struktur terapung tersebut terhadap pelemahan ombak. Kaedah konvensional telah
digunakan dalam kajian ini bagi menguraikan rekod ombak komposit kepada ombak tuju
dan ombak pantulan. Tinggi ombak terhantar diukur di belakang model. Pekali
penghantaran ombak (Ct), pekali pantulan (Cr) dan pekali kehilangan (Cl) dikaitkan
dengan parameter-parameter tanpa dimensi geometri struktur, iaitu lebar relatif (B/L),
draf relatif (D/L) dan sela relatif (G/L), dan parameter-parameter hidraulik, iaitu
kecuraman ombak (H/L) dan kedalaman relatif (d/L). Dua parameter komposit tanpa
dimensi baru, iaitu nombor BD dan nombor BDG telah diperkenalkan dan diperiksa.
Keputusan ujikaji bagi Ct telah dibandingkan dengan hasil keputusan daripada pelbagai
rekabentuk pemecah ombak terapung yang lain. Persamaan empirikal bagi meramal
pekali penghantaran ombak telah dihasilkan bagi setiap sistem yang dikaji dengan
menggunakan Analisis Regresi Linear Berbilang. STEPFLOAT dengan lebar struktur
yang lebih pendek secara relatif mempunyai keupayaan pelemahan ombak yang lebih
baik berbanding dengan kebanyakan pemecah ombak yang lain. Keputusan ujikaji
menunjukkan bahawa sistem komposit STEPFLOAT bertambatan cerucuk dengan
susunan 3-baris, G = b dan G = 2b berupaya mengurangkan tinggi ombak sehingga 80%
daripada tinggi ombak tuju bagi kala ombak kurang daripada 1.33 saat.
vii
TABLE OF CONTENTS
CHAPTER
TITLE
TITLE
i
DECLARATION OF ORIGINALITY AND EXCLUSIVENESS
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xiii
LIST OF FIGURES
xv
LIST OF PLATES
xxii
LIST OF ABBREVIATIONS AND NOTATIONS
xxiv
LIST OF APPENDICES
1
PAGE
xxviii
INTRODUCTION
1
1.1
Overview
1
1.2
Background of the Problem
3
1.3
Statement of the Problem
5
1.4
Objectives of the Study
6
1.5
Scope of the Study
6
1.6
Significance of the Study
8
viii
1.6.1 An Alternative Engineering Solution for Shore
Protection and Coastal Shelter
1.6.2 Multi-Purpose Breakwater Facility
8
8
1.6.3 An Impetus for Future Research and Development
(R & D)
9
1.6.4 References and Guidelines for Future Research
Development
1.6.5 Great Potential for Commercialization
2
9
10
THEORETICAL BACKGROUND AND LITERATURE
REVIEW
11
2.1
Wave Protection
11
2.2
Floating Breakwater Applicability and Advantages
13
2.3
Operation of a Floating Breakwater as a Wave Attenuator
15
2.4
Wave Control and Attenuation Mechanisms
16
2.4.1
Reflection
18
2.4.2
Dissipation
24
2.4.2.1 Wave Breaking and Overtopping
26
2.4.2.2 Turbulence and Eddies
28
Transformation
29
2.4.3
2.5
Mooring Systems
31
2.6
Performance Considerations
37
2.7
Existing Floating Breakwaters
40
2.7.1
Floating Breakwater by Tsunehiro et al. (1999)
43
2.7.2 Floating Dynamic Breakwater by Federico (1994)
44
2.7.3
Cage Floating Breakwater by Murali and Mani (1997)
47
2.7.4
Rapidly Installed Breakwater System (RIBS) by Resio
et al. (1997)
50
ix
3
THE COMPOSITE STEPPED-SLOPE FLOATING
BREAKWATER SYSTEM (STEPFLOAT)
53
3.1
Introduction
53
3.2
The Evolution of the Stepped-Slope Floating Breakwater
System
54
3.2.1 The SSFBW: Fundamental Design of the
Stepped-Slope Floating Breakwater System
55
3.2.2 The STEPFLOAT: Proposed Improved Design of the
Stepped-Slope Floating Breakwater System
58
3.2.2.1 Design Concepts and Practicability
3.3
4
Considerations
58
(a) Shape and Geometry
59
(b) Alternative Features
62
(c) Material Type
65
(d) Mooring System
66
The Composite STEPFLOAT Breakwater Model
67
EXPERIMENTAL SET-UP AND PROCEDURE
72
4.1
Introduction
72
4.2
Laboratory Facilities and Instrumentation
73
4.2.1 Wave Flume
73
4.2.1.1 General Remarks When Using Wave Flume
74
(a) Decay Due to Internal Friction
77
(b) Decay Due to Viscous Boundary Friction
78
4.2.2 Wave Generating System
81
4.2.3 Wave Absorber
83
4.2.4 Wave Probes and Data Acquisition System
83
4.2.4.1 Wave Probe Calibration
85
4.3
Measurement of Incident, Reflected and Transmitted Waves
87
4.4
Determination of Wave Period and Wave Length
89
x
4.5
5
Experimental Tests on STEPFLOAT
93
DIMENSIONAL ANALYSIS AND EXPERIMENTAL RESULTS
99
5.1
Dimensional Analysis
99
5.2
Experimental Results
105
5.2.1
Steel Chain Mooring System
106
5.2.1.1 Two-row System
106
5.2.2 Restrained Case
5.2.3
5.3
109
5.2.2.1 Two-row System
109
5.2.2.2 Three-row System
111
Vertical Pile System
114
5.2.3.1 Two-row System
114
5.2.3.2 Three-row System
118
5.2.3.3 G = b System
119
5.2.3.4 G = 2b System
121
Performance Evaluation Based on Results Comparison
125
5.3.1
125
Performance Evaluation in terms of Mooring Systems
5.3.1.1 STEPFLOAT vs SSFBW vs Rectangular
Pontoon (with Line Mooring)
5.3.1.2 STEPFLOAT (Vertical Piles vs Steel Chains)
5.3.1.3
5.4
136
STEPFLOAT (Vertical Piles vs Restrained
Case)
5.3.2
125
140
Performance Evaluation of Pile-System STEPFLOAT in
terms of System Arrangements
146
5.3.2.1 Two-row vs Three-row
146
5.3.2.2 Three-row vs G = b
154
5.3.2.3 G = 0 vs G = b vs G = 2b
158
Comparison on the Performance between the STEPFLOAT and
Previous Floating Breakwater Studies
163
xi
6
PARAMETRIC ANALYSIS AND EMPIRICAL
RELATIONSHIPS
166
6.1
Introduction
166
6.2
Parametric Analysis and Empirical Relationships
166
Influence of Relative Width, B/L
168
6.2.1.1 Two-row System
168
6.2.1.2 Three-row System
170
6.2.1.3 G = b System
170
6.2.1.4 G = 2b System
173
Influence of Relative Draft, D/L
174
6.2.2.1 Two-row System
174
6.2.2.2 Three-row System
175
6.2.2.3 G = b System
176
6.2.2.4 G = 2b System
177
6.2.1
6.2.2
6.2.3 Influence of Wave Steepness, H/L
178
6.2.3.1 Two-row System
178
6.2.3.2 Three-row System
179
6.2.3.3 G = b System
180
6.2.3.4 G = 2b System
181
6.2.4 Influence of Relative Depth, d/L
182
6.2.4.1 Two-row System
182
6.2.4.2 Three-row System
183
6.2.4.3 G = b System
184
6.2.4.4 G = 2b System
185
6.2.5 Influence of Relative Gap Size, G/L
186
6.2.5.1 G = b System
186
6.2.5.2 G = 2b System
187
xii
7
8
MULTIPLE LINEAR REGRESSION ANALYSIS AND
DIAGNOSTICS
189
7.1
Introduction
189
7.2
Multiple Linear Regression Analysis
190
7.2.1 Examination of the Variables
192
7.2.2 Multiple Linear Regression Models of Ct
201
7.2.2.1 Two-row Equation
202
7.2.2.2 Three-row Equation
205
7.2.2.3 G = b Equation
208
7.2.2.4 G = 2b Equation
211
7.2.3 Multiple Regression Diagnostics
213
7.2.3.1 Two-row Model
214
7.2.3.2 Three-row Model
217
7.2.3.3 G = b Model
219
7.2.3.4 G = 2b Model
222
CONCLUSIONS AND RECOMMENDATIONS
225
8.1
Summary and Conclusions
225
8.2
Recommendations for Future Research
231
REFERENCES
Appendices A1-A4
234
239 - 250
xiii
LIST OF TABLES
TABLE NO.
TITLE
PAGE
4.1
Average wave height at P1, P2, P3 and P4 with various frequencies
76
4.2
Absolute percentage difference between calculated and measured
wave height at different positions
81
4.3
Mean wave period for various frequencies of wave generating motor
90
4.4
Wave period of model and prototype for various frequencies
90
4.5
Determination of wave number, k, by Bi-Section Method
(for T = 1.33 s , d = 45 cm)
92
4.6
Determination of wave length using the linear dispersion relationship
92
4.7
The structure of experimental tests
96
6.1
Summary of regression analysis parameters for the 2-row vertical
pile-system STEPFLOAT breakwater (second order polynomial)
169
Summary of regression analysis parameters for the 2-row vertical
pile-system STEPFLOAT breakwater (exponential)
169
Summary of regression analysis parameters for the 3-row vertical
pile-system STEPFLOAT breakwater (second order polynomial)
171
Summary of regression analysis parameters for the 3-row vertical
pile-system STEPFLOAT breakwater (exponential)
171
Summary of regression analysis parameters for the G = b vertical
pile-system STEPFLOAT breakwater (second order polynomial)
172
6.2
6.3
6.4
6.5
xiv
6.6
Summary of regression analysis parameters for the G = b vertical
pile-system STEPFLOAT breakwater (exponential)
172
Summary of regression analysis parameters for the G = 2b vertical
pile-system STEPFLOAT breakwater (second order polynomial)
174
Summary of regression analysis parameters for the G = 2b vertical
pile-system STEPFLOAT breakwater (exponential)
174
7.1
Variable range for 2-row empirical model
202
7.2
Variable range for 3-row empirical model
206
7.3
Variable range for G = b empirical model
208
7.4
Variable range for G = 2b empirical model
212
7.5
Comparison of predicted and observed Ct for 2-row system
215
8.1
Summary of Ct, Cr and Cl for the STEPFLOAT in terms of various
system arrangements and mooring systems
227
Summary of Ct predictive equations
230
6.7
6.8
8.2
xv
LIST OF FIGURES
FIGURE NO.
2.1
TITLE
PAGE
Wave responses to a line-moored floating structure described by a
single sinusoid wave train
16
2.2
Relationships among (Ȧt)m, F(x) and I(x)
21
2.3
A-Frame Floating Breakwater
24
2.4
Tethered-Float Breakwater [Harms, 1980]
25
2.5
Pole-Tire Breakwater [Harms, 1980]
26
2.6
Variation of the position of the eddies with the movement of the free
surface [Tolba, 1999]
30
2.7
Alaskan floating breakwater [Morey, 1998]
31
2.8
Anchor-and-line mooring system [McCartney, 1985]
32
2.9
Mooring line configurations for a single pontoon-type floating
breakwater [Sannasiraj et al., 1998]
35
The “X” shaped section of Bombardon floating breakwater was beached
near the shoreline near the left center [Normandy Invasion, 1944]
41
2.11
Perspective view of the floating breakwater [Tsunehiro et al., 1999]
44
2.12
Floating dynamic breakwater [Federico, 1994]
45
2.13
Cage floating breakwater [Murali and Mani, 1997
48
2.10
xvi
2.14
Rapidly Installed Breakwater System concept [Resio et al., 1997]
51
3.1
The SSFBW model
56
3.2
Comparison of Ct for different number of rows of SSFBW at water
depths of 20 cm and 30 cm [Teh, 2002]
57
3.3
A single module of a composite STEPFLOAT
60
3.4
3-D view of a composite STEPFLOAT module formed by a pair of top
half and bottom half units
61
3.5
Module assembly of the 2-row STEPFLOAT breakwater
63
3.6
A single module of a suggested solid-type STEPFLOAT breakwater
64
3.7
Proposed horizontal platform as a walkway for pontoons
64
3.8
Proposed synthetic seaweed curtains as wave screens or silt curtains
65
3.9
Schematic sketch of the suggested STEPFLOAT mooring system using
vertical piles
68
3.10
Top half of the STEPFLOAT module
69
3.11
Bottom half of the STEPFLOAT module
70
3.12
The STEPFLOAT system model is formed by a series of composite
single modules
71
4.1
Schematic layout of the wave flume
73
4.2
The measurements of wave decay without the presence of floating
breakwater model
76
Comparison between the calculated attenuated wave heights due to
boundary friction and measured wave heights
79-80
4.3
4.4
Wave prove calibration
86
4.5
Laboratory and STEPFLOAT model set-up in the wave flume
88
4.6
Relationship between wave period and frequency
90
4.7
Plots of d/L vs. T for d = 45 cm
93
4.8
Details of the vertical pile system
98
xvii
5.1
Definition sketch of a pile-system 2-row STEPFLOAT breakwater
102
5.2
Wave profiles of the composite and transmitted waves for 2-row
model system using vertical piles (f = 42 Hz or T = 0.95 sec)
107
Variation of Ct, Cr and Cl against T for 2-row model system using
chain mooring for D/d = 0.104
108
Variation of Ct, Cr and Cl against T for 2-row model system restrained
from moving for D/d = 0.133
111
Variation of Ct, Cr and Cl against T for 3-row model system restrained
from moving for D/d = 0.133
113
Variation of Ct, Cr and Cl against T for 2-row model system using
vertical piles for D/d = 0.133
116
Variation of Ct, Cr and Cl against T for 3-row model system using
vertical piles for D/d = 0.133
118
Variation of Ct, Cr and Cl against T for G = b model system using
vertical piles for D/d = 0.133
121
Variation of Ct, Cr and Cl against T for G = 2b model system using
vertical piles for D/d = 0.133
123
Specifications and test details of a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
127
Ct vs T for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
128
Ct vs B/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
134
Ct vs D/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
134
Ct vs H/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
135
Ct vs d/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
135
Ct vs T for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
137
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
xviii
5.17
5.18
5.19
5.20
5.21
5.22
5.23
[Ct]red vs T between 2-row STEPFLOAT breakwater using vertical
piles and steel chain mooring
138
Ct vs B/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
139
Ct vs D/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
139
Ct vs H/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
141
Ct vs d/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
141
Comparison between restrained case and vertical-pile system on the
effect of heave and limited roll motions on Ct, Cr and Cl for 2-row
STEPFLOAT
142
Comparison between restrained case and vertical-pile system on the
effect of heave and limited roll motions on Ct, Cr and Cl for 3-row
STEPFLOAT
143
5.24
Ct vs B/L - Comparison of 2-row STEPFLOAT between restrained case
and vertical-pile system
147
5.25
Ct vs D/L - Comparison of 2-row STEPFLOAT between restrained case
and vertical-pile system
147
5.26
Ct vs H/L - Comparison of 2-row STEPFLOAT between restrained case
and vertical-pile system
148
5.27
Ct vs d/L - Comparison of 2-row STEPFLOAT between restrained case
and vertical-pile system
148
5.28
Ct vs B/L - Comparison of 3-row STEPFLOAT between restrained case
and vertical-pile system
149
5.29
Ct vs D/L - Comparison of 3-row STEPFLOAT between restrained case
and vertical-pile system
149
5.30
Ct vs H/L - Comparison of 3-row STEPFLOAT between restrained case
and vertical-pile system
150
5.31
Ct vs d/L - Comparison of 3-row STEPFLOAT between restrained case
and vertical-pile system
150
xix
5.32
5.33
5.34
5.35
5.36
5.37
5.38
5.39
5.40
5.41
5.42
5.43
5.44
5.45
5.46
Ct & 'Ct[2-3] vs T - Performance comparison between 2-row and 3-row
STEPFLOAT systems
151
Ct vs B/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
152
Ct vs D/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
153
Ct vs H/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
153
Ct vs d/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
154
Ct & 'Ct[3-b] vs T - Performance comparison between 3-row and G = b
STEPFLOAT systems
155
Ct vs B/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
156
Ct vs D/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
156
Ct vs H/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
157
Ct vs d/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
157
Ct, 'Ct[0-b] & 'Ct[2b-b] vs T - Performance comparison between G = 0,
G = b and G = 2b STEPFLOAT systems
159
Ct vs B/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
160
Ct vs D/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
161
Ct vs H/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
161
Ct vs d/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
162
xx
Ct vs G/L - Performance comparison between G = b and G = 2b
STEPFLOAT systems
162
Comparison of floating breakwaters efficiency between the
STEPFLOAT and those from previous studies
164
6.1
Measured Ct, Cr & Cl versus B/L of 2-row system with D/d = 0.133
169
6.2
Measured Ct, Cr & Cl versus B/L of 3-row system with D/d = 0.133
171
6.3
Measured Ct, Cr & Cl versus B/L of G = b system with D/d = 0.133
172
6.4
Measured Ct, Cr & Cl versus B/L of G = 2b system with D/d = 0.133
173
6.5
Measured Ct, Cr & Cl versus D/L of 2-row system with D/d = 0.133
175
6.6
Measured Ct, Cr & Cl versus D/L of 3-row system with D/d = 0.133
176
6.7
Measured Ct, Cr & Cl versus D/L of G = b system with D/d = 0.133
177
6.8
Measured Ct, Cr & Cl versus D/L of G = 2b system with D/d = 0.133
178
6.9
Measured Ct, Cr & Cl versus H/L of 2-row system with D/d = 0.133
179
6.10
Measured Ct, Cr & Cl versus H/L of 3-row system with D/d = 0.133
180
6.11
Measured Ct, Cr & Cl versus H/L of G = b system with D/d = 0.133
181
6.12
Measured Ct, Cr & Cl versus H/L of G = 2b system with D/d = 0.133
182
6.13
Measured Ct, Cr & Cl versus d/L of 2-row system with D/d = 0.133
183
6.14
Measured Ct, Cr & Cl versus d/L of 3-row system with D/d = 0.133
184
6.15
Measured Ct, Cr & Cl versus d/L of G = b system with D/d = 0.133
185
6.16
Measured Ct, Cr & Cl versus d/L of G = 2b system with D/d = 0.133
186
6.17
Measured Ct, Cr & Cl versus G/L of G = b system with D/d = 0.133
187
6.18
Measured Ct, Cr & Cl versus G/L of G = 2b system with D/d = 0.133
188
7.1
Scatterplot matrix of the Ct and the 4 independent variables for 2-row
193
7.2
Scatterplot matrix of the Ct and the 4 independent variables for 3-row
193
7.3
Scatterplot matrix of the Ct and the 5 independent variables for G = b
194
5.47
5.48
xxi
7.4
Scatterplot matrix of the Ct and the 5 independent variables for G = 2b
194
7.5
Measured Ct versus BD/dL for 2-row STEPFLOAT system
197
7.6
Measured Ct versus BD/dL for 3-row STEPFLOAT system
197
7.7
Measured Ct versus BDG/dL2 for G = b STEPFLOAT system
198
7.8
Measured Ct versus BDG/dL2 for G = 2b STEPFLOAT system
198
7.9
Scatterplot matrix of the Ct and the 2 independent variables for 2-row
199
7.10
Scatterplot matrix of the Ct and the 2 independent variables for 3-row
199
7.11
Scatterplot matrix of the Ct and the 2 independent variables for G = b
200
7.12
Scatterplot matrix of the Ct and the 2 independent variables for G = 2b
200
7.13
Predicted and observed Ct for 2-row system
214
7.14
Studentized deleted residuals versus predicted Ct for 2-row system
217
7.15
Predicted and observed Ct for 3-row system
218
7.16
Studentized deleted residuals versus predicted Ct for 3-row system
219
7.17
Predicted and observed Ct for G = b system [Equation (7.9)]
220
7.18
Predicted and observed Ct for G = b system [Equation (7.10)]
220
7.19
Studentized deleted residuals versus predicted Ct for G = b system
[Equation (7.9)]
221
Studentized deleted residuals versus predicted Ct for G = b system
[Equation (7.10)]
222
7.21
Predicted and observed Ct for G = 2b system
223
7.22
Studentized deleted residuals versus predicted Ct for G = 2b system
224
8.1
Cross-section of the suggested shapes for the bottom layer of the
STEPFLOAT
233
7.20
xxii
LIST OF PLATES
PLATE NO.
2.1
TITLE
PAGE
A floating dock system supported by mooring piles at the Sutera
Harbour Resort, Kota Kinabalu, Sabah, Malaysia
32
4.1
Wave generating system
82
4.2
Electronic analog control panel
82
4.3
Capacitance-type wave probe
84
4.4
HIOKI 8833 MEMORY Hi CORDER data acquisition system
85
4.5
Wave Flume
94
4.6
Various STEPFLOAT model system arrangements
95
4.7
2-row STEPFLOAT model moored to the flume bed by steel chains
96
4.8
A 2-row model as restrained from moving at four steel piles
97
4.9
A 3-row model with vertical pile system
98
5.1
Wave-structure interaction during experimental tests for 2-row system
using chain mooring
110
Wave-structure interaction during experimental tests for the restrained
2-row system
112
Wave-structure interaction during experimental tests for the restrained
3-row system
115
5.2
5.3
xxiii
5.4
5.5
5.6
Wave-structure interaction during experimental tests for 2-row system
using vertical piles
117
Wave-structure interaction during experimental tests for 3-row system
using vertical piles
120
Wave-structure interaction during experimental tests for G = b system
using vertical piles
122
5.7
Wave-structure interaction during experimental tests for G = 2b system
using vertical piles
124
5.8
A 2-row STEPFLOAT breakwater moored with six steel chains
130
5.9
The transition between slack and taut conditions of the STEPFLOAT
system
131
Induced roll and sway motions generate secondary waves at the
leeside of the floating breakwater during the experiments
132
5.10
xxiv
LIST OF ABBREVIATIONS AND NOTATIONS
List of Abbreviations
ANOVA
-
analysis of variance
ASCE
-
American Society of Civil Engineers
CEM
-
Coastal Engineering Manual
COEI
-
Coastal and Offshore Engineering Institute
DC
-
direct current
EPDM
-
ethylene-propylene diene monomer
ERDC
-
U.S. Army Engineer Research and Development Center
GDP
-
gross domestic product
HDPE
-
high-density polyethylene
i.e.
-
that is (Latin id est)
LCD
-
liquid crystal display
MLT
-
mass-length-time system
MPTT
-
modified power transmission theory
PDH
-
principle of dimensional homogeneity
PPD
-
Pusat Pengajian Diploma
PTT
-
power transmission theory
PVC
-
polyvinyl chloride
RIBS
-
Rapidly Installed Breakwater System
RM
-
Malaysian Ringgit
SBR
-
styrene-butadiene rubber
xxv
SS
-
sea state
SSFBW
-
Stepped-Slope Floating Breakwater (fundamental design)
STEPFLOAT -
Stepped-Slope Floating Breakwater (improved design)
SWL
-
still-water level
UTM
-
Universiti Teknologi Malaysia
VIF
-
variance inflation factor
WAMIT
-
Wave Analysis MIT (numerical program developed by the
Massachusetts Institute of Technology)
List of Notations
a
-
wave amplitude
a0, a1, a2
-
regression coefficients for the second order polynomial trend line
b
-
characteristic breakwater pontoon size or dimension
b, c
-
constants for the exponential trend line
B
-
unstandardized coefficient for independent variable
B
-
breakwater width
Bf
-
wave flume width
Bi
-
partial regression coefficient [i = 1, 2, 3, …]
Bo
-
regression constant
B/L
-
relative width
BD/dL
-
§ B ·§ D ·§ L ·
BD number = ¨ ¸¨ ¸¨ ¸
© L ¹© L ¹© d ¹
BDG/dL2
-
§ B ·§ D ·§ G ·§ L ·
BDG number = ¨ ¸¨ ¸¨ ¸¨ ¸
© L ¹© L ¹© L ¹© d ¹
C
-
wave celerity
Cl
-
loss coefficient
Cr
-
reflection coefficient
B
B
B
xxvi
Ct
-
transmission coefficient
[Ct]red
-
percentage of Ct reduction
d
-
water depth
D
-
draft or depth of submergence
d/L or d/gT2
-
relative water depth
D/L
-
relative draft
Ei
-
incident wave energy
El
-
dissipated wave energy or energy loss
Er
-
reflected wave energy
Et
-
transmitted wave energy
f
-
frequency or a mathematical function
F
-
F ratio (= regression mean square/residual mean square)
g
-
gravitational acceleration = 9.81 m/s2
G
-
gap between modules or pontoon spacing
G/L
-
relative gap or indicative of gap size to wave length ratio or
relative pontoon spacing
H
-
wave height
H1
-
wave height at xp = 0
H2
-
wave height after travelling a distance, xp
Hi
-
incident wave height
Hi/gT2
-
wave steepness parameter
Hi/L or H/L
-
wave steepness
Ho
-
deep water wave height
Hr
-
reflected wave height
Ht
-
transmitted wave height
k
-
number of fundamental dimensions
k
-
wave number (=
L
-
wave length
Lo
-
deep water wave length
n
-
number of dimensional variables
2S
L
2S
)
CT
xxvii
n.a.
-
not available
R
-
correlation coefficient
R2
-
square of the correlation coefficient
t
-
time
T
-
wave period
T model
-
wave period of model
T prototype
-
wave period of prototype
W
-
breakwater width
W/L
-
relative width
x
-
horizontal distance or dummy variable representing independent
non-dimensional variable
xp
-
horizontal distance in wave flume
'Ct [2-3]
-
difference of Ct between 2-row and 3-row STEPFLOAT systems
[= Ct 2-row - Ct 3-row]
'Ct [3-b]
-
difference of Ct between 3-row and G = b STEPFLOAT systems
[= Ct 3-row - Ct G=b]
'Ct [0-b]
-
difference of Ct between G = 0 (or 2-row) and G = b
STEPFLOAT systems [= Ct G=0 - Ct G=b]
'Ct [2b-b]
-
difference of Ct between G = 2b and G = b STEPFLOAT systems
[= Ct G=2b - Ct G=b]
İ
-
phase lag induced by reflection process
Ș
-
displacement of the water surface relative to the SWL
Șt
-
total wave surface profile
ș
-
direction of wave advance (=
Q
-
fluid kinematic viscosity
ȡ
-
fluid density
ȡs
-
density of structure
Ȧ or V
-
wave angular or radian frequency (=
2Sx 2St
)
L
T
2S
)
T
xxviii
LIST OF APPENDICES
APPENDIX
A1
A2
A3
A4
TITLE
PAGE
Results of the multiple linear regression analysis for a 2-row
STEPFLOAT system
239
Results of the multiple linear regression analysis for a 3-row
STEPFLOAT system
242
Results of the multiple linear regression analysis for a G = b
STEPFLOAT system
245
Results of the multiple linear regression analysis for a G = 2b
STEPFLOAT system
248
CHAPTER 1
INTRODUCTION
1.1
Overview
Many citizens from maritime nations have settled close along the coast in order
to make a living, engage in trade and access communication links. The coast provides a
source of food and income through fishing activities and recently has provided areas for
recreation. Malaysia and most of the countries in Southeast Asia region are not seen as
countries of extremes, either extremes of climate or extremes of natural events. Hence, it
sometimes escapes attention and awareness that a large proportion of these countries’
population are exposed to wave disturbance and threatened by coastal erosion. Coastal
problems have caused a significant impact on the economy of many countries. As a
result, it is unavoidable that the government and local shore property owners need to
contend with these problems by implementing some programmes of investment in shore
protection and coastal shelter to reduce the risk of loss of life and property.
2
Most sites for small craft harbours, marinas and coastal aquaculture facilities will
be found to need some form of perimeter protection. The physical conditions of a
proposed site may be relatively calm for most of the time due to natural protection.
However, the wave climate of the site could be moderately rough under storm conditions
due to the arrival of far field waves and eventually significant protection may be
required. Competent coastal shelter and shore protection may take the form of stone
barriers, wave screens or vertical barriers, which are either solid or semipermeable such
as floating breakwaters.
Breakwaters, either fixed or floating, are structures constructed to protect the
shoreline, other coastal structures, marinas, etc. by reflecting and/or dissipating the
incident wave energy and thus reduce wave action in the leeside of the breakwater
system. Permanently fixed breakwaters provide a higher degree of protection than
floating breakwaters. However, a fixed breakwater may not be competitive cost wise
with a floating breakwater in relatively deeper water depths and it may also cause a lot
of detrimental effects to the environment.
Increasing construction costs and environmental constraints encourage
alternative considerations to the traditional fixed breakwaters for coastal shelter and
shore protection. Floating breakwaters have later gained wide attention and subsequently
appeared to be a good choice for wave suppression during most weather conditions.
They are considered as cost-effective and environmentally-friendly substitutes for the
conventional type of breakwaters for the perimeter protection. In recent years, many
research institutions such as Indian Institute of Technology Madras, U.S. Army Engineer
Research and Development Center, The University of Auckland, State University of
New York, Sharif University of Technology, University of New Hampshire, Australian
Water and Coastal Studies Pty. Ltd., University of Wuppertal and Suez Canal
University, have been involved with the design and development of floating breakwaters
for application within semi-protected coastal areas from high energy wave condition.
3
1.2
Background of the Problem
The energetic power of water waves are often difficult to deal with and it has
been the most challenging aspect for coastal engineers. Many coastlines of the world are
facing the need for beach stabilization out of the effects of beach erosion. Coastal
erosion has become a more significant environmental issue nowadays as it poses threats
to many lives, valuable resources and properties, as well as commercial activities in
coastal areas. Human lives, sandy beaches, tourism and industrial development,
infrastructure, agriculture, aquaculture, residential and mangroves are among the
examples of the sacrifices of the destructive wave attack.
The increase in the number of private pleasure crafts and small commercial
vessels has generated a demand for convenient and accessible sheltered mooring. Many
naturally protected or semi-sheltered waters along coastlines in established population
centers have been developed to accommodate the influx of vessels. As a result, artificial
man-made structures will be required to provide perimeter protection from incident
waves where nature offers little or no protection.
It is for these reasons that breakwaters of various dimensions and designs have
been widely employed in locations exposed to wave attack. The purpose of installing a
breakwater is to reduce the incident wave heights to a level commensurate with the
intended use of the site in the leeside of the structure. Cost-effective design and the
required degree of wave protection will dictate possible breakwater alternatives.
The rubble mound breakwater offers advantages in the form of excellent
perimeter protection. It provides a high degree of wave protection and has been widely
used to attenuate surface water waves. The breakwater is a fixed gravity structure
4
constructed of organized pile of graded rocks with a sloped surface, a broad base and a
narrow top or crest, consisting of stones which are large enough to prevent or limit
movement under most wave conditions. Nevertheless, there are many sites in marine
setting where the traditional fixed breakwater is not suitable. Construction of fixed
breakwaters are often more expensive in deeper water depth. Poor foundation condition
is another disadvantage of the application of this fixed structure.
An additional negative aspect is that such a structure will not allow the transport
of sediment along the shoreline. It creates unacceptable sedimentation and water quality
problems due to poor water circulation behind the structure. The base of the fixed
breakwater will lead to the bottom loss for plant and animal habitat. As it is a permanent
fixed structure, a rubble mound breakwater must be high enough to provide reasonable
protection under most storm flood level conditions. If it were to be built at a lower level,
its effectiveness could be severely reduced.
In recent years, coastal engineers become more environmentally conscious.
Coastal engineering projects often have a significant effect on natural ecosystems and
the ensuing environmental damage may make things worse for future generations. In
seeking to revolutionize towards softer engineering solutions by encouraging the
provision of technically, environmentally and economically sound and sustainable
perimeter protection measures, a move towards schemes designed to work with nature
rather than against it has begun to emerge. Floating breakwater has later appeared to be a
cost-effective substitute for the conventional type of breakwaters in providing the
required level of protection while working with the power and resources of nature.
The demand made the concept of the first locally designed floating breakwater
technology possible. In 2002, Teh (2002) has completed his study on wave dampening
characteristics of the fundamental design of a stepped-slope floating breakwater, namely
5
SSFBW. Foreseeing the potential of the stepped-slope floating breakwater system to be
commercialized in the market for the benefit of communities, an improved cost-effective
and practical design to suit the local needs is necessary in order to put forward the
system into the industry. Therefore, the present study is carried out as a continuation of
the work done by Teh (2002).
1.3
Statement of the Problem
There has been quite a number of floating breakwaters available in the market
but until the present invention of the STEPFLOAT breakwater, there is no truly
outstanding solution that has been put forward into the local maritime industry. While
attention was given to the preservation and conservation of natural environment, most
floating breakwaters which utilized the concept of wave reflection in their designs, have
neglected the safety of moving vessels in the vicinity of the floating breakwater system.
Therefore, there arises a need for an economical and environmentally-friendly yet viable
floating breakwater that has an acceptably high efficiency in dissipating wave energy,
instead of reflecting it, to provide the required level of tranquility in areas it desires to
protect. As a result, the first locally designed floating breakwater technology has been
developed. However, the fundamental design of the SSFBW was still in the stage of
infancy. Practical requirements such as manufacturing problem, jointing system,
mooring method, material and economics as well as the viability of the system need to
be considered and incorporated into the improved design of the STEPFLOAT. Thus far,
modifications to the fundamental design of the SSFBW system as well as the mooring
method are required not only to enhance the efficiency of the floating breakwater
system, but also to improve the practicability of the system.
6
1.4
Objectives of the Study
1.
The primary objective of this research is to evaluate and predict the wave
attenuation efficiency of the improved design of a composite stepped-slope
floating breakwater system, i.e. STEPFLOAT, as a wave attenuator.
2.
It is also intended to assess the wave reduction capabilities and the stability of the
structure on several system arrangements (i.e. 2-row, 3-row, G = b and G = 2b
systems) and on three types of mooring systems. Analyses of wave-structure
interaction based on measured laboratory data also need to be performed in order
to allow comparisons of results among the STEPFLOAT breakwater model with
different system arrangements and mooring methods.
3.
Also, it is the goal of this study to develop empirical model for each system
arrangement in the form of functional relationship of various dimensionless
parameters of breakwater geometry and wave conditions, to predict the
performance of the STEPFLOAT breakwater system.
1.5
Scope of the Study
The scope of work throughout the study is orderly stated as follows:
1.
Literature review based on various sources of references such as theses, technical
papers, technical reports, books, patents, articles, etc has been conducted to
provide sufficient knowledge and understanding on wave attenuation concepts,
wave protection systems, laboratory and field studies for the design and
investigations on the performance of a floating breakwater system.
7
2.
A review of the previous design of the SSFBW system and the mooring method
in order to produce modifications and improved design of the STEPFLOAT
system has been carried out.
3.
Planning and design of appropriate and suitable research methodology to conduct
the laboratory experiments.
4.
Fabrication of the STEPFLOAT model and the construction of the composite
STEPFLOAT system with an assembly of several modules connected to one
another by a stainless steel bolt-and-nut system. This part of the study was
conducted in collaboration with the industry, i.e. SEGINIAGA Rubber Industries
Sdn. Bhd.
5.
Design and building of the vertical piling system with aluminium rods, steel
pipes and U-shape steel bars.
6.
Setting up of the equipment and apparatus as well as setting up the model of the
floating breakwater system in the laboratory.
7.
A series of laboratory tests on the STEPFLOAT with different mooring systems
and various system arrangements under wave only condition was conducted.
8.
Experimental data on wave reduction capabilities, the physical mechanism of the
wave-structure interaction and stability of the structure, were observed, recorded
and systematically documented.
9.
Dimensional analysis and parametric analysis were performed. Measured
laboratory data was further analyzed using Multiple Linear Regression Method to
yield empirical wave-structure relationships for pile-supported STEPFLOAT to
predict the performance of the floating breakwater system.
10.
Assessment and comparisons of results of the STEPFLOAT with the previous
study on the SSFBW design as well as studies on other floating breakwaters done
by other researchers.
8
1.6
Significance of the Study
1.6.1
An Alternative Engineering Solution for Shore Protection and Coastal
Shelter
The STEPFLOAT system may provide an alternative solution for coastal
protection with a functional cost-effective engineering design while protecting and
enhancing the environment. The amount of money spent on imported technologies and
products or conventional breakwater construction for coastal protection would therefore
be greatly minimized. Long-term dependence on costly imported technologies would be
an impractical solution and it is not worthwhile. Therefore, a locally designed floating
breakwater system would be an alternative engineering solution to minimize the
unnecessary loss to the country’s resources.
1.6.2
Multi-Purpose Breakwater Facility
The design and development of the multi-purpose STEPFLOAT breakwater
system would eventually benefit the communities, especially those shore property
owners or citizens who reside near the coastal area, as the STEPFLOAT system has
multi-purpose functions such as wave attenuator, walkway platform and encourage
marine habitats. Other advantages that can be provided by the STEPFLOAT system as a
multi-purpose breakwater facility will be further discussed in Chapter III.
9
1.6.3 An Impetus for Future Research and Development (R & D)
The rubble mound breakwater has found frequent application in Malaysia’s
coastal water due to its durability and the high degree of wave protection. Even though it
has been proven as an effective wave attenuation structure, the rubble mound breakwater
is limited to its potential application in certain regions and it causes environmental
degradation. It is for these reasons that floating breakwater designs are of interest for
perimeter protection. The STEPFLOAT system is the first floating breakwater
technology designed locally. Its promising results with good wave attenuation capability
have gained momentum for further research and development. It is believed that this
potential floating breakwater system would be the impetus for continuing future research
and development in Malaysia in this particular engineering design and other coastal and
marine engineering aspects, especially technologies for shore protection and coastal
shelter.
1.6.4
References and Guidelines for Future Research Development
Laboratory experiments have been carried out to gather some information about
the performance of the new design and improved floating breakwater system to provide
data and information for the preliminary design of the prototype-scale field version of
the STEPFLOAT system. Results and findings as well as empirical models from the
laboratory investigations in the present study could be very useful information,
references and guidelines for future research development by other researchers, who
attempt to investigate this particular field of study.
10
1.6.5
Great Potential for Commercialization
The present study on STEPFLOAT system aims to assess the performance of the
improved floating breakwater system design and does not involve any commercial
interest. However, the success of this study, with encouraging results and findings on the
performance of the system, would determine the potential of the STEPFLOAT system to
be commercialized in the market in future. An increasing demand for mooring in coastal
water in Malaysia and simultaneous shortage of suitable construction sites that are
naturally sheltered from wave action generate a need for artificial cost-effective
perimeter protection devices. Keizrul Abdullah (2005) reported that Malaysia with an
extensive coastline of 4809 km has a total eroding coastline of 1372 km. Coastal erosion
and wave attack on other coastal facilities for aquaculture activities, leisure purposes,
etc. have also fostered the development of the environmentally-friendly floating
breakwater system to ameliorate the risk of livelihood and properties of the coastal
communities. It is believed that for these reasons, the potential use of floating
breakwaters in Malaysia and perhaps in South East Asia countries would boom a vast
popular demand for perimeter protection from the more traditional harder defences to
solutions that we now term as “soft engineering”.
CHAPTER 2
THEORETICALBACK
GROUND AN
2.1
D ILTERATURE REV
IEW
W
ave Protection
The destructive power of sea waves has been one of the most challenging tasks
for coastal engineers to contend for decades. The development in coastal waters
generally depends on the anticipated wind-generated wave climate at specific site and
most sites will be found to need some form of wave protection. In some cases, protection
is readily established for on-site conditions because of existing natural wave and wake
protection. Nevertheless, few sites have this natural protection. As a result, man-made
structures are installed to prevent the coastal lines of defence or project sites from being
affected by severe hydraulic loadings from the sea. Fixed breakwaters and floating
breakwaters are often constructed to reduce surface water wave energy entering the
sheltered area.
There are no universal standards or guidelines to define the maximum acceptable
wave height within the proposed project sites. The degree of wave protection needed
will depend on the owner’s or engineer’s perception of acceptable costs and damage
12
risks. These considerations often limit marina and any other small or medium-scale
coastal projects feasible to naturally sheltered or semi-sheltered waters.
The rubble mound breakwaters and berm breakwaters have been widely used to
create protected berthing areas for boats and ships. Rubble mound breakwater is one of
the most conventional fixed breakwaters. The breakwater is a fixed, pervious gravity
structure constructed of graded rock material, with one or more stone underlayers and a
cover layer composed of stone or specially shaped concrete armour units, to minimize
the transmission and overtopping of wave energy. The permanently fixed breakwaters
may vary in profile from vertical to the gently sloping, usually no flatter than 1 to 10
(Teh, 2002). It has found popular application in coastal regions as an effective wave
attenuator due to its durability and the high degree of wave energy suppression it can
provide.
However, the rubble mound breakwater is limited in its potential application
when the water depth at a given site is getting deeper as it will ultimately increase the
breakwater cost significantly. Besides, a firm subsurface soil capable of providing
adequate foundations is necessary to withstand the mass of the rubble mound
breakwater. The breakwater may also interrupt littoral transport including local silt or
scour problems and may disrupt water circulation and cause water quality deterioration
within a given site, especially a marina which requires clean and clear aesthetic view of
its surrounding water. It is for these reasons that alternative breakwater designs,
primarily floating breakwaters, are of interest for a more environmentally-friendly and
cost-effective application.
13
2.2
Floating Breakwater Applicability and Advantages
Environmental and financial restrictions on marina and any other coastal
facilities development have fostered alternative engineering designs and development for
shore protection and coastal shelter. An alternative attenuator or substitute to traditional
fixed rubble mound breakwater is essential to the future of coastal engineering. A
floating breakwater is one such alternative, a concept which utilizes reflection,
dissipation and/or transformation to reduce wave energy and therefore attenuating
incident waves to an acceptable level (Morey, 1998).
A floating breakwater is a floating structure of finite draft and relies on wavestructure interaction in the upper portion of the water column. Generally, a floating
breakwater consists of a floating pontoon connected to the bed of the sea by line
moorings such as cables and chains. Floating breakwaters can act as the primary source
of wave protection or supplemental protection where partial shelter is given by reefs,
shoals or conventional fixed structures. They are commonly installed at sites such as
marinas, yacht clubs, small craft harbours, recreational areas and aquacultural facilities.
While fixed breakwaters provide some environmental and financial restrictions,
floating breakwaters possess several distinct advantages. These include lower capital
cost, shorter construction time, suitability for deep water sites, minimal impact on water
circulation and marine habitat, accommodation for a variety of bottom conditions, and
effective performance where large tidal variation exists.
While providing medium degree of wave protection, floating breakwaters offer
cost-effective and economical alternatives, especially in deeper water regions. Based on
the state-of-the-art literature review of floating breakwaters by Hales (1981), in water
14
depths greater than about 10 feet (or 3.05 meters), a fixed breakwater may not be
competitive cost wise with a floating breakwater (depending on the incident wave
period). Another attractive benefit of floating breakwaters is that they are flexible,
removable, and movable from a location to another. They can be realigned into new
layout as desired with minimum effort as facilitated by the buoyant and mobile
characteristics of the structures.
Floating breakwaters also tend to work with Nature and are environmentally
friendly. As the breakwater does not extend to the full depth of the water, there is very
little or negligible interference of floating breakwater with littoral transport, shore
processes, local water circulation and flushing currents that are essential for the
maintenance of water quality. While the construction cost of a fixed breakwater
increases with depth and requires firm foundation, the cost of a floating system is
relatively less sensitive to water depth and subsurface soil conditions at a site. Floating
breakwaters can act as multi-purpose breakwater facility and permit greater multi-use
potential than fixed structures. Apart from the main function as a wave attenuator,
floating breakwater can be used to serve as walkway, marine habitat, pier and boat dock.
Floating breakwaters, however, pose some drawbacks which require careful
consideration in their evaluation. The engineering involved in the research, design and
development of floating breakwaters for coastal regions present major challenges. The
design of a floating breakwater system must be carefully matched to the site specific
conditions and must be determined after the analysis of the anticipated wave climate at a
specific location. Some disadvantages include the limitation to short fetches, shorter
service life (10-20 years) and a portion of the incident wave is transmitted (Morey,
1998). Hales (1981) stated that uncertainties in the magnitude and types of applied
loading on the system, and lack of maintenance cost information, dictate conservative
design practices which naturally increase the initial project cost. A major disadvantage is
15
that floating breakwaters move in response to wave action and thus are more prone to
structural-fatigue problems.
2.3
Operation of a Floating Breakwater as a Wave Attenuator
The main focus in this study of wave-structure interaction is about the wave
attenuation of a newly developed floating breakwater system. Figure 2.1 schematically
illustrates wave responses to a line-moored floating breakwater. As an incident wave
approaches the floating breakwater, the breakwater is subjected to incident wave energy.
A portion of the incident wave energy is reflected seaward as a reflected wave while
another portion is transmitted past the breakwater to the leeside of the structure as
transmitted wave. Part of the wave energy is dissipated at the structure or passes beneath
it. The remaining energy excites the motions of the breakwater, i.e. heave, roll and sway
motions.
The oscillating motions of the floating breakwater in turn generate waves at both
sides of the structure in the direction of the reflected and transmitted waves. The portions
generated by the breakwater motions, overtop and pass beneath the structure form the
total transmitted wave while the total reflected wave is composed by the portions
generated by the breakwater motions and the reflected components. Wave attenuation is
therefore accomplished by the mechanisms of wave reflection, interference due to wave
radiation, and energy extraction/dissipation due to wave breaking, absorption,
turbulence, eddies and friction (Briggs, 2001).
16
Floating breakwaters anchored with chains or cables have some disadvantages
such as large roll motion and secondary waves generation at the leeward side of the
structure due to sway motion. Therefore, a replacement of the mooring system with piles
instead of mooring lines can be beneficial. Such a system may overcome the problem of
sway motion, which is prevented in this case by piles, and in addition the roll motion is
limited due to the existence of the piles.
Incident wave
height, Hi
Reflected wave
height, Hr
Heave
Roll
Transmitted wave
height, Ht
SWL
Sway
Anchor
Figure 2.1 : Wave responses to a line-moored floating structure described by a
single sinusoid wave train
2.4
Wave Control and Attenuation Mechanisms
Waves on the surface of the ocean with periods of 3 to 25 seconds or surface
gravity waves are primarily generated by winds and are a fundamental feature of coastal
regions of the world (USACE, 2002). The most elementary wave theory developed by
Airy, namely the small-amplitude or linear wave theory, gives a reasonable
approximation of wave characteristics for a wide range of wave parameters. Equation
17
(2.1) describes the free surface as a function of time t and horizontal distance x for a
simple periodic, sinusoidal, progressive wave travelling in the positive x-direction
(USACE, 2002):
K
a cos( kx Zt )
H
2Sx 2St
cos(
)
L
T
2
a cos T
(2.1)
where Ș is the elevation of the water surface relative to the still-water level SWL, and
H
is one-half the wave height equal to the wave amplitude a.
2
Engineers and designers need to arm with the fundamental concepts of wave
control and attenuation in order to use variations on the ideas and principles to match
specific site conditions with the appropriate design of wave attenuation system. From the
Equation (2.1) of wave profile, it can be observed that there are four basic physical
parameters of water wave, namely wave height, wave length, wave period and wave
direction, dictate the water surface elevation. Hence, in order to control the waves, these
basic parameters of water wave need to be controlled.
According to Tobiasson and Kollmeyer (1991), the ultimate wave attenuator is a
wide, low sloping shoreline beach made up of coarse sands and gravels. With the proper
absorption and drainage characteristics, along with the room for reshaping its profile,
this attenuator may be able to withstand any wave system, especially for shorter period
wind-generated waves. However, “We are part of Nature. How can the part conquer the
whole?” as stated by a great Indian philosopher named Osho (2001). It implies that no
human invention can utterly stop the existence of Nature. The floating breakwater is not
a panacea for wave protection. There are no systems designed today that will offer
complete protection and each of them has its limitations relative to wave intensity. Once
it is accepted that the system will offer much reduced protection at some wave intensity,
it must then be accepted that the system may also fail at certain extent of wave intensity.
18
A properly designed floating breakwater is capable to provide adequate wave
attenuation for the normal conditions and it becomes less protective as the waves
increase in size while meeting the primary requirement of at least surviving through the
storm. Floating breakwaters are generally ineffective when subjected to incident waves
of extreme height or long periods. The action of floating breakwater may be considered
in terms of basic wave processes which may result in wave attenuation. Therefore, the
control of basic physical wave processes such as reflection, dissipation and
transformation should be optimized in a wave attenuation system in order to reduce
incident wave height through the conversion of wave energy. These energy reduction
mechanisms operate in a singular nature or in a combination of two or more modes and
will be discussed in the following sections.
2.4.1 Reflection
Wave dampening through the process of reflection is accomplished when the
wave energy is redirected, with a change in wave direction, to somewhere else by an
obstacle. Each component wave of a random sea is assumed to be reflected at an angle
equal to the angle of incidence and to continue to propagate in that direction, as in the
theory of geometrical optics.
For the case of pure standing wave with perfect reflection of an incident wave
from an impermeable vertical wall, the reflected wave, with the same height as the
incident wave, propagates in the reverse direction off the wall and meets the incident
wave. The two waves of the same height and period momentarily meet and superimpose,
resulting in a new wave twice as high as the original incident wave. However, often in
nature, not all of the incident wave energy is perfectly reflected from obstacles. Some is
19
absorbed by the obstacle and some is transmitted past the obstacle. Normally partial
standing waves are formed in front of an obstacle. An incident wave with a height of Hi
and a reflected wave with a smaller height Hr but with different phase than the incident
wave, both with the same wave periods, give the total wave profile seaward of the
obstacle (Dean and Dalrymple, 2000) as in the Equation (2.2a):
Kt
Hi
H
cos(kx Zt ) r cos(kx Zt H )
2
2
(2.2a)
where İ is the phase lag induced by the reflection process.
It is quite often that reflections occur during the performance test on floating
breakwater system when measuring wave heights in a wave flume or tank. It is necessary
to separate out the incident and reflected wave heights from the total wave profile
seaward of the floating breakwater structure. Using trigonometric identities, Equation
(2.2a) is rewritten as follows:
nt
Hi
H
(cos kx cos Zt sin kx sin Zt ) r (cos(kx H ) cos Zt sin(kx H ) sin Zt )
2
2
Grouping similar time terms,
Kt
Hr
Hr
º
ª Hi
º
ª Hi
« 2 cos kx 2 cos(kx H )» cos Zt « 2 sin kx 2 sin(kx H )» sin Zt
¼
¬
¼
¬
or, for convenience, denoting the parenthetical terms by I(x) and F(x),
Kt
I ( x) cos Zt F ( x) sin Zt
(2.2b)
20
Thus Șt is a sum of standing waves with the first term as a regular incident wave
propagating in the positive-x direction and the second term as the reflected wave
traveling in the reverse direction. To find the extreme values of Șt for any x, it is
necessary to find the maximums and minimums of Șt of the envelope of the wave heights
with respect to time t. By taking the first derivative and setting it equal to zero to find the
extremes, yields
wK t
wt
I ( x)Z sin Zt F ( x)Z cos Zt
0
or
tan(Zt ) m
F ( x)
I ( x)
(2.3)
where the subscript m indicates either maximum or minimum. Examining Figure 2.2, the
relationships among (Ȧt)m, F(x) and I(x) are deduced that
I ( x)
cos(Zt ) m
I 2 ( x) F 2 ( x)
F ( x)
sin(Zt ) m
I ( x) F 2 ( x)
2
Substituting into Equation (2.2b) yields
(K t ) m
I 2 ( x) F 2 ( x)
I ( x) F ( x)
2
2
r I 2 ( x) F 2 ( x)
Substituting for I(x) and F(x) from Equation (2.2b) into Equation (2.4), the extreme
values of Șt for any location x are
(2.4)
21
I 2 ( x) F 2 ( x)
F(x)
(Ȧt)m
I(x)
Figure 2.2 : Relationships among (Ȧt)m, F(x) and I(x)
2
>K t ( x)@m
2
HH
§H · §H ·
r ¨ i ¸ ¨ r ¸ i r cos(2kx H )
2
© 2 ¹ © 2 ¹
(2.5)
[Șt(x)]m varies periodically with x. At the phase positions (2kx1 + İ) = 2nʌ (n = 0, 1,…),
[Șt(x)]m becomes a maximum of the envelope:
(K t ) max
1
(H i H r ) ,
2
the quasi-antinodes
(2.6)
Whereas at the phase positions (2kx2 + İ) = (2n + 1)ʌ (n = 0, 1,…), the value of [Șt(x)]m
becomes a minimum of the envelope:
(K t ) min
1
(H i H r ) ,
2
the quasi-nodes
(2.7)
For laboratory experiments, where wave reflection from a floating breakwater is
present, if the amplitude of the quasi-antinodes and nodes are measured by slowly
moving a wave probe along the wave flume, the incident and reflected wave heights are
found simply from Equations (2.8) and (2.9) as follows:
22
Hi
(K t ) max (K t ) min
H max H min
2
(2.8)
Hr
(K t ) max (K t ) min
H max H min
2
(2.9)
The reflection from the floating breakwater or obstacle can be defined as the ratio
of the reflected wave height to the incident wave height, which is termed the reflection
coefficient:
Cr
Hr
Hi
2K t
2K t
max
2K t
min
max
2K t
min
(2.10)
or it can also be defined in terms of wave energies as follows:
Cr
Er
Ei
(2.11)
Most breakwaters function primarily as wave reflectors. Although some of the
intercepted wave energy is indeed dissipated upon the structure, the larger portion is
generally redirected seaward again. The application of reflection method must always be
given careful consideration as reflection may end up on a neighbouring shoreline and
pose erosion problem. The interaction of the reflected and incident waves often creates
messy and unstable conditions of water surface which are particularly dangerous to boats
or small crafts at the seaward side of a floating breakwater system that is at its proximity
to a navigational watercourse.
23
Therefore, vertical impermeable and rigid walls are less favourable if compared
to sloping walls which are capable to considerably reduce wave run-up and wave impact
on the walls. A fairly efficient reflector needs to be designed to withstand rather large
forces. When the reflecting slope becomes very flat, the incident wave will break on the
slope, causing an increase in energy dissipation and commensurate decrease in the
reflection coefficient. In some cases, wave energy may be transformed into secondary
wave trains.
Reflective floating breakwaters utilize large vertical or inclined surfaces to
reflect incident wave energy back seaward. According to Morey (1998), the performance
of reflective structures is most sensitive to incident wave height and period, depth and
angle of the reflecting surface and the overall structure stability while Teh (2002) stated
that, in general, the reflection characteristics are governed primarily by surface slope,
roughness, permeability and geometrical slope of the structure and also dependent on the
still water depth, incident wave condition, wave steepness and angle of wave advance.
As the slope of the wall decreases or the wall roughness or permeability increases, the
reflected wave height decreases (Sorensen, 1978). Also, for a given obstacle, wave
reflection decreases with an increase in incident wave steepness, especially when wave
breaking occurs.
A typical floating breakwater design utilizing the concept of reflection is the AFrame as shown in Figure 2.3. The A-Frame floating breakwater has been used
extensively in British Columbia (Morey, 1998). A centerboard made of timber is
combined with stabilizers such as steel, plastic or wood and steel framing members to
form a large moment of inertia in order to increase the mass of the entire system. The
light-weighted structural and simplistic design of the A-Frame, however, have some
drawbacks which include high cost, corrosion of steel frames and damage to ends
through collisions with other modules, resulting in loss of buoyancy.
24
Cylinder Pontoon
Steel
Wood Sheet
Figure 2.3 : A-Frame Floating Breakwater
2.4.2 Dissipation
Dissipative floating breakwaters convert incident wave energy into heat and
sound through the mechanisms of fluid turbulence, vortices or/and eddies as well as
breaking on sloping surfaces or against structural members. The process of wave
breaking occurs naturally on shelving coasts where most wind-generated wave energy is
eventually dissipated. The efficiency of the dissipative structures is dominated mainly by
structural geometry and mooring restraints. These have limited use in attenuating waves
of any significant heights but have been used extensively in quelling wind-generated
waves (EDCL, 1991).
Concepts grouped under this attenuation method include tethered-float and poletire breakwaters as shown in Figures 2.4 and 2.5, respectively. Most of the incident wave
energy is transformed into turbulence within and around the many components of these
structures, while only a small portion is reflected. The pole-tire and tethered-float
breakwaters are technically feasible solutions to wave protection problems in short-fetch
(e.g. less than 10 km) or semi-protected locations. However, according to Harms (1980),
25
the pole-tire breakwater is a more effective wave energy filter than a tethered-float
breakwater of equal size.
Ht
Floats
d
H, L
Frame
Anchor
ELEVATION
PLAN VIEW
Figure 2.4 : Tethered-Float Breakwater [Harms, 1980]
The most common dissipative floating breakwater is the Goodyear. Goodyear
scrap tire floating breakwater uses a modular building-block design of 18 tires bound
together with flexible belting such as unwelded open-link galvanized chain. It has
overall length, width and height dimensions of 2.0 by 2.2 by 0.8 m, respectively (Morey,
1998). Units, with 18 tires each (3-2-3-2-3-2-3 combination), are connected to form a
floating breakwater system with 3, 4 or 5 units in width.
26
Chain
Tire-string
Anchor
Pole
Tire
mooring
damper
PLAN VIEW
B
d
D
ELEVATION
Figure 2.5 : Pole-Tire Breakwater [Harms, 1980]
2.4.2.1 Wave Breaking and Overtopping
Wave breaking is a phenomenon where all or part of a wave is caused to break,
i.e., tumbles or trips over itself when reaching a critical state. Basically, wave breaking
conditions for wave-structure interaction depend on the slope, geometrical shape and
27
permeability of structures, wave steepness and bottom slope or configuration. According
to Tobiasson and Kollmeyer (1991), whatever wave part cannot be blocked will break or
tumble over the top of the structure, perhaps disturbing the water on the other side of the
structure and creating significantly smaller wave than the original wave. Waves
transmitted by overtopping tend to have shorter periods, because the impact of the
falling water mass often generates harmonic waves with periods of one-half and onethird the incident wave period (Goda, 2000). The volume of water carried over the
structure or overtopping wave and water depth are among the basic factors that
determine the wave form after the wave overtopping. The resulting waves after
reforming will have smaller wave heights and shorter period.
Sorensen (1978) explained the wave breaking phenomena stating that the crest
particle velocity is typically much lower than the wave celerity. In deep water regions,
for a given wave period, the particle velocity of a wave crest is proportional to the wave
height. Thus with increasing wave heights the particle velocity will eventually reach a
point that is equal to the wave celerity where the wave becomes unstable and break. As a
wave shoals or propagates from a greater to a lesser depth of water, the increasing crest
particle velocity becomes equal to the decreasing phase velocity causing the wave to
break.
There are a few relationships for wave breaking quoted in the literature by
Sorensen (1978). Miche’s equation determined the limiting condition for wave breaking
in any water depth as follows:
§H·
¨ ¸
© L ¹ max
1
§ 2Sd ·
tanh¨
¸
7
© L ¹
(2.12)
Equation (2.12) was later found to be sufficiently accurate for engineering purposes and
Danel had reduced the equation to
28
§ H0
¨¨
© L0
·
¸¸
¹ max
1
,
7
in deep water
(2.13)
Equation (2.13) indicates that the wave will break when the deep water wave height
becomes one-seventh of the wave length. In shallow water, the Equation (2.14) gives the
condition for breaking of an ideal wave on a horizontal bottom of zero slope.
§H·
¨ ¸
© L ¹ max
1 2Sd
7 L
or
§H·
¨ ¸
© d ¹ max
0.9 ,
in shallow water
(2.14)
2.4.2.2 Turbulence and Eddies
Frictional dissipation of wave energy is rather an ambitious endeavour. It is
realized that some form of frictional effects play a role in many wave attenuation
processes. However, the losses are too small to be used exclusively in attenuator designs.
Nonetheless, the concept of turbulent disorientation is possible to envision some well
organized fluid movement under the action of a water wave being subverted into even
smaller chaotic motions which first destroy the organization of the fluid under a surface
wave, and therefore the wave itself, and then further break down into turbulent motions
(Tobiasson & Kollmeyer, 1991). The concept of turbulence is to get the larger
threatening wave form to cascade its energy down into more numerous, but smaller
physical systems that can easily be dealt with. The turbulent disorientation is capable to
destroy the wave over a short distance and this principle offers some innovative floating
breakwater design possibilities.
29
Another kind of energy loss is the formation of eddies around the corners of
floating structure due to the fact that the floating body is in the domain of the waves.
Eddy has a swirling flow pattern and is an important feature in many research and
engineering fields. In floating breakwater design, eddies are generally desirable and
design is optimized to promote the occurrence of eddies in order to enhance wave
dissipation. A qualitative study was performed by Tolba (1999) to describe how the
incident waves lose some of its energy when it passes through a floating breakwater. The
study included the description of water particles movement around the edges of the
structure by using small artificial particles and watching their motion using the video
camera. The study showed that some of the incident wave energy is dissipated due to the
formation of eddies around the two sharp edges of the body. The area of the eddy formed
in front of the body is bigger in size than the second eddy formed at the lee of the body.
Furthermore, the position of the eddy changes with the movement of the free surface of
the wave as shown in Figure 2.6
2.4.3
Transformation
One of the methods of wave attenuation is the application of the mechanism of
transformation, which convert incident wave energy through induced motion response
into secondary wave trains of various heights and periods. Highest efficiencies occur
when the secondary transmitted wave trains are out of phase with the incident waves.
Attenuation is influenced by mass, natural periods of motion, and the relative width. A
typical design using the concept of transformation is the Alaskan floating breakwater as
shown in Figure 2.7. The Alaskan floating breakwater, which is currently used in several
harbours along the Alaskan coast, is a double pontoon system constructed from concrete
and polystyrene foam (Morey, 1998). The two large pontoons are held in position using
a series of braces to provide additional stiffness and floatation.
30
Heave motion
Incident wave
Transmitted wave
Reflected wave
S.W.L.
Eddy (1)
Eddy (2)
d
Heave motion
Incident wave
Transmitted wave
Reflected wave
S.W.L.
Eddy (4)
d
Eddy (3)
Figure 2.6 : Variation of the position of the eddies with the movement of the free
surface [Tolba, 1999]
31
Polystyrene filled
Reinforced
concrete
Figure 2.7 : Alaskan floating breakwater [Morey, 1998]
2.5
Mooring Systems
Mooring systems are an integral part of any successful floating breakwater
design. A highly efficient wave attenuator is no consolation if the mooring system fails
to keep the floating breakwater in position during survival storm conditions. In addition,
mooring design has a direct influence on wave transmission performance and breakwater
structural design. Prior to the specific design of mooring system, it is important to have
an investigation on the types of subsoil that will hold the mooring system, the applied
loads and the types of material best suited for mooring structures (Tobiasson &
Kollmeyer, 1991). Most floating breakwaters would in most instances be moored using a
conventional catenary anchor leg mooring system or by rigid guide pile-type mooring as
shown in Figure 2.8 and Plate 2.1, respectively.
32
Precast concrete block
Chain
Figure 2.8 : Anchor-and-line mooring system [McCartney, 1985]
Plate 2.1 : A floating dock system supported by mooring piles at the Sutera
Harbour Resort, Kota Kinabalu, Sabah, Malaysia
There is increased interest in the use of chain, cable and synthetic line spanning
from the surface to seabed anchors as mooring method due to deep water and aesthetic
considerations. When mooring line is connected to a floating breakwater, the weight of
the line causes it to hang vertically downward due to gravity. With sufficient length of
the line, the other end of the line may be laid along the bottom due to gravity and
33
connected to an anchor. As the horizontal force applies to the floating breakwater, this
force is then translated into a tension force in the anchor line, and ultimately becomes a
horizontal force translocated to the anchor. The translocation of forces along the
mooring line is attained by the catenary shape formed by the chain. According to
Tobiasson and Kollmeyer (1991), anchoring systems with a chain should use a scope of
three to five times the depth of water, with the smaller scope ratios related to the heavier
chains while anchoring systems with nylon should use a scope of at least seven times the
depth of water.
The principle of catenary provides the proper force lead to an anchor by
translocating the horizontal pull down the line and it allows some effective flex. Chains
are preferred for catenary moorings because of their greater weight, durability and
robustness. Chains also have the advantage of ideal bending properties and good seabed
abrasion qualities with predictable maintenance intervals. Pretension of the mooring
system should be considered so that the floating breakwater remains close to the fully
loaded locations, even in an unloaded situation. The length of the mooring line should be
such that maximum line tension does not give uplift forces on an anchor. Sufficient
excess line length needs to be stored in the catenary reservoir so that the floating
breakwater will not be pulled under the surface under storm tidal conditions. Clump
weight or heavy block weight can be positioned at some midpoint along the mooring line
to form an auxiliary anchor. It will greatly provide more horizontal pull on the main
anchor because of its downward pull on the line.
Mooring systems are no better than their anchoring devices. Dead weight anchors
such as concrete blocks or ship anchors rely primarily on anchor mass with additional
help from frictional resistance between the anchors and soil whereas penetration anchors
rely on soil shear strength to resist “pullout” under load conditions (USACE, 2002).
Dead weight anchors can be used in any water depth, but work best in sand or mud
34
bottoms to allow some embedment. Most anchors hold best if the pull on the anchor is
close to the horizontal and parallel to the bottom. There is conservative belief that if a
heavy anchor does not dig into the seabed, the substantial weight of the anchor may
provide certain extent of safety. However, it should be noted that in salt water an iron or
steel anchor’s weight will only be 87% of its weight in air and it will be only slightly
heavier in fresh water (Tobiasson & Kollmeyer, 1991).
Sannasiraj et al. (1998) have studied the behaviour of a single pontoon-type
floating breakwater with three different types of mooring configurations, viz. mooring at
water level, mooring at base bottom and cross moored at base bottom level as shown in
Figure 2.9. In all cases, the length of the mooring line is fixed at twice the water depth
and four mooring chains are used for each configuration. The transmission coefficient is
not significantly affected by the mooring configurations studied. However, among the
three configurations, cross-moored floating pontoon yields a higher Ct compared to the
other two configurations. The configuration with mooring at base bottom was observed
to be efficient in attenuating the incident wave energy and in addition the forces on the
mooring chains for this are found to be less. Based on their investigation on responses,
transmission characteristics and mooring forces, crossed mooring for floating breakwater
is not efficient. Therefore, mooring configuration is one of the primary factors that need
to be taken into serious consideration in floating breakwater design.
35
Mooring at
water level
Mooring at
base bottom
Cross moored at
base bottom
Figure 2.9 : Mooring line configurations for a single pontoon-type floating
breakwater [Sannasiraj et al., 1998]
Pile guide systems are probably the most commonly used form of floating dock
mooring support system. Cantilever piles are driven into the subsoil and used as mooring
system with an adequate connection between the piles and the floating system to transfer
the applied load to the piles. Most pile guide systems will be designed to allow the
application of horizontal load by use of a bearing or rolling guidance system. Piling
system allows the breakwater to rise and fall with the tide but not move laterally. Pileanchored breakwaters are limited to fairly shallow sites with water depth of about 30 feet
or 9.1 m and require suitable bottom material to allow adequate pile penetration and
sufficient lateral strength (McCartney, 1985).
The two main structural materials used to manufacture piles are concrete and
steel, which have already been extensively used in many applications at sea. Timber
might also be employed. Reinforced concrete ships of the 1914-18 war, with lives of 50
years or more, and concrete forts of the 1939-45 war have demonstrated the durability
and corrosion resistance of concrete (Shaw, 1982). For spun concrete pile, except where
piles are to be jetted, SAI (2001) has recommended that the lower end of the pile should
be sealed with a plug or driving shoe to prevent continuing ingress of salt water into the
36
inside of the pile (a requirement for saline water). As for steel, it is a well-proven
material for use at sea although there may be fatigue problems and appropriate corrosion
protection must be provided. Where steel piles are used, consideration shall be given to
their protection. Some commonly used methods include HDPE sleeves, epoxy coating
and paint systems.
Catenary mooring systems have significant cost advantages over guide pile
systems in deepwater, i.e. greater than 10 m (Headland, 1995). Furthermore, catenary
mooring systems are softer than guide pile systems which generally result in lower
mooring forces. The mooring forces for the guide pile system could be about 10 times
greater than those for the catenary mooring system. However, catenary mooring systems
have some drawbacks. Line-moored floating breakwaters are dynamic structures that are
more prone to fail at connecting joints between units and at mooring line connectors.
Also, if the mooring lines or anchors fail, a floating breakwater can damage nearby
vessels, piers and other structures. Another disadvantage of such a system could be the
potential rectilinear and angular motions affecting the performance of the floating
breakwater, and the wear of the mooring lines at the seabed touch down point.
For a permanent floating breakwater such as floating docks in marina, it is
recommended that pile guide system is used as mooring method to restraint the floating
structure if the water depth is not too deep. Moreover, most of the floating breakwaters
are designed for relatively shallower water region. Piling system is a solution to
overcome the above mentioned disadvantages of those floating breakwaters moored with
chains or cables. Such a system may overcome the problem of sway motion and in
addition the roll motion is limited due to the existence of the piles. Therefore, the pilesystem floating breakwater can move freely with the tides and probably avoid the
problems of the overtopping due to the effect of tides on the mooring lines when the
mooring lines reach their maximum length.
37
2.6
Performance Considerations
A wave’s influence extends down into the water to depths which approximate
one-half of its wave length (Tobiasson and Kollmeyer, 1991). As a wave passes a point
in shallow and transitional water regions, the entire water column under the crest moves
forward in the direction of wave travel. As the trough passes, the water reverses itself
and moves backwards. Hence an elliptical orbit of water particle as a wave passes a
fixed point. For deepwater conditions, particle paths are circular. No closed orbit is
formed because the crest movement is generally greater than the trough movement. As a
result, wave current is induced in the direction of wave travel. The amplitude of the
water particle displacement decreases exponentially with depth and in deepwater regions
becomes small relative to the wave height at a depth equal to one-half the wave length
below the free surface (USACE, 2002). Thus the power of waves varies from the top of
the crest to the bottom. The greatest power occupies the SWL and it diminishes rapidly
with depth.
The wave power, or rate of energy transport, is what must be attenuated. A wave,
which is basically described in terms of size and shape, can be expressed mathematically
in terms of power relationships. The effects of a floating breakwater on a wave system
can be determined by observing the structure’s effect on the power of waves. An
incident wave that is going to strike on the floating breakwater is described in terms of
pure power. As it interacts with the structure, some of this power is blocked and
attenuated by the wave attenuator. The remaining power that flows past the structure will
reconstitute back into a new wave behind the structure and can be used to determine the
wave attenuation efficiency. The power of the wave increases with the square of the
wave height. Thus, a general rule of thumb is that to attenuate one-half of the wave
power is to reduce the wave height by about one-quarter (Tobiasson and Kollmeyer,
1991). Therefore, incident wave height should be attenuated as much as possible by
38
redirecting the wave energy back to the sea or converting the wave energy into other
forms of energy such as heat and sound.
Briggs (2001) in his technical report has made a reference to the breakwater
performance characteristics obtained by Jones (1971), reported that an ideal wave barrier
will have the following performance characteristics: (a) good performance or attenuation
of wave energy, (b) high mobility, (c) quick installation and removal, (d) survivability in
a ‘design’ storm, (e) economic, and (f) reusable. Of the criteria for evaluating a floating
breakwater, the most important is performance or wave attenuation, as quantified by
wave transmission. Wave transmission is an important aspect in the determination of the
effectiveness of the floating breakwater to protect the targeted region. The generally
accepted criterion for evaluating a floating breakwater’s performance is the transmission
coefficient, Ct, a ratio of the transmitted wave height, Ht to the incident wave height, Hi:
Ct
Ht
Hi
(2.15)
Ct can also be defined in terms of the total incident wave energy, Ei, and transmitted
wave energy, Et:
Ct
Et
Ei
(2.16)
According to Hales (1981), this definition is satisfactory as long as the waves are
regular. However, in wave climates consisting of short-crested irregular waves, the
definition may need to reflect the amount of energy transmission instead of wave height
39
transmission. Accordingly, a transmission coefficient is frequently formulated as the
ratio of the transmitted wave height squared, Ht2 to the incident wave height squared,
Hi2:
H t2
H i2
Ct
(2.17)
The design of a floating breakwater system is always site-specific. Waves
favourably attenuated by a floating breakwater usually do not exceed 4 feet (or 1.22 m)
in height and periods usually do not exceed 4 seconds (Hales, 1981). Note that these
waves are relatively short period, it is pertinent to remember that the average waves are
not the ones which cause the destruction of the floating structure. The peak waves or the
extreme waves are the parameters the structure must be designed to withstand.
The incident wave energy is split up into reflected, dissipated (taking into
account the wave energy dissipation by breaking and friction) and transmitted wave
energy. The relationship can be expressed mathematically as:
Ei
E r El Et
(2.18a)
or
UgH i2
UgH r2
8
8
UgH l2
8
UgH t2
8
(2.18b)
As the density of water, ȡ and the gravitational acceleration, g are constant, Equation
(2.18b) can be rewritten as:
H i2
H r2 H l2 H t2
(2.19)
40
However, Hl, as appears in Equation (2.19), does not exist in reality. Therefore, it would
be more appropriate to correlate the equation in terms of coefficients by the relationship:
§ Hr
¨¨
© Hi
2
2
· § Hl · § Ht ·
¸¸
¸¸ ¨¨
¸¸ ¨¨
H
H
¹ © i¹ © i¹
2
(2.20a)
1
or
C r2 C l2 C t2
with C r
1
Er
, Cl
Ei
(2.20b)
El
and C t
Ei
Et
Ei
The parameter relating to the energy loss due to wave dissipation may readily be used in
the form of dimensionless parameter as Cl to describe the floating breakwater
performance:
Cl
1 C r2 C t2
2.7
Existing Floating Breakwaters
(2.21)
Little attention was paid to the use of floating breakwaters until the expedient
harbour design for the invasion of Normandy of World War II had included moored
floating breakwaters (“Bombardon”) to dissipate wave energy and provide shelter for
invading troops. The Bombardon had a cross section similar to a Maltese cross in shape;
each unit was 61 m (200 feet) in length, 7.6 m (25 feet) in beam and depth with 5.8 m
41
(19 feet) draft (USACE, 2002). According to Hales (1981), the Bombardon was
designed to withstand a wave of 10 feet (3.05 m) high and 150 feet (45.72 m) long, and
was successful during the invasion. However, the structure collapsed during an
unexpected storm when the seas grew to 15 feet (4.57 m) in height with lengths of 300
feet (91.44 m), thus generating stresses more than eight times those for which the
structure had been designed. Shown in Figure 2.10 is the transfer point on Omaha Beach
where the cargos of amphibious trucks were reloaded on other trucks for transfer to
supply and ammunition depots (Normandy Invasion, 1944). Note the “X” shaped section
of Bombardon floating breakwater beached near the shoreline in the left center.
Bombardon
floating
breakwater
Figure 2.10 : The “X” shaped section of Bombardon floating breakwater was
beached near the shoreline near the left center [Normandy Invasion, 1944]
42
Use of floating breakwaters declined over the following years until 1957, when
the U. S. Navy Civil Engineering Laboratory saw the potential of floating units to
protect small, moored craft and marine structures (Hales, 1981). The need to protect
boats and structures increased as there were more and more people settled along the
coasts. Over the years, studies were performed on the use of many different types of
floating breakwaters to protect beaches, harbours, pleasure craft and other coastal
structures from the effects of waves. Some of these studies include Adee (1976),
Agerton et al. (1976), Armstrong & Peterson (1978), Harms (1979), Harms (1980), Cox
et al. (1991), Mani (1991), Murali & Mani (1997), Archilla (1999), Farmer (1999),
Tolba (1999), Briggs et al. (2000), Christian (2000), Briggs et al. (2002), Hadibah Ismail
& Teh (2002a) and Hadibah Ismail & Teh (2002b).
In recent years, many types of floating breakwaters have been model-tested and
some have been constructed. Hales (1981) provides a comprehensive survey of floating
breakwater types. An inventory of typical floating breakwater types, model test
information, prototype installations and design considerations can be found in
McCartney (1985). There have also been a lot of different types of floating breakwater
design patents available today. Patent is a way to protect the ideas and rights of inventors
apart from its original purpose to provide more information available to the public. Some
of the floating breakwater design patents include those from Federico (1994), Resio et
al. (1997), Tsunehiro et al. (1999), Bishop & Bishop (2002) and Meyers & Brown
(2002).
There has been research conducted in the area of floating breakwaters of various
types with different design concepts. There were plenty of works have been done in this
area for floating structures utilizing different wave attenuation mechanism. Therefore, it
is beneficial to have a background in this field due to the fundamental relationship with
the composite stepped-slope floating breakwater system being considered in the present
43
study. Several relevant research studies and floating breakwater designs will be
discussed in the following section.
2.7.1 Floating Breakwater by Tsunehiro et al. (1999)
Tsunehiro et al. (1999) have patented their floating breakwater design as shown
in Figure 2.11. The floating breakwater was designed primarily to provide protection for
fishing port, marina and shoreline erosion. Conventionally, line-moored floating
breakwaters are used for wave attenuation. However, these floating breakwaters have
caused rectilinear and angular motions which in turn affecting the performance of the
system. For this reason, a floating breakwater system consists of a breakwater body
supported by piles was developed in order to allow the floating structure to move freely
in vertical direction following the tides. Rollers are used as rolling guidance system to
provide a smooth vertical movement. The structure is being restrained from horizontal
movement due to the existence of piles.
The floating breakwater was designed such that it has another structure with
opening area below the floating breakwater body. Partition walls with necessary spacing
are installed within the opening area to enhance the strength of the structure as well as to
hold the water within the opening area. As the floating breakwater body moves vertically
up and down, the hollow structure with partition walls built below it moves
simultaneously and accordingly with the movement of the body. Trapped water between
the partition walls inside the opening area are lift together with the structure, thus
creating eddies and turbulence around the structure. As a result, wave energy is
dissipated through the formation of eddies. However, part of the incident wave energy is
reflected back seaward due to the floating breakwater body.
44
Incident wave
Breakwater body
Transmitted wave
Base
Pile
Partition wall
Figure 2.11 : Perspective view of the floating breakwater [Tsunehiro et al., 1999]
Experimental results on wave attenuation have shown that the floating
breakwater is capable to attenuate the height of short-period waves up to 60% with Ct
ranging from 0.4 to 0.6. In order to achieve the threshold value of Ct = 0.5, the minimum
B/L value has to be approximately 0.13. The Cr value was reported to be in the range of
0.45 to 0.55 while Cl remains fairly constant at approximately 0.5. Therefore, it can be
concluded that the floating breakwater achieved its purpose as a wave attenuator
utilizing the hybrid method of wave reflection and dissipation.
2.7.2 Floating Dynamic Breakwater by Federico (1994)
A dynamic floating breakwater or dock device designed by Federico (1994) is
generally T-shaped in cross-section, having a platform beam, a vertical beam and a keel
member. Figure 2.12 shows a perspective view of the floating dynamic breakwater with
an exposed cross-sectional slice (a), cross-sectional view of two alternative embodiments
45
of the floating breakwater (b and c), and an illustration showing the pivoting motion
induced by the incoming waves (d).
(a)
Platform beam
Keel member
Vertical beam
(c)
Weight adding member
Floatation member
(b)
Incoming wave
Closed
hollow
member
(d)
Anti-wave
Figure 2.12 : Floating dynamic breakwater [Federico, 1994]
The density of the floating structure varies from top to bottom, such that the
platform beam is the least dense (approximately 0.5 g/ml to 0.7 g/ml), the keel member
46
is the densest (approximately 2.5 g/ml) and the density of the vertical beam falls in
approximately 1.0 g/ml. The floating member as a whole must have an average density
(approximately 0.6 g/ml to 0.8 g/ml) remaining less than that of water (1 g/ml) so that
the structure will float at or near the surface of the water. The variation in densities of
the components creates a low center of gravity for the floating breakwater, such that the
center of gravity is located in the vicinity of the junction of the keel member and the
lower portion of the vertical beam. Therefore, the keel member will remain relatively
stable and motionless, acting as a pivot point when encountering incoming waves. The
pivoting motion in conjunction with the vertical motion act to create anti-waves which
cancel or lessen the incoming waves.
As a wave approaches the floating breakwater, the high density keel member
causes the structure to have a slow response to the vertical component of the
approaching wave, causing the tip of the wave to break onto itself and across the
platform beam. In conjunction with the incoming waves, the structure begins pivoting
about its low center of gravity, the frequency of the pivoting motion corresponding to the
frequency of the incoming waves. After an initial incoming wave has pivoted the
structure in the direction of wave advance, the structure pivots back in the direction
opposite to the incoming waves. The T-shape of the platform beam and vertical beam
creates an anti-wave in the opposite direction of the incoming waves. These anti-waves
meet the incoming waves with the troughs of the anti-waves canceling the peaks of the
incoming waves, thus significantly reducing the size of the incoming waves and creating
a calmer condition on the other side of the breakwater. The particular dimension of the
dynamic breakwater is a function of the expected size, shape and length of waves to be
encountered by the structure. Proper dimension will result in the breakwater creating the
proper anti-waves to dampen the incoming waves.
The platform beam is preferably rectangular in surface configuration, although
variations in configuration are possible. The cross-sectional width of the depending
47
vertical beam is preferably the same from top to bottom but may also be flared such that
the cross-sectional width is wider at the bottom than at the top. The keel member may
match the vertical beam in cross-section but is preferably wider so as to concentrate
more mass in a shorter vertical distance. The keel member may also be bulbous, circular,
flared or triangular to accomplish the concentration of mass.
The platform beam and vertical beam of the floating breakwater are preferably
constructed of a lightweight concrete created by using polystyrene beads or other similar
light-weight material as the aggregate filler. By varying the proportion of polystyrene
beads relative to the proportions of cement and sand in the mixture, the density of the
finished product can be controlled. This material is lightweight in its finished form,
having a controllable density of between 0.4 g/ml and 1.5 g/ml. The keel member may
be formed of standard concrete having gravel or rock as the aggregate fill material to
provide the high density value required. The platform beam can be constructed with low
density floatation members made of a lightweight, closed-cell foam material, such as
polystyrene, recycled plastic or closed hollow members such as PVC pipe. Additionally,
various strength enhancing reinforcement members such as wire mesh or weight adding
member such as high density materials may be incorporated into the design to improve
structural integrity or add weight to the keel member.
2.7.3 Cage Floating Breakwater by Murali and Mani (1997)
To meet the demand for a cost-effective floating breakwater, a new, improved
cage floating breakwater system has been developed by Murali and Mani (1997). The
basic configuration reported for a cost-effective Y-frame floating breakwater (Mani,
1991) has been adopted for the system. The details of the cage floating breakwater are
48
shown in Figure 2.13. The breakwater comprises two trapezoidal pontoons of width B
spaced at a clear distance b and fixed with two rows of equally spaced piles with a
certain gap to diameter ratio (G/D). It should be specified here that the symbols or
nomenclatures used in this particular section are based on the definition given in Figure
2.13. The space between the two pontoons might serve as a cage by enclosing it with
suitable nylon mesh. The model was moored to the flume bed by eight taut mooring
lines: four connected to the bottom of the pontoons and four to the bottom of the pipes.
TRAPEZOIDAL FLOAT
PVC pipes
Net
Mooring lines
Flume bed
ELEVATION
SECTION A-A
Figure 2.13 : Cage floating breakwater [Murali and Mani, 1997]
Experiments were conducted in a 30 m x 2 m x 1.5 m wave flume with a constant
water depth of 1 m to study the performance characteristics of the cage floating
breakwater. A geometrical scale of 1:15 was adopted for the model study. The pontoon
width, B was designed to be 0.2 m. The diameter of the pipes was 0.09 m. A wave gauge
mounted on to a trolley was towed slowly against the direction of incident waves for a
distance of about 6-8 m from a position 5 m away from the model to record the wave
envelopes.
49
From the laboratory study results, Ct is more or less constant for G/D less than
0.22 irrespective of Hi/gT2 (Hi/gT2 ranging between 0.0014 and 0.016). A 15-20%
reduction in Ct was observed when attempts were made to reduce G/D from 0.33 to 0.22.
This is attributable to higher reflection characteristics and wave energy dissipation due
to turbulence created in the vicinity of the pipes. For increase in d/h from 0.36 to 0.46,
considerable reduction in Ct value is observed (about 10-15%). An increase in d/h
beyond 0.46 would not be beneficial as the reduction in Ct value is of the order of 3% for
increase in d/h from 0.46 to 0.56. The results suggest that an optimal value of G/D =
0.22 and d/h = 0.46 for the floating breakwater to obtain the desired results of Ct below
0.5.
Results on the effect of increasing gap between the pontoons b on Ct
recommended that a b/B ratio not greater than 1.0 for the cage floating breakwater.
Moreover, the size and hence the cost of the breakwater would significantly increase for
b/B > 1. Attempts to reduce the b/B ratio to 0.5 resulted in a drastic increase in water
surface oscillations inside the cage due to multiple reflections, leading to frequent surge
of water level from within the cage.
By using the optimal combination of parameters mentioned earlier, for large
Hi/gT2 > 0.010, the system was found to be able to effectively restrict the Ct below 0.1.
This performance is comparable with those of conventional breakwaters like a rubble
mound, vertical wall, etc. For the experimental range of Hi/gT2, the Cr varies between
0.46 and 0.77. Results comparison revealed that the cage floating breakwater is 10-20%
more efficient in controlling the Ct compared to the Y-frame floating breakwater by
Mani (1991). To restrict Ct below 0.5, the studies carried out by Yamamoto (1981) on
rectangular pontoons suggest a W/L ratio of about 0.25, whereas the study on the cage
floating breakwater indicates that by fixing a row of pipes below the floating body, the
W/L requirement can be reduced to 0.15 without any compromise in the performance.
The two rows of pipes in the floating breakwater act as a barrier for the incident waves
50
and any attempt to increase the number of rows would no doubt increase the efficiency
of the system but at the same time would result in an increase in size, force and cost.
Therefore, the number of rows is restricted to a maximum of two.
2.7.4
Rapidly Installed Breakwater System (RIBS) by Resio et al. (1997)
The RIBS is a floating breakwater with two legs in a “V” shape in plan view that
provide a sheltered region from waves and currents. This new concept of the RIBS
promises to expand floating breakwater technology by allowing operations in SS3
conditions (waves with peak periods in the range of 3 to 6 sec and significant wave
heights between 1 to 1.5 m). The driving force behind RIBS concept has been the fact
that SS3 conditions seriously diminish or halt force projection plans. The objective of
the RIBS is to reduce the waves from SS3 to SS2 or wave heights up to 1 m within the
lee of the structure to facilitate military and civilian operations along exposed portions of
the world’s coastlines. Figure 2.14 is an artist’s rendition of what the full-scale RIBS
will look like.
According to the patent (Resio et al., 1997), the legs of the RIBS are composed
of two structural members where a first member provides a means for floating the
structure and sufficient freeboard for the structure to minimize wave overtopping.
Second member provides a means for ballasting the structure and a curtain member
connecting the first and second members that extend through a water column between
the first and second members. The suspended curtain deflects and redirects incident
wave front energy. According to Briggs (2001), the RIBS legs need to be of the order of
1.5 to 3 wavelengths in length and extend through the water column a depth sufficient to
deflect most of the wave energy (i.e. order of 0.5 times the water depth).
51
Figure 2.14 : Rapidly Installed Breakwater System concept [Resio et al., 1997]
An integrated study of analytical, numerical, laboratory and field experiments
has been performed to predict the performance of the RIBS (Briggs, 2001). Several
RIBS geometries and configurations were tested in the directional spectral wave basin at
the ERDC in 1997 to investigate the effects of draft, interior angle and shape on
performance. In these Fundamental Laboratory experiments (FLab), the RIBS was
idealized as a thin, rigid, fixed, vertical barrier corresponding to a 293-m-long RIBS in
15-m-deep water. The physical model scale was 1:32. Wave transmission was measured
and evaluated. The Ct decreased as wave period decreased and D/d increased, and was
fairly uniform with distance in the lee of the RIBS. From the experimental results, it was
determined that the RIBS should be positioned so that the D/d is greater than 50% of the
water depth and an interior angle greater than 45 degrees should be used.
The ERDC deployed the first ocean-scaled version of the RIBS off Port
Canaveral, Florida, during 20-30 May 1999. This XM99 prototype was approximately
52
77 m long, 2.4 m wide and 7.3 m deep. Water depth was 13.41 m. The novel
construction technique consisted of rigid steel truss frames and flexible membrane
panels in a “Venetian Blind” arrangement. Incident directional and transmitted waves
were measured with buoys. The measured Ct were less than the Ct = 0.5 desired level of
efficiency for several hours every day. Incident wave heights up to 0.83 m were
efficiently reduced during the field trial. The XM99 performed very well considering
that it experienced some minor tearing in the curtains after the first two days, resulting in
a reduced barrier to wave transmission.
The analytical models used in this study are based on linear wave theory and
idealize the RIBS as a fixed, rigid, vertical barrier. They are designated the power
transmission (PTT) and modified power transmission (MPTT) models. However, these
models were never intended for a multi-legged structure such as the RIBS. Because of
some shortcomings in their formulation, they are compared to the measured data only to
assess their range of applicability to the RIBS. The numerical program WAMIT (Wave
Analysis MIT) was the numerical model used in the study of RIBS. The average
WAMIT-predicted Ct was 0.13 higher than the corresponding measured value.
CHAPTER 3
THE COMPOSITE STEPPED-SLOPE FLOATING BREAKWATER SYSTEM
(STEPFLOAT)
3.1
Introduction
Wave can be detrimental to beaches and marinas and their patron vessels.
Unprotected beaches are subject to soil erosion and are open to waves at times strong
enough to cause danger to picnickers. Marinas need protection from waves or otherwise
the patron vessels may bump into one another to cause damage. There have been quite a
number of floating breakwaters but until the present invention of the STEPFLOAT, there
is no truly outstanding solution that has been put forward into the local industry. While
attention was given to the preservation and conservation of natural environment, most
floating breakwaters which utilized the concept of wave reflection in their designs, have
neglected the safety of moving vessels in the vicinity of the floating breakwater system.
Recent years have seen a boom in marina development and other developments
in coastal region not only in Malaysia but other maritime countries. In Malaysia, about
RM40 million has been allocated for the construction of public marinas in every state.
54
Therefore, there arises a need for a new economical and environmentally-friendly
floating breakwater that has an acceptably high efficiency in dissipating wave energy,
instead of reflecting it, to provide shelter or protection to those particular regions. The
demand made the concept of the first locally designed floating breakwater technology
possible. In 2002, Teh (2002) has completed his study on wave dampening
characteristics of the fundamental design of the SSFBW.
Bearing in mind that the SSFBW provides promising results in wave reduction
capability and foreseeing the potential of the SSFBW system to be commercialized in
the market for the benefit of communities, the author has decided to make some
modifications to the fundamental design of the SSFBW system as well as the mooring
method in order to improve the performance of the system. In 2003, with due respect to
the several floating breakwater designs available in the market, the author has committed
to further study the efficiency of the improved design of the stepped-slope floating
breakwater, namely STEPFLOAT, in order to put forward the system into the industry.
Therefore, this study is carried out as a continuation of the work done by Teh (2002).
3.2
The Evolution of the Stepped-Slope Floating Breakwater System
Malaysia is not a country of extremes of climate and natural events. However, we
are a maritime nation whose coastal problems are well recognized. It is difficult to be
precise about the impact of coastal events on the Malaysia economy. Certainly the risk is
not as great as in Holland where probably 50% of the GDP is at risk (Purnell, 1996).
Many residents on the coast live behind eroding and potentially eroding coastlines. Some
kind of engineering solutions to meet the need of coastal problems is necessary. A
change towards softer engineering solutions, instead of more traditional harder defences,
55
to reduce the risk to people and the developed and natural environment from wave
impact is recommended. For these reasons, the SSFBW has been designed and
developed and it has been evolved to a more cost-effective system as the STEPFLOAT
marks the state of the art in floating breakwater technology. The evolution of the design
and development of the stepped-slope floating breakwater system will be discussed in
the following sections.
3.2.1 The SSFBW: Fundamental Design of the Stepped-Slope Floating
Breakwater System
Teh (2002) has pioneered a research to study the effectiveness of applying the
fundamental design of the SSFBW as a wave attenuator by conducting flume tests for
the determination of wave attenuation characteristics. A series of unidirectional regular
wave only condition were generated on a 1-row, 2-row and 3-row SSFBW for water
depths of 20 cm and 33 cm. A 1-row SSFBW model of 0.80 m x 0.25 m x 0.13 m is
depicted in Figure 3.1. Basically, the SSFBW is a trapezoidal structure that is composed
of an impermeable stepped slope at both the windward and the leeward of the upper
layer of the floating module. The surface area of the stepped slope is somewhat larger
than the smooth slope. Hence, it is expected that the wave energy can be greatly reduced
as the wave action impinges on the stepped slope of the structure. The portion at the
bottom is relatively bulky and is totally immersed into water for better stability. The
SSFBW model was made of a mixture that consists of cement, sand, polystyrene and
water with ratio in volume of 1:1.5:7:2. In static freshwater condition, the SSFBW has a
5 cm freeboard and an 8 cm draft and it was cross-moored to the bottom of the flume by
four nylon ropes to keep the body in position.
56
13 cm
80 cm
25 cm
Figure 3.1 : The SSFBW model
The SSFBW is principally a wave energy dissipator, which causes the wave to
break over it, expending the upper part of the wave energy across its stepped slope
surface. If the system consists of multiple modules in rows, the remaining wave energy
is expected to be suppressed to the least. The SSFBW was tested to be hydrodynamically
stable under wave action. Figure 3.2, as reported by Teh (2002), shows the comparison
of Ct for different number of rows of SSFBW at water depths of 20 cm and 33 cm.
Detailed design drawings, system description, laboratory set-up and procedures, related
calculations, results and interpretation on the SSFBW can be found in Teh (2002).
From Figure 3.2, for d = 20 cm, the performance of the SSFBW improves with
the increase in number of rows when wave period is below 1.4 sec. The improvement of
efficiency is not subjected to the increment of row when wave period is greater than 1.4
sec. The Ct values of a single row of the SSFBW maintain at approximately 0.7 when the
period periods increase to 1.1 sec and beyond. The wave attenuation performance of 2row and 3-row systems is found to be better than the single row system for the wave
periods smaller than 1.4 sec. For d = 33 cm, the single row of SSFBW has the lowest
performance on wave attenuation whereas the performance for the 2-row and 3-row
systems are more encouraging.
57
d = 20 cm
1.0
0.8
0.6
Ct
0.4
0.2
0.0
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.4
1.5
1.6
1.7
T (s)
d = 33 cm
1.0
0.8
0.6
Ct
0.4
0.2
0.0
0.8
0.9
1.0
1.1
1.2
1.3
T (s)
1 ROW
3 ROWS
Poly. (2 ROWS)
2 ROWS
Poly. (1 ROW)
Poly. (3 ROWS)
Figure 3.2 : Comparison of Ct for different number of rows of SSFBW at water
depths of 20 cm and 30 cm [Teh, 2002]
58
The effect of water depth is found to be less influential on Ct for the case of
single row of SSFBW. It was also observed that Ct varies very little with wave period for
the single row system. Most of the plotted data points are seen to be scattered around Ct
= 0.7. In general, Ct for the 1-row system in 20 cm water depth is slightly lower than the
one in 33 cm for the given range of wave periods.
3.2.2 The STEPFLOAT: Proposed Improved Design of the Stepped-Slope
Floating Breakwater System
3.2.2.1 Design Concepts and Practicability Considerations
The objectives of the development of the STEPFLOAT are primarily to design
and develop a floating breakwater system in order to provide a multi-purpose breakwater
facility apart from its main function of attenuating waves. The composite STEPFLOAT
breakwater system is designed to provide several advantages as follows:
i.
It is a modular structure. Therefore it is removable and it provides flexibility,
i.e. the system can be rearranged into new layout with minimum effort.
ii.
Multi-purpose functions (as a wave attenuator, walkway platform and
encourage marine habitats).
iii.
Low construction cost as compared to other fixed breakwater systems.
iv.
Shorter construction period.
v.
Adaptable to water level change.
vi.
Local design and made of locally-available material.
vii.
Materials are of polymer rubber blend or/and polyethylene that are heat and
weather resistant under marine conditions.
59
viii.
Materials are non-toxic and strong, i.e. durable against wear and tear, hence
do not break easily leaving rubbish or debris in the water.
ix.
Anti-fungal compounds in the material are capable to prevent marine fouling.
x.
Environmentally and ecologically friendly.
xi.
Provide a safe habitat for fish and other marine life.
xii.
Aesthetics provided.
The concept of the STEPFLOAT breakwater system mainly rests on the shape
and geometry of the structure, the durability and robustness of the materials making up
the structure, the mooring system design and the provisional designs that may be added
on to complete the structure as a whole and cost-effective system that satisfy the
practical requirements of the users.
(a)
Shape and Geometry
In order to improve the fundamental design of the SSFBW to be a more practical
and cost-effective floating breakwater system with enhanced wave attenuation
capability, STEPFLOAT is designed to be a modular composite floating breakwater type
which consists of a series of modules with impermeable stepped slopes (to remain the
fundamental stepped slope feature of the SSFBW) that are neatly connected to one
another. Each single module takes up the geometrical shape of a polygonal structure as
shown in Figure 3.3.
The top layer of the composite STEPFLOAT module is designed to be squarish
and having an upwardly tapering wall looking side view with two upper side walls come
60
Figure 3.3 : A single module of a composite STEPFLOAT
in the form of square steps (as in Figure 3.4). Impermeable stepped slopes at both the
seaward and leeward of the upper portion of the floating module provide a shape that
would effectively attenuate waves impinging on the structure. The portion at the bottom
layer (as in Figure 3.4) is designed to be in an octagonal shape and is made squarish
having a downwardly tapering wall looking side view in order to create multiple sharp
edges for better wave attenuation through the formation of eddies and turbulence. The
idea can also minimize the cross sectional area of the floating breakwater immersed
under water, thus reducing the construction costs. Besides, the bottom layer remains its
bulky and massive features in order to enhance structure stability.
The central part of the module is hollow inside and acts as a ballast tank so that
the required draft of the system can be controlled by filling in or removing the seawater
or any other suitable substances in the tank. The top half being adapted to be mated with
the bottom half to thereby form a single module of a composite STEPFLOAT as shown
in Figure 3.4. The stepped-slope and multiple sharp-edge features of the STEPFLOAT
serve to intercept waves by dissipating (rather than reflecting) the wave energy as the
wave action impacts on the surface of the structure, thus reducing wave action in the
area it is desired to protect in a more environmentally-friendly manner. The geometrical
Figure 3.4 : 3-D view of a composite STEPFLOAT module formed by a pair of top half and bottom half units
Bottom half
Top half
61
62
configuration of a polyhedron is designed to allow the STEPFLOAT to operate mainly
on the physical principles of wave breaking, frictional dissipation, eddies and turbulence
and partly by wave reflection.
The STEPFLOAT is designed in modular form so that they can be joined
together into a larger and longer unit as the condition requires. Figure 3.5 shows a 2-row
STEPFLOAT system consists of an assembly of modules attached to one another by a
jointing system. The top portion can be securely fixed to the bottom portion by a simple
jointing system and is not described in detail in the present specification. Each module of
composite STEPFLOAT is manufactured in parts so that they can be dismantled and
transported in small units and assembled in-situ. Alternatively, the STEPFLOAT module
can be fabricated as a solid-type unit as shown in Figure 3.6 using lightweight concrete
with HDPE coating. Post-tensioning cables running through hollow tubes shall be
moulded inside the modules with cables clamped from one end to the other in order to
unite and strengthen the joined modules. Hence, the composite-type or solid-type
breakwater system acts as one solid body to dampen the incoming waves. The system
can also be expanded into as many rows as required, depending on the strength of the
approaching waves.
(b)
Alternative Features
Apart from the modules that make up the main system, additional modules are
also suggested to complete the multi-purpose tasks that may be required. An additional
inverted stepped-slope unit can be locked between two jointing rows of the system as
shown in Figure 3.7. This unit will thus act as a walkway platform for users who need to
stand or walk on the structure. An indigenous design unit is also proposed to be
Figure 3.5 : Module assembly of the 2-row STEPFLOAT breakwater
1.0 Material for upper portion is polymer blend
between EPDM and SBR or equivalent.
2.0 Material for lower portion is polyethylene.
3.0 Excellent impact strength at low temperatures.
4.0 Material for bolts and nuts must be made from
stainless steel (galvanize iron) and coated with
marine paint.
5.0 Part assembly must be sealed and free from any
hole which can be accessed by water.
6.0 The central hollow ballast tank must be filled
with sand or other alike material which is free
from vapourization process (where necessary).
Notes:
63
64
Figure 3.6 : A single module of a suggested solid-type STEPFLOAT breakwater
Figure 3.7 : Proposed horizontal platform as a walkway for pontoons
incorporated as an option. It takes up the shape of artificial seaweeds hanging from the
bottom of the modules acting as several sheet of ‘grass’ keels or curtains (as in Figure
3.8) to further reduce the wave transmission as well as to minimize the oscillating
currents or kinetic energy beneath the structure. It is anticipated that the artificial
seaweed curtains beneath the structure will also serve as possible breeding grounds and
habitat for fish and other marine life. Alternatively, old tires could also be utilized to
serve the same purpose as a wave screen.
65
Figure 3.8 : Proposed synthetic seaweed curtains as wave screens or silt curtains
(c)
Material Type
There are many types of material that could be used for the STEPFLOAT
breakwater. The common ones are fibre-glass, hardwood, PVC-based and polystyrene.
However, all these materials have the disadvantage of not being robust or durable
enough to remain exposed under the hot sun in marine conditions. The top module of the
composite STEPFLOAT will be of polymer rubber blend between ethylene-propylene
diene monomer (EPDM) and styrene-butadiene rubber (SBR) or equivalent. This
polymer rubber blend has high strength or resistance to long-term wear and tear.
However, the bottom module is subjected to abusive abrasion from floating wood, coral
reefs, sand and rocks and thus made from material having high elasticity and abrasive
resistance such as rubber or the like. High-density polyethylene (HDPE) with an
excellent balance of stress cracking resistance, stiffness and melt strength was chosen to
fabricate the bottom part of the composite module. The material will also be injected
66
with anti-fungal compounds that will prevent marine fouling on the structure. For solidtype STEPFLOAT breakwater, lightweight concrete with HDPE coating is preferable as
the density of lightweight concrete is adjustable in order to control the draft of the
STEPFLOAT.
(d)
Mooring System
A floating breakwater generally requires some form of mooring or restraint
system to maintain its position and limit excursions within certain operational
constraints. This may generally be provided by mooring line system or by piles. When a
floating breakwater is subjected to the attack of incident waves, it will experience the
rectilinear motions of heave, sway and surge, and the angular motions of yaw, pitch and
roll. These potential motions represent the structure’s six degrees of freedom. However,
hydraulic model test in a wave flume has limited the possible degree of freedom to allow
only the motions of heave, sway and roll. Generally, a floating breakwater moored with
chains or cables are subjected to these oscillatory motions of the structure due to wave
excitation.
The line-moored SSFBW has some disadvantages which need to be remedied in
order to provide a better floating breakwater system for the benefits of users. Large roll
motion may affect the performance of the floating breakwater to be used as a pier or
pedestrian walkway. The horizontal sway motion generates secondary waves that are not
recommended on the leeward side of the structure. In addition, the sway motions of the
floating breakwater allow the structure to impact with the boats when it is used as a pier
or mooring. The line-moored floating breakwater moves upward and downward with the
wave surface according to the tide. During flood tide, the floating breakwater moves
67
upward and the mooring lines may reach its maximum length. In this case, the designed
draft may increase and excessive overtopping may occur. On the contrary, when the
structure moves downward with the ebb tide, the mooring lines become slack. This
slackness increases the sway motion and may affect the performance of the floating
breakwater. If the floating breakwater is exposed to strong hydrodynamic loads, cracks
may occur at the connection between the chains and the floating structure and failure
may occur at the connection zone.
With those identified disadvantages and considering that the primary application
of the STEPFLOAT is in relatively shallower water, a suggested mooring method using
vertical piles as a modification to the classical mooring system using chains or cables is
recommended for the STEPFLOAT system. Such a system may overcome the problem
of sway motion, which is prevented in this case, and in addition the roll motion is limited
due to the existence of the piles. The STEPFLOAT is allowed to move freely in heave
and limited roll motions (two degrees of freedom), thus eliminating or reducing those
aforementioned disadvantages and eliminating any chance the floating breakwater
getting adrift. Schematic sketch of the suggested STEPFLOAT mooring system concept
using vertical piles is given in Figure 3.9.
3.3
The Composite STEPFLOAT Breakwater Model
In the present study, only composite STEPFLOAT breakwater is considered. A
geometrical scale of 1:10 is adopted. The composite STEPFLOAT module used in the
present study is constructed in parts with the top portion being adapted to be mated with
the bottom portion to form a floating module. The size of each module is 100 mm long,
100 mm wide and 100 mm high. The top layer is fabricated from polymer rubber blend
68
heave
Ht
limited
roll
Hr
Hi
circular ring or roller
vertical
piles
seabed
Figure 3.9 : Schematic sketch of the suggested STEPFLOAT mooring system using
vertical piles
between EDPM and SBR while the lower layer is made from HDPE. Figures 3.10 and
3.11 show the drawings of the top half and bottom half of the STEPFLOAT module,
respectively.
The model of the composite STEPFLOAT breakwater system (as in Figure 3.12)
is constructed as a modular system with an assembly of eight modules in each row
connected to one another side by side by a stainless steel bolt-and-nut system to U-shape
aluminium bars. Special sealant is used between the top and bottom modules to avoid
water going into the cavity. The STEPFLOAT breakwater may be made with several
rows of modules attached side by side according to requirements of the tests. The length
of the model with eight modules in each row nearly crosses the wave flume width to best
approximate the one dimensional wave condition. With the density of fresh water, the
model has a freeboard of 4.0 cm and immersed depth of 6.0 cm in static water condition.
Figure 3.10 : Top half of the STEPFLOAT module
69
Figure 3.11 : Bottom half of the STEPFLOAT module
70
Figure 3.12 : The STEPFLOAT system model is formed by a series of composite single modules
Lower portion
Upper portion
71
CHAPTER 4
EXPERIMENTAL SET-UP AND PROCEDURE
4.1
Introduction
This chapter describes the test procedures, experimental facilities and equipments
used to physically model and analyze wave transmission over the floating breakwater
system. The experimental investigations that have been carried out relate to laboratory
tests of the STEPFLOAT breakwater system to gather some information relative to
fundamental questions about the STEPFLOAT performance and to provide generic
performance characteristics which could subsequently be applied to specific design
situations as well as for the preliminary design of the field version of the STEPFLOAT.
These laboratory experiments were conducted in the unidirectional wave flume at the
Coastal and Hydraulic Laboratory, Coastal and Offshore Engineering Institute (COEI),
Universiti Teknologi Malaysia City Campus, Kuala Lumpur. The experiments were carried
out in a simplified environment and controlled condition where monochromatic wave
only condition was used in the wave flume throughout the experiments.
73
4.2
Laboratory Facilities and Instrumentation
4.2.1 Wave Flume
The laboratory tests on hydraulic performance of the STEPFLOAT were
performed in a unidirectional wave flume. The total length of the flume is approximately
16.6 m while the width is 0.92 m and the height is 0.7 m. It has a rigid flat bed, made of
marine plywood, and raised 1.16 m above the laboratory ground level. Both sides of the
wall boundary were built of a 5 mm thick glass and 5 mm thick plastic Perspex panels
fixed in steel frames. The wave flume is supported by a piston-type wave generator. At
the other end of the flume is a flexible wave absorber to reduce the turbulent reflection
velocities. The wave flume is equipped with a carriage which moves on two steel rails at
the top of the side walls. On this carriage, a wave probe was fixed to measure the water
level variation. The longitudinal section of the wave flume is shown in Figure 4.1.
Fly
Wheel
Wave Probes
Motor
Carriage
Steel Rail
Water Level
0.7 m
Wave
Absorber
Marine Plywood
Wave
Paddle
16.6 m
Note: All units are not to scale
Figure 4.1 : Schematic layout of the wave flume
Flume width: 0.92 m
74
4.2.1.1 General Remarks When Using Wave Flume
It is recognized that the quantitative definition of some of the limiting
requirements of the wave flume is very difficult in many instances and only
approximations are presently available. Surface tension tends to increase the velocity of
propagation of surface waves. According to Hughes (1993), surface tension effects must
be considered when wave periods are less than 0.35 sec and when water depth is less
than 2 cm. At these small parameter values, the restoring force of surface tension begins
to be significant and the model will experience wave motion damping that does not
occur in the prototype. Since the physical model tests in the present study were carried
out in a constant water depth of 45 cm and with wave period greater than 0.35 sec, the
effects of surface tension is considered negligible.
Water waves are also attenuated by internal friction and by viscous boundary
layer friction caused by the water viscosity. An expression to estimate wave height
attenuation due to internal friction in waves in deep water where boundary shear is
negligible is shown in Equation (4.1) (Hughes, 1993).
d § H 2C 2 ·
¸
¨ SU
dt ¨©
4 L ¸¹
H 2C 2
16S UQ
4 L3
3
(4.1)
where H is defined as wave height (decays in time). The left hand side represents the
time rate of change of total wave energy per unit surface area in a linear wave and the
right hand side is the average rate of energy conversion per unit area due to internal
shearing stresses.
By rearranging and canceling variables, Equation (4.1) can be integrated as in
Equation (4.2) to give Equation (4.3).
75
³
H
Ht
0
2
³ 16LS Q dt
1
d H2
2
H
H t Ht 0
t
0
(4.2)
2
e 8S Qt / L 2
2
(4.3)
where H(t) is the attenuated wave height at time t. The above formulation assumes
uniform, regular waves travelling over a horizontal bottom and can be used to examine
the range of potential scale effect arising from internal friction.
Over short distances in the wave flume, internal friction is minimal and viscous
dissipative effects in non-breaking waves are limited to the thin boundary layer. A
formula for estimating wave attenuation of regular waves in a rectangular wave channel
having a uniform and constant cross-section is found in Hughes (1993). The expression
for viscous boundary layer damping of a small amplitude linear wave in a wave flume of
constant cross-section is defined as in Equation (4.4). Equation (4.4) for wave height
attenuation due to viscous boundary layer dissipation is for constant water depths.
H2
H1
e
Dx p
(4.4)
where
D
ª
§ 4Sd · 2SB f
¸
« sinh ¨
2
SQ «
L
© L ¹
Bf C T «
§ 4Sd · 4Sd
« sinh ¨ L ¸ L
©
¹
¬
º
»
»
»
»
¼
(4.5)
For the present work, the effect of the viscosity in wave motion and the boundary
friction on the decay of water waves was studied by measuring the incident wave height
76
at four different positions (i.e. P1, P2, P3 and P4) spaced 1.5 m from each other in the
middle of the wave flume as shown in Figure 4.2. The measurements were conducted
without the presence of floating breakwater model in a constant water depth of 45cm.
Table 4.1 shows the average wave height measured at P1, P2, P3 and P4 with various
frequencies, ranging from 36 Hz to 56 Hz. The decay was also calculated theoretically
using Equations (4.3) and (4.4).
P4
P3
P2
Fly
Wheel
P1
Motor
Steel Rail
Water Level
Wave
Paddle
45 cm Wave
Absorber
1.5 m
Note: All units are not to scale
1.5 m
1.5 m
Flume width: 0.92 m
Figure 4.2 : The measurements of wave decay without the presence of floating
breakwater model
Table 4.1 : Average wave height at P1, P2, P3 and P4 with various frequencies
Average wave height (cm) with various frequencies
Probe
Horizontal
36 Hz
40 Hz
44 Hz
48 Hz
52 Hz
56 Hz
position distance, x p (m)
P1
0.0
3.8112
6.6025
6.7007
6.9456
6.8495
5.4435
P2
1.5
3.7729
6.5313
6.5669
7.0300
6.6681
5.5944
P3
3.0
3.7854
6.5233
6.4719
6.8735
6.4600
5.5176
P4
4.5
3.7240
6.4871
6.3967
6.5761
6.1385
5.1300
1.1113
1.0034
0.9098
0.8305
0.7655
0.7148
T (s)
1.7747
1.5006
1.2629
1.0659
0.9109
0.7962
L (m)
0.25
0.30
0.36
0.42
0.49
0.57
d/L
Water region
transitional transitional transitional transitional transitional deep water
77
(a)
Decay Due to Internal Friction
From Table 4.1, the only water wave in deep water region (d/L > 0.50) is given
by wave with f = 56 Hz. The corresponding wave period and wave length are 0.7148 sec
and 0.7962 m, respectively. Average wave height at P1, H t x p
0.0 m
was measured as
5.4435 cm. For temperature equals to 27°C, the kinematic viscosity for water,Q equals
to 8.5680 (10)-7 m2/s (Lienhard IV & Lienhard V, 2003). Further calculations are
performed as follows to yield the attenuated wave heights:
The wave celerity, C
L
T
0.7962
0.7148
1.1139m / s . Therefore, the time required
for the wave to travel 1 m is 0.8978 sec. The time used to calculate the attenuated wave
height at the second, third and the fourth positions are calculated as follows:
txp
1.5 m
1.5(0.8978) 1.3467 s
txp
3.0 m
3.0(0.8978)
2.6934 s
(for the third position at P3)
txp
4.5 m
4.5(0.8978)
4.0401s
(for the fourth position at P4)
(for the second position at P2)
Using the time calculated, the attenuated wave height can be calculated at the different
position using Equation (4.3) as follows:
H t
H t
H t
H txp
0.0 m
x p 1.5 m
x p 3.0 m
x p 4.5 m
5.4435cm
5.4435e 8S
5.4435e 8S
5.4435e 8S
2
8.5680 10 7 1.3467 / 0.7962 2
5.4427cm
(0.014% decay)
2
8.5680 10 7 2.6934 / 0.7962 2
5.4419cm
(0.029% decay)
2
8.5680 10 7 4.0401 / 0.7962 2
5.4412cm
(0.043% decay)
78
(b)
D
Decay Due to Viscous Boundary Friction
2
S (8.5680)10
0.921.1139
0.7148
7
ª
§ 4S (0.45) · 2S (0.92) º
« sinh ¨ 0.7962 ¸ 0.7962 »
©
¹
«
»
§ 4S (0.45) · 4S (0.45) »
«
« sinh ¨© 0.7962 ¸¹ 0.7962 »
¬
¼
0.003788
and using Equation (4.4) yields:
H 1 x
p
0.0 m
5.4435cm
H 2 x
p
1.5 m
5.4435e 0.0037881.5 5.4127cm
(0.57% decay)
H 2 x
p
3.0 m
5.4435e 0.0037883.0 5.3820cm
(1.13% decay)
H 2 x
p
4.5 m
5.4435e 0.0037884.5 5.3515cm
(1.69% decay)
The calculations showed that wave attenuation due to internal friction is very
small and nearly negligible while the effects of boundary friction on wave attenuation
are relatively more significant. Plots in Figure 4.3 show the comparison between the
calculated attenuated wave heights due to boundary friction and measured wave heights
with various frequencies. The absolute percentage difference between calculated and
measured wave height at different positions for f ranging from 36 Hz to 56 Hz are
tabulated in Table 4.2. The maximum percentage error, i.e. 8.99%, was found to occur at
xp = 4.5 m for f = 52 Hz. In general, most absolute percentage errors are below 5% and
therefore the errors are deemed to be insignificant.
79
4.2
Wave height (cm)
f = 36 Hz
4.0
3.8
3.6
3.4
3.2
0
1.5
3
4.5
6
7.0
Wave height (cm)
f = 40 Hz
6.8
6.6
6.4
6.2
6.0
0
1.5
3
4.5
6
7.0
Wave height (cm)
f = 44 Hz
6.8
6.6
6.4
6.2
6.0
0
1.5
3
4.5
6
Horizontal distance, xp (m)
Measured
cal (Boundary friction)
Figure 4.3 : Comparison between the calculated attenuated wave heights due to
boundary friction and measured wave heights
80
7.2
Wave height (cm)
f = 48 Hz
7.0
6.8
6.6
6.4
6.2
0.0
1.5
3.0
4.5
6.0
7.0
f = 52 Hz
Wave height (cm)
6.8
6.6
6.4
6.2
6.0
0.0
1.5
3.0
4.5
6.0
5.8
Wave height (cm)
f = 56 Hz
5.6
5.4
5.2
5.0
4.8
0.0
1.5
3.0
4.5
6.0
Horizontal distance, xp (m)
Measured
cal (Boundary friction)
Figure 4.3 (continued) : Comparison between the calculated attenuated wave
heights due to boundary friction and measured wave heights
81
Table 4.2 : Absolute percentage difference between calculated and measured
wave height at different positions
f (Hz)
36
40
44
48
52
56
4.2.2
Absolute % error at different positions
x p = 0.0 m
x p = 1.5 m
x p = 3.0 m
x p = 4.5 m
0.00
0.00
0.00
0.00
0.00
0.00
0.69
0.72
1.60
1.68
2.15
3.36
0.04
0.49
2.63
0.13
4.71
2.52
1.35
0.69
3.37
4.01
8.99
4.14
Wave Generating System
The wave generator used in the present study is a piston type in order to displace
the water at a rate matching the requirements of the wave train being generated. The
mechanical response of the piston-type wave generator is based on simple piston
principle that creates the waves by using an electric motor to drive a paddle forward and
backward. The wave generating system consists of a motor, electronic analog control
panel, pulleys, fly wheel and wave paddle as shown in Plate 4.1. An analog control
panel, as shown in Plate 4.2, was used to control the frequency (ranging from 0.1 Hertz
to 60.0 Hertz) of a DC motor in order to generate monochromatic waves with various
wave periods in the flume. The pulleys were responsible for reciprocating the wave
paddle forward and backward corresponding to the motor’s speed. The iron fly wheel is
a disk of 46 cm in diameter, 1.5 cm thick and has a groove of length equals to 32 cm.
The wheel was rotated by a shaft connected to the motor. It was also connected to the
wave paddle plate by means of a 1 m metal crank. The wave paddle is an aluminium
plate of 89 cm in length and 56 cm in width. An opening of 2 cm between the lower end
of the paddle and the flume bed was provided to reduce shear friction as well as to
minimize turbulence. The desired wave height was associated with both paddle
82
movement and water depth. The maximum attainable wave height at a particular
frequency was limited by wave breaking or the limitation of the wave flume itself.
Plate 4.1 : Wave generating system
Plate 4.2 : Electronic analog control panel
83
4.2.3 Wave Absorber
In order to make efficient use of the facility in wave tests, it is necessary to
prevent reflection of the waves from the far end of the flume. Unwanted reflections can
alter significantly the incident wave field, which in turn may impact test results.
Therefore, at the end of the wave flume, a wave absorber was constructed consisting of
L-shaped steel bar screen at a slope of approximately 1:7. Teh (2002) has tested the
wave absorbing performance of various types of wave absorbers and reported that the
performance of the L-shaped steel bar screen is capable to achieve 89.11% to 98.90% of
wave energy absorption for the tested wave period range of 0.87 sec to 1.66 sec. A layer
of sponge covers the walls behind the wave absorber was also constructed to further
absorb wave energy and reduce reflection. In general, the average reflection was found
to be less than 6% of the incident wave energy and was neglected during analysis.
4.2.4 Wave Probes and Data Acquisition System
Instantaneous wave surface elevation was measured by two capacitance-type
wave gauges. The sensor portion of the wave probe consisted of a thin insulated wire
held taut by a supporting rod. The rod is constructed of stainless steel with a minimum
cross-section to reduce flow disturbance. Plate 4.3 shows the capacitance-type wave
probe used in the experiments. The wire insulation served as a capacitor between the
inside conducting wire and the water. The capacitance varies linearly as the water
surface elevation changes. One of the main advantages of the single-wire capacitance
wave probe is that the gauge exhibits good linearity and dynamic response over a
reasonable length so that it can be used for fairly large waves (Hughes, 1993). The
capacitance wave probe is also stable over sufficiently long times. Therefore, gauge
84
“drift” is not a significant problem. It has minor obstruction to the wave front, no
distortion of the wave shape and low construction costs.
Plate 4.3 : Capacitance-type wave probe
The variations in sensor capacitance as the water level changes were converted
by an amplifier into voltage signals and the signals were transferred to the HIOKI 8833
MEMORY Hi CORDER data acquisition system (as shown in Plate 4.4). Wave probe
calibration yields the voltage-elevation conversion graph with associated coefficients.
With the coefficients input into the data acquisition system, the voltage signals will be
converted instantaneously to water surface elevation during the experimental run. Realtime wave observation is possible on the LCD screen of the HIOKI 8833 MEMEORY
Hi CORDER wave recorder or wave profiles can be printed on the recording papers
from the thermal printer of the wave recorder for further analysis.
85
Plate 4.4 : HIOKI 8833 MEMORY Hi CORDER data acquisition system
4.2.4.1 Wave Probe Calibration
A frame supporting the probe was suspended over the wave flume. This
mounting greatly facilitated the daily calibration of the probes and also helped to
maintain the probes in the proper position relative to the static water level. Each day
before experimental tests began, wave probes were calibrated at several different water
levels to account for variations in water level changes.
Capacitance wave probes feature a linear (or nearly linear) relationship between
the sensor output and the elevation of the water level on the gauge. This relationship is
determined by calibrating the probe statically. In static calibration, the wave probe is
vertically raised and lowered in known incremental distances relative to the still water
86
level and the gauge output at each location is recorded. The calibration relationship is
obtained as a mathematical curve-fit between the recorded gauge outputs (in voltage
units) and the corresponding elevations in length units as shown in Figure 4.4 as one of
the examples of the wave probe calibration. Ideally, the relationship is linear and a leastsquares linear regression is applied to obtain the necessary conversion equation. All the
probes responded linearly with a correlation coefficient of 0.9995 and greater and were
deemed acceptable for calibration.
During the static calibration, it is very important not to disturb the water because
even the slightest water level fluctuation will impact the quality of the wave probe
calibration. The static calibration obtained for capacitance wave gauges is considered
sufficient for most laboratory wave conditions.
12
Water level (cm)
8
y = 3.689685x - 18.182769
4
2
R = 0.999905
0
-4
-8
-12
0
1
2
3
4
5
6
Voltage (V)
Figure 4.4 : Wave prove calibration
7
8
87
4.3 Measurement of Incident, Reflected and Transmitted Waves
As incident waves propagate shoreward and impinge on an obstacle, part of the
incident wave energy is reflected, hence posing wave reflection at the seaward of the
structure. However, quite often in nature, when waves are reflected from structures, not
all of the wave energy is reflected; some is absorbed by the obstacle and some is
transmitted past the obstacle. Incident and reflected waves, propagating in opposite
directions, will superimpose. For this reason, in measuring wave heights in the wave
flume where reflection from a floating breakwater is present, the amplitude of the quasiantinodes and nodes are measured by slowly moving a wave probe along the wave
flume. It requires slowly traversing a wave probe along the direction of wave
propagation in the wave flume.
Figure 4.5 shows the position of the wave probes used for the wave height
measurement. A wave probe was located at the seaward and leeward of the floating
breakwater, respectively. For the measurement of incident and reflected wave heights, a
wave probe was fixed at the middle of the carriage that moved on two steel rails. It
would be traversed along the direction of wave propagation for a distance of 2L at the
seaward side of the STEPFLOAT. The scattered wave system, which occurs near the
STEPFLOAT, decays exponentially beyond the imaginary boundaries of 3d away from
the structure (Dean and Dalrymple, 2000). Therefore, at leeward side of the floating
breakwater, another wave probe was fixed at a distance of 3d away from the floating
structure to measure the transmitted wave heights as well as to avoid the scattered effect
near the floating breakwater.
The duration of the recording was selected after allowing a few seconds for the
waves to stabilise so as to ensure that measurement of waves upwave of the breakwater
were not contaminated by reflected waves which were re-reflected by the wave paddle;
0.7 m
Marine Plywood
16.6 m
Wave
Probes
8.5 m
Vertical pile
STEPFLOAT
Carriage
2L
Figure 4.5 : Laboratory and STEPFLOAT model set-up in the wave flume
Note : All units are not to scale
Wave
Absorber
Water Level
Steel Rail
3d
Flume width: 0.92 m
Wave
Paddle
Motor
Fly
Wheel
88
89
and that the measurement of waves downwave of the breakwater were not contaminated
by the waves reflected by the wave absorber. Also, the time origin of the data recording
was selected such that the shorter period waves reach the location of the model and there
were sufficient numbers of wave cycles within a chosen record length. Hence, a
recording length of 20 seconds was chosen. However, the length of wave record for
analysis was chosen carefully and must be suitable so as to exclude those contaminated
waves. Data were recorded at a sampling frequency of 25 Hz.
It is necessary to be able to separate out the incident and reflected wave heights
from the co-existing waves in front of the model. To do this, the incident and reflected
wave heights are found simply from Equations (2.8) and (2.9) in Section 2.4.1. To
calculate the transmitted wave height, a number of transmitted waves recorded by the
second wave probe at the leeward side of the structure were analyzed and the average of
those wave heights was taken to be the transmitted wave.
4.4 Determination of Wave Period and Wave Length
In the present study, the wave period is varied by means of controlling the
frequency of the motor. In order to obtain the wave period T, a point was marked on the
fly wheel. Time taken by the marked point to revolve 10 revolutions was recorded. The
time measurement for 10 revolutions was repeated for six times for various frequencies.
An average wave period for each frequency was calculated. Table 4.3 indicates the mean
wave period for each frequency of the wave generating system. A relationship can be
found when the wave period is plotted against the frequency as shown in Figure 4.6.
Based on the equation, wave period for various frequencies ranging from 30 Hz to 58 Hz
was calculated and would be applied throughout the experiments. Table 4.4 shows the
90
Table 4.3 : Mean wave period for various frequencies of wave generating motor
Frequency
(Hz)
29
32
35
38
41
44
47
50
53
56
59
Time for 10 revolutions (s)
2
3
4
5
13.85
13.85
13.88
13.79
12.54
12.54
12.60
12.55
11.43
11.42
11.50
11.48
10.48
10.54
10.53
10.54
9.66
9.79
9.79
9.76
9.11
9.16
9.10
9.10
8.48
8.51
8.48
8.50
7.94
7.92
8.02
7.98
7.57
7.56
7.54
7.54
7.23
7.12
7.16
7.23
6.82
6.82
6.85
6.82
1
13.79
12.57
11.41
10.48
9.75
9.12
8.50
8.01
7.57
7.12
6.81
Average wave
period (s)
1.38
1.25
1.14
1.05
0.97
0.91
0.85
0.80
0.76
0.72
0.68
6
13.85
12.48
11.44
10.62
9.66
9.05
8.47
7.93
7.56
7.11
6.85
1.5
Wave Period, T (s)
1.2
0.9
0.6
2
T = 0.0006 f - 0.0717 f + 2.9820
0.3
2
R = 0.9988
0.0
26
29
32
35
38
41
44
47
50
53
56
59
62
Frequency, f (Hz)
Figure 4.6 : Relationship between wave period and frequency
Table 4.4 : Wave period of model and prototype for various frequencies
f (Hz)
30 32 34 36 38 40 42 44 46 48 50 52 54 56 58
T model (s) 1.33 1.26 1.19 1.12 1.06 1.01 0.95 0.91 0.86 0.82 0.79 0.76 0.74 0.72 0.70
*T prototype (s) 4.21 3.98 3.76 3.55 3.36 3.18 3.02 2.87 2.73 2.61 2.50 2.41 2.33 2.26 2.21
* Based on geometrical scale of 1:10
91
wave period of model and its corresponding prototype wave period (based on a
geometrical scale of 1:10) for various frequencies. According to Froude Scaling, based
on a geometrical scale of 1 : x, the prototype wave period, T prototype was obtained by
multiplying the wave period of model, T model with
x.
In water wave modelling, a direct measurement of wave length is inconvenient.
Therefore, wave length was determined by the linear dispersion relationship:
V2
gk tanh kd
(4.6)
V and d were known from the measurement of wave period and water depth (in this
case, d = 45 cm), hence k could be computed by using the Bi-Section Method.
Subsequently, the wave length could be determined by L
2S
. For instance, a wave
k
with T = 1.33 sec is propagating in a water depth d = 0.45 m. The corresponding angular
frequency, V
2S
T
2S
1.33
4.72rad / s .
Transforming Equation (4.6) into a functional equation:
f k gk tanh kd V 2
Bi-Section Method is applied to obtain the k value. The exact value for k will be found
when f(k) equals zero as shown in Table 4.5.
Hence, the value of k is 2.7045 rad/s. Therefore, the corresponding wave length,
L can be determined by L
2S
k
2S
2.7045
2.32m . The results of wave length for all
92
wave periods in water depth of 0.45 m are tabulated in Table 4.6. From Figure 4.7, it is
found that the d/L for all the cases are in the range between 0.19 and 0.59. Hence, the
tests are carried out in transitional and deep water region. It should be noted that the test
results are only applicable for this condition.
Table 4.5 : Determination of wave number, k, by Bi-Section Method
(for T = 1.33 s , d = 45 cm)
k (rad/s)
2.900000000
2.710000000
2.704000000
2.704400000
2.704480000
2.704486000
2.704486700
2.704486620
2.704486540
2.704486535
2.704486538
2.704486537
f(k)
2.297351428
0.064858246
-0.005723512
-0.001017998
-0.000076897
-0.000006314
0.000001921
0.000000980
0.000000039
-0.000000020
0.000000015
0.000000000
Table 4.6 : Determination of wave length using the linear dispersion relationship
d (m)
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
f (Hz)
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
T (s)
1.33
1.26
1.19
1.12
1.06
1.01
0.95
0.91
0.86
0.82
0.79
0.76
0.74
0.72
0.70
V (rad/s)
4.72
4.99
5.29
5.60
5.92
6.25
6.59
6.93
7.28
7.62
7.95
8.26
8.54
8.78
8.99
k (rad/s)
2.70
2.93
3.19
3.48
3.81
4.17
4.57
5.01
5.48
5.97
6.48
6.98
7.45
7.88
8.25
f(k)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
L (m)
2.32
2.14
1.97
1.80
1.65
1.51
1.37
1.25
1.15
1.05
0.97
0.90
0.84
0.80
0.76
d/L
0.19
0.21
0.23
0.25
0.27
0.30
0.33
0.36
0.39
0.43
0.46
0.50
0.53
0.56
0.59
93
0.8
0.7
Deep water
0.6
d/L = 0.50
d/L
0.5
0.4
0.3
Transitional water
0.2
0.1
d/L = 0.04
0.0
0.6
0.8
1.0
1.2
1.4
T (s)
Figure 4.7 : Plots of d/L vs. T for d = 45 cm
4.5 Experimental Tests on STEPFLOAT
A series of laboratory experiments were conducted in wave only condition in a
unidirectional wave flume (Plate 4.5) for the composite STEPFLOAT breakwater system
with four different types of model system arrangements, i.e. 2-row, 3-row, G = b and G
= 2b as shown in Plates 4.6 (a) to 4.6 (d), respectively. Steady monochromatic nonbreaking waves were generated in all the experiments. The structure of the experiments
is summarized in Table 4.7. The tests were generally divided into three groups of
experiments. In the first group of experiments, the 2-row STEPFLOAT breakwater
system moored with chains was studied in order to allow performance comparison to the
SSFBW system and rectangular pontoon. The STEPFLOAT model, in a constant water
depth of 53 cm, was allowed to move freely in three degree of freedom, i.e. heave, sway
94
and roll motions, with the model moored by steel chains to the bed of the flume as
shown in Plate 4.7.
Plate 4.5 : Wave flume
In the second group of experimental tests, the wave attenuation efficiency of the
restrained floating breakwaters was studied for 2-row and 3-row systems in a constant
water depth of 45 cm. The models were restrained at four piles (Plate 4.8) in the upper
column of water with their original depth of submergence when they were allowed to
float freely without any external disturbance. They were restrained such that no motions
of the structures were allowed. The results of the coefficients of transmission were to be
used as reference values for the results of the pile-system STEPFLOAT breakwater and
to evaluate the effect of heave and limited roll motions of the floating body on wave
attenuation.
(d) G = 2b
(c) G = b
Plate 4.6 : Various STEPFLOAT model system arrangements
(b) 3-row
(a) 2-row
95
96
Table 4.7 : The structure of experimental tests
System arrangments
2-row
3-row
Chain mooring
¥
Restrained
Heave & limited roll
(using vertical piles)
¥
¥
¥
¥
G=b
¥
G = 2b
¥
Plate 4.7 : 2-row STEPFLOAT model moored to the flume bed by steel chains
97
Plate 4.8 : A 2-row model as restrained from moving at four steel piles
The third group of experiments studied the performance of the 2-row and 3-row
STEPFLOAT breakwater system using four vertical steel piles (Plate 4.9) as mooring
method. The floating breakwater models were allowed to move in heave and limited roll
motions (two degrees of freedom). This will improve the stability of the STEPFLOAT
breakwater system when used as a walkway or pier. The effect of the motions on wave
attenuation can be evaluated when compared to the restrained case. Besides 2-row and 3row systems, another two different system arrangements with G = b and G = 2b were
studied using vertical piles with heave and limited roll motions in order to investigate the
effect of pontoon spacing on wave attenuation. All experimental tests on vertical pilesystem STEPFLOAT breakwater were carried out in a constant water depth of 45 cm.
The vertical pile system consists of two horizontal bars restrained at the steel
rails on top of the side walls of the flume. Four external hollow pipes were welded to the
horizontal bars. Eight slide bearings were fixed inside the external hollow pipes for
smooth linear motion of the aluminium internal rods. The four aluminium internal rods
were fixed to the STEPFLOAT body and extended inside the external hollow pipes,
98
allowing the floating breakwater to move in heave and limited roll motions when
exposed to the attack of the incident regular waves. The details of the vertical pile
system used in the present study are given in Figure 4.8.
Plate 4.9 : A 3-row model with vertical pile system
Aluminium
internal rods
Horizontal bar
Steel rail
External
hollow pipes
STEPFLOAT
0.70 m
Heave motion
Side wall of flume
Flume bed
0.92 m
Figure 4.8 : Details of the vertical pile system
CHAPTER 5
DIMENSIONAL ANALYSIS AND EXPERIMENTAL RESULTS
5.1
Dimensional Analysis
Dimensional analysis is an algebraic technique of dimensionally homogeneous
functions that makes use of the study of dimensions. It is based on the principle of
dimensional homogeneity (PDH) that all mathematical equations that relate to physical
quantities must be dimensionally homogenous. Dimensional analysis is particularly
helpful in experimental work as it indicates the direction in which experimental work
should go and it can efficiently shorten laboratory procedures. If we divide all the terms
in a dimensionally homogenous equation by a quantity that has the same dimensions,
then all the terms will become dimensionless. As a result, we can express the equation
more simply as relationship between dimensionless groups or numbers. Dimensional
analysis is a powerful scientific procedure that formalizes this process.
The pi theorem or the Buckingham pi theorem is a generalized method of
dimensional analysis and is the most popular now (Finnemore and Franzini, 2002). If an
equation satisfies the PDH, then the pi theorem arranges or reduces the variables into a
100
lesser number of dimensionless groups of variables. Thus if an equation has a number of
n dimensional variables composed of k fundamental dimensions, (n-k) dimensionless
terms, often referred to as ɉ terms, may be derived.
Previous studies on floating breakwater (Hadibah Ismail & Teh, 2002b;
Christian, 2000 and Tobiasson & Kollmeyer, 1991) have revealed that breakwater width
is a dominant parameter that affects the floating breakwater performance. In some cases,
the floating breakwater can be made wide enough in comparison to the wave that it is
designed for so as to allow all of the breaking energy to expend itself on the surface of
the breakwater. With waves breaking over a floating breakwater, particularly if this
occurs in a random manner, the waves created in the water space behind the structure
may differ considerably from the incident waves (Tobiasson and Kollmeyer, 1991).
The B/L ratio is a dimensionless parameter which characterizes the size of a
floating breakwater. Based on literature survey done by Mani (1991), to achieve a Ct
value of 0.5, the B/L ratio for most of the floating breakwaters should be in the range of
0.45-1.70. While for floating tire breakwaters, they offer excellent protection for marinas
and effectively stop the short length waves which can be very destructive to boats
berthed at pier (Armstrong and Petersen, 1978). Sorensen (1978) stated that Ct = 0.1-0.2
when the wave length was equal to the major structure length in direction of wave
propagation. But when the wave length was increased to four times the structure length,
Ct increased to 0.6. For floating breakwaters aligned normal to the wave direction, the
relative width (i.e., breakwater width in the direction of wave travel) should be greater
than unity (Briggs, et. al, 2002). Hales (1981) reported that wave transmission decreased
as the breakwater width increased to more than one-half the wave length.
Christian (2000) in his experimental tests on floating breakwaters which have
various vertical plates attached to improve their efficiency, found out that a deeper skirt
101
or draft generally reduces the Ct values. Teh (2002) reported that a slow decrease in Ct is
observed as D/L increase for a single row of SSFBW. For the two-row and the three-row
systems, improvement in wave attenuation is found considerable when D/L is less than
0.7. In general, for a given wave length, if the draft is small, the relative width must be
larger and vice versa. Thus, if the draft is relatively shallow, the breakwater width needs
to be longer to compensate.
Mani (1991) in his study on Y-frame floating breakwater concluded that Ct
decreased for an increase in Hi/gT2. Murali and Mani (1997) in their study on the
performance of cage floating breakwater reported that for large Hi/gT2 (>0.010) the
system effectively restricts the Ct (below 0.1). Flat waves with a low value of Hi/gT2 are
transmitted with ease. However, steep waves with greater value of Hi/gT2 are arrested
effectively. Period, being an extremely important and basic characteristic of a parameter,
is intrinsically incorporated within the wave steepness parameter, as smaller wave
periods produce waves with smaller wave lengths and smaller wave lengths produce
steeper waves.
Based on Teh (2002)’s work, Ct values of a single row system are less dependent
on the given range of d/gT2. However, the two-row and three-row systems have strong
transmission dependence upon d/gT2. As d/gT2 increases, Ct decreases from as high as
0.9 to as low as 0.1. The results indicate that the deeper the water depth is, the greater
the wave dampening ability of the system it makes.
With some literature background on parameters affecting the floating breakwater
performance, a dimensional analysis using pi theorem was performed in order to develop
a design relationship for Ct in terms of hydraulic characteristics and the geometry of the
STEPFLOAT. The main function of a floating breakwater is to minimize the height of
102
waves transmitted, Ht past the structure. The physical processes of wave-structure
interaction are visualized in order to consider relevant physical factors that probably
influence the wave transmission processes. Figure 5.1 provides a definition sketch of the
problem under consideration. In general, the transmitted wave height, Ht may be
B
heave
limited
roll
Ht
Hr
Hi
D
L
d
vertical
piles
seabed
Figure 5.1 : Definition sketch of a pile-system 2-row STEPFLOAT breakwater
assumed to be a function of the following independent variables of hydraulic parameters,
structure’s geometry parameters, fluid and structure properties:
Ht = f (Hi, L, d, B, D, ȡ, ȡs)
(5.1)
where f is a function. The resulting basic functional Equation (5.1) contains eight
dimensional variables. That means n = 8. Since the floating breakwater system is a
surface dominated phenomenon, the viscous effect has been considered negligible and
103
excluded from the analysis. Based on Equation (5.1), six of the eight terms already have
linear dimensions while the remaining two terms of water density and density of the
structure include dimension of mass. The dimensions of the physical quantities are
subsequently expressed in terms of the mass-length-time (MLT) system as follows:
Ht
>L@ ,
Hi
>L@ ,
L
>L@ , d >L@ ,
B
>L@ ,
D
>L@ ,
U
ªM º
«¬ L3 »¼ , U s
ªM º
«¬ L3 »¼
It is obviously shown that only M and L are involved, so the number of fundamental
dimensions, k = 2. Therefore, the number of dimensionless ɉ groups needed is (n-k) =
(8-2) = 6.
From the list of dimensional variables, k of them are selected to be primary
(repeating) variables, which must contain all of the fundamental dimensions and must
not form a ɉ among themselves. In this case, k = 2. Therefore, L and ȡ are selected as
the primary (repeating) variables. The ɉ groups are formed by multiplying the product
of the primary variables, with unknown exponents, by each of the remaining variables,
one at a time. In order to satisfy dimensional homogeneity, the exponents of each
dimension are equated on both sides of each pi equation, and so solve for the exponents
and the forms of the dimensionless groups. Since the ɉs are dimensionless, they can be
replaced with M0L0T0. In the case of this study, only M and L are involved, thus M0L0.
Working with ɉ1,
31
( L ) a1 ( U ) b1 H t
M 0 L0
( L) a1 ( ML3 ) b1 L
M:
0 = b1
L:
0 = a1 - 3b1 + 1
104
Solving for a1 and b1,
a1 = -1,
Thus
31
b1 = 0
Ht
L
(5.2)
Working in a similar fashion with ɉ2, ɉ3, ɉ4, ɉ5 and ɉ6,
32
Hi
L
(5.3)
33
d
L
(5.4)
34
B
L
(5.5)
35
D
L
(5.6)
36
Us
U
(5.7)
The basis for the selection of the terms is based on the particular system under
investigation and should reflect the understanding of the process involved. A common
dimensionless parameter used to quantify wave attenuation is the coefficient of
transmission, Ct which is defined as a ratio of the transmitted wave height to the incident
§H
wave height ¨¨ t
© Hi
·
¸¸ . Note that Ct
¹
Ht
Hi
§ 31
¨¨
© 32
Ht L ·
. ¸ . Therefore, it follows that the
L H i ¸¹
105
above dimensionless pi groups are further rearranged and combined as desired to yield
the following non-dimensional relationship:
Ct
Ht
Hi
§B D H d·
f¨ , , i , ¸
© L L L L¹
(5.8)
The most common parameters are shown in Equation (5.8) and are considered
representative for a variety of floating breakwater types. The four governing
dimensionless pi terms have been used by researchers as the fundamental parameters for
the study of floating breakwaters. However, some complicated and unique systems may
not be completely explained by the results of the analysis. Thus, additional parameters
would be required to analyze the system. Note that
Us
will remain constant for a given
U
floating breakwater, therefore it is not included in the final relationship in Equation
(5.8). For the case of the STEPFLOAT breakwater with gap between rows of modules,
i.e. G = b and G = 2b, it follows the same method of dimensional analysis above with
similar working fashion and pattern for all variables with an additional variable of G to
arrive at the final functional Equation (5.9) as follows:
Ct
5.2
Ht
Hi
§B D H d G·
f¨ , , i , , ¸
©L L L L L¹
(5.9)
Experimental Results
This section describes the results of the experimental work which was done in the
laboratory to evaluate the efficiency of the STEFPLOAT breakwater system by
106
measuring the incident and transmitted wave heights in order to obtain the transmission
coefficients. During the experimental tests, incident and transmitted wave profiles were
measured by two wave probes and printed simultaneously on the recording papers from
the thermal printer of the HIOKI 8833 MEMORY Hi CORDER wave recorder. Figure
5.2 (a) and 5.2 (b) give examples of the wave profiles of the composite wave, which is a
combination of incident and reflected waves, and transmitted wave, respectively.
The attenuation efficiency of the STEPFLOAT floating breakwater is quantified
by the coefficients of transmission. The results of the experimental study in terms of
reflection coefficients and loss coefficients are also included in this chapter and directly
related to the wave periods. Therefore, wave dampening characteristics of the
STEPFLOAT breakwater, in terms of Ct, Cr and Cl, for the 2-row, 3-row, G = b and G =
2b floating breakwater systems are presented in the following sections, according to
different types of mooring systems.
5.2.1 Steel Chain Mooring System
5.2.1.1 Two-row system
Figure 5.3 shows the plots of Ct, Cr and Cl versus wave period for 2-row
STEPFLOAT breakwater model system moored with steel chains in a constant water
depth of 53 cm. The system was tested in the laboratory under regular incident waves
with T ranging from 0.68 sec to 1.05 sec. For the given range of wave period, it was
observed that the Ct value increases from 0.67 to 0.93 as the wave period increases while
the Cr curve shows a slowly decreasing trend from a Cr value of 0.30 to 0.21. The curve
107
Wave elevation (cm)
7.0
0.0
-7.0
0
2
4
6
8
10
12
14
16
Time (sec)
(a) The recorded signal of the composite wave in front of the model
Wave elevation (cm)
7.0
0.0
-7.0
0
2
4
6
8
Time (sec)
(b) The recorded signal of the transmitted wave behind the model
Figure 5.2 : Wave profiles of the composite and transmitted waves for 2-row model
system using vertical piles (f = 42 Hz or T = 0.95 sec)
108
1.0
Ct, Cr & Cl
0.8
Ct
0.6
Cl
0.4
0.2
Cr
0.0
0.6
0.7
0.8
0.9
1.0
1.1
Time (sec)
Figure 5.3 : Variation of Ct, Cr and Cl against T for 2-row model system using chain
mooring for D/d = 0.104
as in Figure 5.3 shows the decreasing trend of loss coefficient as a function of wave
period. It is seen that the 2-row model system using chain mooring does not provide
promising results with all its data points above Ct = 0.5. Small variation in Cr values
within the tested range of wave period shows that wave period or wave length has little
influence on Cr. The low values of Cr indicate that little wave reflection occurred when
the floating breakwater is subjected to the incident wave. However, most of the incident
wave energy is attenuated by the mechanism of wave dissipation with higher Cl values.
Plates 5.1 (a) and 5.1 (b) show the digital images taken during the experiments
with relatively shorter and longer waves for 2-row system with chain mooring for
performance comparison. It is seen in Plate 5.1 (a) that shorter incident wave gives better
wave attenuation compared to relatively longer incident wave in Plate 5.1 (b). A calmer
wave environment with smaller transmitted wave height behind the structure is observed
in Plate 5.1 (a) while relatively longer incident wave does not have significant impact on
wave attenuation as visualized in Plate 5.1 (b). In fact, these phenomena have readily
shown in Figure 5.3 with the Ct trend line. From the curve, it is noticeable that shorter
109
wave length (or smaller wave period) gives lower Ct whereas longer wave length (or
greater wave period) has resulted in higher Ct. Plates 5.1 (c) and 5.1 (d) show some wave
dissipation phenomena during experiments.
5.2.2 Restrained Case
5.2.2.1 Two-row System
Figure 5.4 shows the plots of measured Ct, Cr and Cl versus wave period for 2row STEPFLOAT model that is restrained at the vertical piles to be prevented from
moving in a constant water depth of 45 cm. The system was tested under incident waves
with T ranging from 0.69 sec to 1.11 sec. As expected, Ct decreases as wave period
decreases. Ct value decreases from 0.55 to 0.17 as the wave period decreases for the
given range of tested wave period. The threshold level of Ct = 0.5 is achieved for T <
1.06 sec for D/d = 0.133. In general, the trend line of Cr is fairly uniform around the
value of 0.4 with incident wave period. There is a tendency to decrease slightly as the
wave period increases. It was also observed that Cl varies little with wave period. The
lowest and highest values of Cl are 0.72 and 0.93, respectively, with a difference of 0.21.
It was observed that the restrained 2-row model system provides effective
performance on wave attenuation, especially in higher frequency waves, with moderate
wave reflection. Again, like the 2-row model with chain mooring, small variation in Cr
values for this restrained case of 2-row system shows that wave period or wave length
has little influence on Cr. Similarly, most of the incident wave energy is attenuated
through wave dissipation with even higher Cl values.
(d)
(c)
Plate 5.1 : Wave-structure interaction during experimental tests for 2-row system using chain mooring
(b)
(a)
110
111
1.0
Ct , Cr & Cl
0.8
Cl
0.6
Ct
0.4
Cr
0.2
0.0
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Time (sec)
Figure 5.4 : Variation of Ct, Cr and Cl against T for 2-row model system restrained
from moving for D/d = 0.133
Plate 5.2 shows the digital images taken during the experiments for the restrained
2-row system. A calmer wave environment at the leeward side of the floating breakwater
[see Plate 5.2 (a)] was observed with its short incoming wave stroke on the structure.
Relatively longer incident wave does not give significant influence on wave attenuation
[see Plate 5.2 (b)]. Plates 5.2 (c) and 5.2 (d) show the chaotic phenomena of wave
breaking and turbulence during the experiments that have contributed considerably to the
wave attenuation.
5.2.2.2 Three-row System
Figure 5.5 shows the plots of measured Ct, Cr and Cl versus wave period for the
restrained 3-row STEPFLOAT breakwater system in water depth of 45 cm. Incident
waves with T ranging from 0.70 sec to 1.33 sec were generated throughout the
(d)
(c)
Plate 5.2 : Wave-structure interaction during experimental tests for the restrained 2-row system
(b)
(a)
112
113
experiments. For the given range of tested wave period, Ct decreases from 0.65 to 0.15
as wave period decreases. For the restrained 3-row system, only when the wave period T
<1.07 sec is the threshold level of Ct = 0.5 is attained for D/d = 0.133. It is noted that the
effect of wave length (or wave period) has less influential to wave reflection. The trend
line of Cr is fairly uniform around the value of 0.4 with a tendency to decrease slightly as
the wave period increases. Nevertheless, it is found that Cl varies from 0.67 to 0.88 as
the wave period decreases.
In general, the restrained 3-row model system provides effective performance on
wave attenuation with moderate wave reflection and better attenuation efficiency is
achieved in higher frequency wave region. Analogously to the previous discussed
floating breakwater systems, most of the incident wave heights are reduced through
wave dissipation, resulting in high values of Cl.
1.0
Cl
Ct , Cr & Cl
0.8
0.6
Cr
0.4
0.2
Ct
0.0
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Time (sec)
Figure 5.5 : Variation of Ct, Cr and Cl against T for 3-row model system restrained
from moving for D/d = 0.133
114
Plate 5.3 shows the digital images taken during the experiments for the restrained
3-row system. Plates 5.3 (a) and 5.3 (b) compare the effect of wave attenuation for the
model with the relatively shorter and longer incident waves, respectively. Smaller wave
height at the leeward of the floating breakwater was observed with its short incident
wave stroke on the structure. Relatively longer incident wave, however, does not pose
significant influence on wave attenuation. Plates 5.3 (c) and 5.3 (d) show the formation
of eddies and wave breaking or turbulence during the experiments that have enhanced
and accelerated the capability of wave dissipation by the model.
5.2.3
Vertical Pile System
5.2.3.1 Two-row System
Plots of measured Ct, Cr and Cl versus wave period for 2-row STEPFLOAT
breakwater model system using vertical piles in a constant water depth of 45 cm are
shown in Figure 5.6. The system was allowed to move in heave and limited roll motions.
The model was tested under incident waves with T ranging from 0.70 sec to 1.33 sec. Ct
decreases from 0.63 to 0.29 as wave period decreases, as expected. The threshold level
of Ct = 0.5 is attained for T < 0.97 sec for D/d = 0.133. The curve of Cr is rather uniform
around the value of 0.2 with incident wave period. It was observed that Cl varies from
0.72 to 0.93 as the wave period decreases. Most of the incident wave energy is dissipated
instead of reflected seaward, hence high values of Cl and lower values of Cr are obtained.
The vertical pile-system floating breakwater for 2-row model provides promising result
on wave attenuation with little wave reflection.
(d)
(c)
Plate 5.3 : Wave-structure interaction during experimental tests for the restrained 3-row system
(b)
(a)
115
116
1.0
Cl
Ct , Cr & Cl
0.8
0.6
0.4
Ct
0.2
Cr
0.0
0.6
0.8
1.0
1.2
1.4
Time (sec)
Figure 5.6 : Variation of Ct, Cr and Cl against T for 2-row model system using
vertical piles for D/d = 0.133
Plate 5.4 shows the digital images taken during the experiments for the 2-row
system using vertical piles. It could be seen obviously from the Plates 5.4 (a) and 5.4 (b)
that different effects of wave attenuation resulted when the structure was subjected to
shorter and longer incident waves, respectively. Calmer wave environment was observed
at the lee of the floating breakwater when shorter incident waves were generated in the
wave flume. Incident waves with longer period, however, do not have significant impact
on wave attenuation. Plates 5.4 (c) shows the phenomenon of wave dissipation by
breaking and turbulence over the structure during the experiments while the formation of
eddies was observed around the structure in heave and limited roll motions as shown in
Plate 5.4(d).
(d)
(c)
Plate 5.4 : Wave-structure interaction during experimental tests for 2-row system using vertical piles
(b)
(a)
117
118
5.2.3.2 Three-row System
Figure 5.7 shows the plots of measured Ct, Cr and Cl versus wave period for 3row STEPFLOAT breakwater model using vertical piles as mooring method. The system
was tested under incident waves with T ranging from 0.70 sec to 1.33 sec. Ct decreases
from 0.67 to 0.23 as the wave period decreases. The threshold level of Ct = 0.5 is
achieved for T < 0.99 sec for D/d = 0.133. The trend line of Cr lies between 0.2 and 0.3
with incident wave period. It was also observed that Cr varies very little with wave
period. This implies that wave period or wave length has little influence on Cr. Plots of
Cl lie above 0.6 with the minimum and maximum values as 0.69 and 0.93, respectively,
corresponding to the greatest and smallest wave periods tested. It shows that most of the
incident wave energy is lost through wave dissipation. This 3-row model system gives
effective performance on wave attenuation with little wave reflection.
1.0
Cl
Ct , Cr & Cl
0.8
0.6
0.4
Cr
0.2
Ct
0.0
0.6
0.8
1.0
1.2
1.4
Time (sec)
Figure 5.7 : Variation of Ct, Cr and Cl against T for 3-row model system using
vertical piles for D/d = 0.133
119
Plates 5.5 (a) and 5.5 (b) compare the wave transmission due to the 3-row model
using vertical piles with shorter and longer incident waves. It was observed that shorter
incident waves result in smaller wave heights at the leeward side of the structure after
the wave transmission while longer incoming waves have lesser impact on wave
attenuation. Plates 5.5 (c) and 5.5 (d) show the chaotic phenomena of turbulence and
wave breaking during the experiments. It was also observed that there is greater rolling
motion occurred during the experiments for 3-row model if compared to the 2-row
system.
5.2.3.3 G = b System
Plots of measured Ct, Cr and Cl versus wave period for G = b breakwater model
using vertical piles are shown in Figure 5.8. The system was tested under incident waves
with T ranging from 0.70 sec to 1.33 sec. Ct decreases from 0.62 to 0.23 as the wave
period decreases. The threshold level of Ct = 0.5 is obtained for T < 1.02 sec for D/d =
0.133 with the best efficiency occurring at T = 0.70 sec for the given range of tested
wave period. The trend line of Cr lies fairly uniform around the value of 0.2, varying
very little with wave period. Plots of Cl lie above 0.7 with the minimum and maximum
values as 0.71 and 0.94, respectively. Therefore, it is clear from this figure that the
model system provides effective performance on wave attenuation with most of the
incident wave energy is lost through wave dissipation with minimum wave reflection.
Plates 5.6 (a) and 5.6 (b) compare the wave transmission due to the G = b model
using vertical piles with shorter and longer incident waves. It was observed that
performance of the system with shorter incident waves distinguishes the one with longer
incident waves as smaller wave heights were observed at the leeward side of the
(d)
(c)
Plate 5.5 : Wave-structure interaction during experimental tests for 3-row system using vertical piles
(b)
(a)
120
121
1.0
Cl
Ct , Cr & Cl
0.8
0.6
Ct
0.4
Cr
0.2
0.0
0.6
0.8
1.0
1.2
1.4
Time (sec)
Figure 5.8 : Variation of Ct, Cr and Cl against T for G = b model system using
vertical piles for D/d = 0.133
structure after the wave transmission of shorter incident waves. Plates 5.6 (c) and 5.6 (d)
show the chaotic phenomena of wave breaking, turbulence and eddies during the
experiments.
5.2.3.4 G = 2b System
Figure 5.9 shows the plots of measured Ct, Cr and Cl versus wave period for G =
2b STEPFLOAT breakwater model system using vertical piles in a constant water depth
of 45 cm. The model was tested under incident waves with T ranging from 0.70 sec to
1.33 sec. As expected, the curve of Ct decreases from 0.64 to 0.20 as wave period
decreases. The threshold level of Ct = 0.5 is attained for T < 0.98 sec for D/d = 0.133.
The curve of Cr lies within the values of 0.1 to 0.4 with a “trough” occurring between
(d)
(c)
Plate 5.6 : Wave-structure interaction during experimental tests for G = b system using vertical piles
(b)
(a)
122
123
1.00 sec and 1.05 sec. It was observed that the trend line of Cl varies from 0.69 to 0.92 as
the wave period decreases. Most of the incident wave energy is dissipated rather than
reflected seaward, hence high values of Cl and lower values of Cr are obtained. The
vertical pile-system floating breakwater for G = 2b model provides promising result on
wave attenuation with little wave reflection.
Plate 5.7 shows the digital images taken during the experiments for the G = 2b
system using vertical piles. It could be seen obviously from the Plates 5.7(a) and 5.7 (b)
that distinct effects of wave attenuation resulted when the structure was subjected to
shorter and longer incident waves, respectively. Calmer wave environment was observed
at the lee of the floating breakwater when shorter incident waves were generated in the
wave flume. Nevertheless, incident waves with longer period do not have significant
impact on wave attenuation. Plates 5.7 (c) shows the phenomenon of wave dissipation
through the formation of turbulence and eddies. Wave breaking was also observed
during the experiments as shown in Plate 5.7 (d).
1.0
Cl
Ct , Cr & Cl
0.8
0.6
0.4
Cr
0.2
Ct
0.0
0.6
0.8
1.0
1.2
1.4
Time (sec)
Figure 5.9 : Variation of Ct, Cr and Cl against T for G = 2b model system using
vertical piles for D/d = 0.133
(d)
(c)
Plate 5.7 : Wave-structure interaction during experimental tests for G = 2b system using vertical piles
(b)
(a)
124
125
5.3
Performance Evaluation Based on Results Comparison
Floating breakwater performance is usually defined by the transmission
coefficient. A value of Ct = 0.5 or less is indicative of very good performance as the
transmitted wave height is reduced to one half of its incident value which is also
equivalent to only 25 percent of its incident wave energy. Analytical, numerical or/and
laboratory experiments have been conducted to predict the performance of various
floating breakwater designs as discussed in Chapter II. Besides Briggs (2001)’s RIBS
XM99 field trials, Nelson and Hemsley (1988) have also completed their field
monitoring studies on six floating breakwaters in Puget Sound, Washington and have
reported the performance and durability of the six floating breakwaters in a brief report.
Wave transmission for a floating breakwater is a function of many wave and structural
parameters. The author intends to present the performance evaluation of the
STEPFLOAT in terms of mooring systems and system arrangements.
5.3.1
Performance Evaluation in terms of Mooring Systems
5.3.1.1 STEPFLOAT vs SSFBW vs Rectangular Pontoon (with Line Mooring)
Previously, a series of laboratory experimental tests on the wave dampening
characteristics of a fundamental geometrical shape of a stepped-slope floating
breakwater system (SSFBW) were carried out by Teh (2002). The experiments included
a single row of SSFBW, two-row SSFBW and three-row SSFBW systems. The three
sets of experiments were run in wave only condition in the flume and steady
unidirectional regular non-breaking waves were generated throughout the tests. Each
126
model was moored to the flume bed by four nylon ropes to keep the body in position.
Nylon lines were crossed for all the experiments in order to provide additional keel
clearance. The mooring scopes, which are technically defined as the ratio of length of
anchor rode in use relative to the vertical depth of the anchor, were 1.5 and 2.1 for water
depth of 20 cm and 33 cm, respectively.
Additional tests on rectangular pontoon were also carried out by Teh (2002) in
order to compare the wave attenuation characteristics of a SSFBW to the rectangular
pontoon which has similar design criteria to the SSFBW. Rectangular pontoon is known
to be the most common and simplest design in the history of floating breakwaters. It was
reported that rectangular pontoons performed satisfactorily and gave high degree of
wave attenuation than most of the existing types of floating breakwaters. Among many
researchers who had investigated the performance of the rectangular pontoon in the past
were Kato et al. (1966), Carve (1979), Nece & Skjelbreia (1984), Isaacson & Byres
(1988), Tolba (1999) and so on.
In order to evaluate the wave dampening characteristics of the 2-row
STEPFLOAT breakwater system moored with steel chains, a comparison was made
between the results of the present work for the 2-row chain-moored STEPFLOAT
breakwater system with the work of Teh (2002) for a single row SSFBW and the
rectangular pontoon. The specifications of a 2-row STEPFLOAT, a single row of
SSFBW and rectangular pontoon models as well as the details of respective
experimental tests are listed in Figure 5.10. Results comparisons were performed in
terms of Ct against specific variables, which include T, B/L, D/L, H/L and d/L.
Figure 5.11 shows that the results of the STEPFLOAT tend to be at the shorter
range of the tested wave period that is smaller than 1.05 sec while the tested wave period
range of the SSFBW and rectangular pontoon is between 0.87 sec and 1.66 sec.
2-row STEPFLOAT
A single row of SSFBW
Rectangular pontoon
Length
Width
Height
Freeboard
Draft
Mass
Material
86.2 cm
80 cm
80 cm
21 cm
25 cm
25 cm
10 cm
13 cm
10 cm
4 cm
5 cm
2.5 cm
5.5 cm
8 cm
7.5 cm
4.5 kg
16 kg
15.5 kg
Top module: Polymer rubber blend
Sand, cement, polystyrene and water Sand, cement, polystyrene and water
(EDPM and SBR or equivalent)
(in ratio 1.5 : 1 : 7 : 2)
(in ratio 1.5 : 1 : 7 : 2)
Bottom module: Polyethylene (HDPE)
Steel chains (spread mooring pattern)
Nylon lines (cross mooring pattern)
Nylon lines (cross mooring pattern)
Mooring system
1.0 (for d = 53 cm)
1.5 (for d = 20 cm)
1.5 (for d = 20 cm)
Mooring scope
2.1 (for d = 33 cm)
2.1 (for d = 33 cm)
53 cm
20 cm and 33 cm
20 cm and 33 cm
Water depth
0.68 - 1.05 sec
0.87 - 1.66 sec
0.87 - 1.66 sec
Wave period
Water region
0.09 < d/L < 0.20 (for d = 20 cm)
0.09 < d/L < 0.20 (for d = 20 cm)
0.32 < d/L < 0.73
0.12 < d/L < 0.29 (for d = 30 cm)
0.12 < d/L < 0.29 (for d = 30 cm)
(transitional to deepwater region)
(transitional water region)
(transitional water region)
Figure 5.10 : Specifications and test details of a 2-row STEPFLOAT, a single row of SSFBW and rectangular pontoon models
Details
127
128
1.0
Pontoon (d=33cm)
STEPFLOAT
(d=53cm)
SSFBW (d=33cm)
Pontoon
(d=20cm)
Ct
0.8
0.6
SSFBW (d=20cm)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Time (sec)
Figure 5.11 : Ct vs T for comparisons among a 2-row STEPFLOAT, a single row of
SSFBW and rectangular pontoon models
Nevertheless, it was demonstrated that the performance of the present study of the chainmoored STEPFLOAT breakwater system in water depth of 53 cm is not as effective as
the line-moored SSFBW and rectangular pontoon (except the rectangular pontoon tested
in d = 33 cm) tested in water depths of 20 cm and 33 cm. As can be seen in the trend
lines of SSFBW or rectangular pontoon for d = 20 cm and 33 cm, as the water depth
increases from 20 cm to 33 cm, Ct was found to increase. Note that the STEPFLOAT
system was tested in a constant water depth of 53 cm, which is much higher than d = 20
cm and 33 cm, therefore, as expected, higher Ct was observed in the 53-cm water depth
if compared to the wave attenuation of the other two systems in shallower water depths
of 20 cm and 33 cm.
Comparing the wave attenuation efficiency of the STEPFLOAT and rectangular
pontoon (d = 33 cm), it was noticed that the STEPFLOAT breakwater has better
performance. However, no conclusive remarks could be made as the tested water depth
for the STEPFLOAT was different from the other two. Nonetheless, the comparisons
129
pose a general picture of the level of wave attenuation efficiency given by the chainmoored STEPFLOAT system.
It was believed that the floating breakwater with improved geometrical shape,
namely the STEPFLOAT, is capable to attenuate waves with better performance. With a
model to prototype scale of 1:10, a higher water depth of 53 cm, instead of 20 cm or 33
cm, was chosen for the chain-moored STEPFLOAT to be tested in the laboratory,
considering the practicability and feasibility of the STEPFLOAT system in deeper water
zone (transitional and deep water region) in the real world in future.
From the Figure 5.10, it shows an obvious comparison of the structural
geometrical configuration dimensions among the three compared models. It is indicated
that parameters, which significantly influence the wave attenuation, such as breakwater
width, height, freeboard and draft of the STEPFLOAT model have smaller dimensions
compared to the other two. For instance, the STEPLOAT breakwater width is 21 cm
compared to the 25 cm for the SSFBW and pontoon. The STEPFLOAT has smaller
draft, i.e. 5.5 cm, if compared to the SSFBW and pontoon with 8-cm and 7.5-cm draft,
respectively. The comparison of the structural geometrical dimensions has shown that
even though with minimum structural width and draft, the STEPFLOAT system in
relatively deeper water depth is capable to provide wave attenuation to a certain extent
as compared to the other two systems.
Besides, it was also observed that the scope of mooring lines used in the
experiments were different between the STEPFLOAT breakwater and the other two
systems. The STEPFLOAT breakwater was moored to the flume bottom with six steel
chains as shown in Plate 5.8. The mooring scope is approximately 1.0. Taut legs, i.e. the
chains rise from the anchor under normal pretension, were used in the experiments. Taut
moorings work in a fundamentally different way to catenary moorings because taut
130
Plate 5.8 : A 2-row STEPFLOAT breakwater moored with six steel chains
moorings have a much more linear stiffness than the progressively stiffening catenary
systems where offsets under wave load can be better controlled. However, the taut
mooring used in the experiments had not been stretched out completely and had not
reached their total natural length.
As the waves impinge upon the floating breakwater structure, snap loads or
snatch loads arise when chains go through a transition from a relatively slack condition
to a fully taut condition suddenly over a small interval of time and the transition between
slack and taut conditions occur repeatedly (see Plate 5.9). This phenomenon of
becoming taut suddenly is like a violent impact due to excessive drift of the
STEPFLOAT breakwater model. This may be attributed to the limited scope of the
mooring lines. As a result, the induced motions of the floating breakwater such as sway
and roll occurred which in turn generate secondary waves behind the structure, thus
affecting the performance of the STEPFLOAT breakwater (see Plate 5.10).
131
(a) STEPFLOAT breakwater in slack condition
(b) STEPFLOAT breakwater in taut condition
Plate 5.9 : The transition between slack and taut conditions of the STEPFLOAT
system
132
Plate 5.10 : Induced roll and sway motions generate secondary waves at the leeside
of the floating breakwater during the experiments
Sannasiraj et al. (1998) in their study on the behaviour of pontoon-type floating
breakwaters with different types of mooring configurations, viz. mooring at water level,
mooring at base bottom and cross moored at base bottom level, has reported that Ct
values are not significantly affected by the mooring configurations studied. According to
Bhat (1998) in his study on twin-pontoon floating breakwaters using chain and nylon
moorings, the choice of mooring line does not appear to significantly affect the overall
performance of the breakwater. Therefore, it was assumed that the mooring
configuration and type of mooring lines used in the present study on 2-row
STEPFLOAT system and the study done by Teh (2002) did not significantly influence
the overall performance of the systems. Therefore, it is strongly believed that with a little
more improvement on the mooring system of the STEPFLOAT breakwater to eliminate
those unfavourable motions, the whole system of the floating breakwater would give a
promising result on wave attenuation with better performance, i.e. with low Ct values.
133
Figure 5.12 to Figure 5.15 show the results comparison for the 2-row chainmoored STEPFLOAT breakwater system with the SSFBW and rectangular pontoon in
terms of Ct versus four different specific dimensionless variables, i.e. B/L, D/L, H/L and
d/L. As in Figure 5.12, it was observed that all results show decreasing trends as B/L
increases. The larger the relative width of the floating breakwater, the lower the Ct
values obtained. Similar trends were also observed for the plot of Ct versus D/L as
shown in Figure 5.13. Ct decreases as the relative draft increased. Both results of the
transmission coefficient as a function of B/L or D/L indicated that the SSFBW system
tested in water depths of 20 cm and 33 cm gives lower Ct as compared to the
STEPFLOAT system tested in the water depth of 53 cm, for the given tested range of
B/L and D/L. However, the STEPFLOAT system performs better than the rectangular
pontoon in d = 33 cm but d = 20 cm.
Figure 5.14 indicates that slightly decreasing trends were observed as the H/L
increased. Generally, the three line-moored floating breakwater systems did not show
distinctly huge variations of Ct with respect to the change of H/L. This has indicated that
the wave steepness does not have significant influence on the wave attenuation.
However, steeper waves are expected to give lower values of Ct. As the experiments on
the SSFBW and the rectangular pontoon were tested in transitional water region while
the study on the STEPFLOAT system was modeled and tested in transitional and deep
water zone (as indicated in Figure 5.10), the Ct trend line of the STEPFLOAT
breakwater was segregated in a higher d/L from the trend lines of the SSFBW and
rectangular pontoon with d/L below 0.29, as shown in Figure 5.15. As a result, no
comparison could be made between the STEPFLOAT system with the other two.
Anyway, the result shows that the greater the d/L is, the better the wave dampening
ability of a system it makes.
134
1.0
Pontoon (d=33cm)
0.8
Ct
SSFBW
(d=33cm)
STEPFLOAT (d=53cm)
0.6
Pontoon
(d=20cm)
SSFBW (d=20cm)
0.4
0.05
0.10
0.15
0.20
0.25
0.30
B/L
Figure 5.12 : Ct vs B/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
1.0
Pontoon (d=33cm)
0.8
Ct
STEPFLOAT (d=53cm)
0.6
Pontoon
(d=20cm)
SSFBW (d=33cm)
SSFBW (d=20cm)
0.4
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D/L
Figure 5.13 : Ct vs D/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
135
1.0
Pontoon (d=33cm)
Ct
0.8
STEPFLOAT
(d=53cm)
SSFBW (d=33cm)
0.6
SSFBW (d=20cm)
Pontoon (d=20cm)
0.4
0.00
0.02
0.04
0.06
0.08
H/L
Figure 5.14 : Ct vs H/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
1.0
Pontoon
(d=33cm)
0.8
Ct
STEPFLOAT (d=53cm)
Pontoon
(d=20cm)
0.6
SSFBW (d=33cm)
SSFBW (d=20cm)
0.4
0.0
0.2
0.4
0.6
0.8
d/L
Figure 5.15 : Ct vs d/L for comparisons among a 2-row STEPFLOAT, a single row
of SSFBW and rectangular pontoon models
136
The floating breakwaters are normally designed with an efficiency of about 50%
in attenuating the incident wave energy. With this in mind, it was clearly seen that the
performance of the 2-row chain-moored STEPFLOAT breakwater considered in the
present study was found to be not satisfactory as the Ct reached the minimum of only
0.6. Better efficiency on wave attenuation with lower values of Ct is required to attenuate
waves effectively.
Based on the observations during the laboratory experiments, it was observed
that the mooring system using chains has caused the motions of sway and roll, which
subsequently has contributed to the secondary wave formation at the leeside of the
floating breakwater. Therefore, the Ct values do not go below 0.6. In order to avoid the
formation of sway and roll motions, another mooring method was suggested by using
vertical pile system, which limits the floating breakwater system to only heave and
limited roll motions. Besides, this system is suggested to overcome the problems of
mooring lines at area with significant tidal range and also to improve the performance of
the floating breakwater system as a walkway or pier in fishing harbors and marinas. For
the latter use, a detailed study on the floating structure motions needs to be investigated,
especially the motion of limited roll which might affect the stability of the structure, thus
the safety of the pedestrians. However, the present study has excluded this specific scope
of work.
5.3.1.2 STEPFLOAT (Vertical Piles vs Steel Chains)
In order to avoid the formation of sway and roll motions, STEPFLOAT
breakwater using vertical pile system was suggested. In order to allow comparison with
the same model system using chain mooring, a series of experimental tests on two-row
137
STEPFLOAT using vertical pile system was conducted under similar wave conditions as
discussed in previous section. However, the water depth for all experiments on pilesupported STEPFLOAT breakwater remains constant at 45 cm.
For performance evaluation, the variations of Ct with T, B/L, D/L, H/L and d/L,
for two-row STEPFLOAT breakwater using vertical piles (for a constant D/d = 0.133)
from the present study is compared with that for chain-moored STEPFLOAT breakwater
(for D/d = 0.104). It is seen from Figure 5.16 that for the given tested range of wave
period, the minimum Ct for the STEPFLOAT using vertical piles and steel chains is 0.25
and 0.60, respectively. It is observed that Ct varies from 0.25 to 0.69 for the pile-system
STEPFLOAT and from 0.60 to 0.97 for the chain-moored STEPFLOAT. Furthermore,
regardless of types of mooring system, the value of Ct is found to decrease with decrease
in wave period showing that the STEPFLOAT system performs more effective with
incoming wave of relatively shorter period. A greater amount of energy gets transmitted
for longer period waves and more energy gets dissipated for shorter period waves and
hence Ct is lower for smaller T.
1.0
2
Ct = 0.9573T - 0.9614T + 0.8803
steel chains
2
(D/d = 0.104)
R = 0.7958
0.8
Ct
0.6
vertical piles
0.4
(D/d = 0.133)
2
Ct = -0.6407T + 1.8403T - 0.6807
0.2
2
R = 0.8425
0.0
0.6
0.8
1.0
1.2
1.4
Time (sec)
Figure 5.16 : Ct vs T for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
138
The variations of percentage of Ct reduction, [Ct] red, for the given tested range of
wave periods from 0.68 sec to 1.05 sec, between pile-system and chained-moored
STEPFLOAT are calculated based on the second order polynomial regression equations
(see Figure 5.16) and are projected in Figure 5.17. It is seen that the [Ct] red ranges from
33.42% to 39.30% with wave period. The results clearly indicate that significant wave
attenuation capability is provided by the pile-system STEPFLOAT compared to the
chain-moored system.
100
[Ct] red (%)
80
60
40
20
0
0.6
0.7
0.8
0.9
1.0
1.1
Time (sec)
Figure 5.17 : [Ct]red vs T between 2-row STEPFLOAT breakwater using vertical
piles and steel chain mooring
Regardless of mooring systems, both curves in Figure 5.18 show a decreasing
trend of Ct as B/L increases. The relatively longer period waves are expected to have
larger transmission in the direction of wave propagation and the larger relative width
would lead to substantial dissipation, leading to a smaller Ct for larger B/L. It is evident
from the results that vertical pile-system floating breakwater has lower Ct compared to
the similar system moored with chains. Plots of Ct against D/L, as shown in Figure 5.19,
shows similar trends as the previous one. Ct is seen to decrease with increase in D/L.
139
1.0
steel chains
(D/d = 0.104)
0.8
Ct
0.6
vertical piles
0.4
(D/d = 0.133)
0.2
0.0
0.05
0.10
0.15
0.20
0.25
0.30
B/L
Figure 5.18 : Ct vs B/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
1.0
steel chains (D/d = 0.104)
0.8
Ct
0.6
0.4
vertical piles
(D/d = 0.133)
0.2
0.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
D/L
Figure 5.19 : Ct vs D/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
140
For the plots of Ct against H/L and d/L, as shown in Figures 5.20 and 5.21
respectively, decreasing trends of Ct are also observed as the independent dimensionless
variables of H/L and d/L increase. Figure 5.21 shows that Ct decreases as d/L or as the
water region goes from transitional water to deepwater. For example, for d/L = 0.4
(transitional water zone), Ct is as high as 0.82 for the chain-moored system while for
vertical-pile system, Ct value is much lower, i.e. 0.42. Ct values for d/L = 0.6 (deep water
zone) for steel chain system and pile-system floating breakwaters are 0.69 and 0.28,
relatively. This shows that the vertical pile-system floating breakwater performs
approximately 40% better than the chain-moored floating breakwater in reducing the
incident wave height.
5.3.1.3 STEPFLOAT (Vertical Piles vs Restrained Case)
The pile-system floating breakwater is designed to allow the structure to move
only in heave and limited roll motions when exposed to the attack of the incident regular
waves. This idea can eliminate the problem of sway motion and reduce the roll motion
which momentarily generates waves at the lee of the structure. In order to establish the
efficiency of the pile-system STEPFLOAT breakwater, a comparison was made between
the results for the pile-system STEPFLOAT and the restrained body, for 2-row and 3row model systems. All experiments were carried out in a constant water depth of 45 cm
for D/d = 0.133.
Figures 5.22 and 5.23 show the values of Ct, Cr and Cl for both cases of verticalpile system and restrained body plotted versus wave period for 2-row and 3-row
systems, respectively. The results compare the effect of heave and limited roll motions
on Ct, Cr and Cl for both tested cases. The figures show that the values of Ct of the pile-
141
1.0
steel chains
(D/d = 0.104)
0.8
Ct
0.6
0.4
vertical piles
(D/d = 0.133)
0.2
0.0
0.00
0.02
0.04
0.06
0.08
H/L
Figure 5.20 : Ct vs H/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
1.0
0.8
steel chains
(D/d = 0.104)
Ct
0.6
0.4
vertical piles
(D/d = 0.133)
0.2
0.0
0.15
0.30
0.45
0.60
0.75
d/L
Figure 5.21 : Ct vs d/L for comparison of 2-row STEPFLOAT breakwater using
vertical piles and steel chain mooring
142
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.6
0.8
1.0
1.2
1.4
1.0
0.8
0.6
Cr
restrained case
0.4
0.2
vertical piles
0.0
0.6
0.8
1.0
1.2
1.4
1.0
vertical piles
0.8
restrained case
Cl
0.6
0.4
0.2
0.0
0.6
0.8
1.0
1.2
1.4
Time (sec)
Figure 5.22 : Comparison between restrained case and vertical-pile system on the
effect of heave and limited roll motions on Ct, Cr and Cl for 2-row STEPFLOAT
143
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.6
0.8
1.0
1.2
1.4
1.0
0.8
0.6
Cr
restrained case
0.4
0.2
vertical piles
0.0
0.6
0.8
1.0
1.0
1.2
1.4
1.2
1.4
vertical piles
0.8
restrained case
Cl
0.6
0.4
0.2
0.0
0.6
0.8
1.0
Time (sec)
Figure 5.23 : Comparison between restrained case and vertical-pile system on the
effect of heave and limited roll motions on Ct, Cr and Cl for 3-row STEPFLOAT
144
system STEPFLOAT are slightly higher than the results for the restrained structure for
2-row and 3-row systems. The explanation of these results is that much of the incident
wave energy was reflected back offshore for the case of restrained body if compared to
lower wave reflection by the vertical pile-system STEPFLOAT. The Cr curves for 2-row
and 3-row systems show a difference of about 20% between the restrained case and the
pile-system breakwater.
It is obviously seen in the figures that the restrained case has higher values of
coefficient of reflection and thereby contributed to the lower values of coefficient of
transmission. Much of the incident wave energy was reflected back offshore, for the case
of restrained body, rather than transmitted over or beneath the structure. For the case of
vertical pile-system STEPFLOAT, lesser wave reflection occurred during wave-structure
interaction as the floating breakwater moved up and down continuously in heave motion
which allowed partial incident wave energy to be transmitted over or beneath the
floating structure. Thus, higher Ct was observed for the case of vertical pile-system
STEPFLOAT.
There is another possible explanation for the Ct curves of the restrained case and
the vertical pile-system breakwater. During wave-structure interaction, limited roll
motion exerted. This motion might have generated secondary radiated waves in the lee
of the structure. Thus higher Ct was observed for the pile-system breakwater than the
one prevented from moving. However, it is believed that the limited roll motion does not
have significant influence on secondary waves formation.
While most of the wave attenuation was contributed by energy lost as clearly
seen from the Cl curves, no significant difference of Cl was observed between the
restrained case and the pile-system STEPFLOAT. In general, however, it was observed
that the values of Cl of the restrained case for both 2-row and 3-row systems are slightly
145
lower than the Cl of the pile-system breakwater. In the case of the restrained structure,
the total energy of the incident wave is divided into three proportions, i.e. the energy of
the transmitted wave, the energy of the reflected wave and the energy lost. In the case of
the vertical pile-system STEPFLOAT breakwater, there is an additional source of toss
energy, which is the energy lost in exerting the heave motion of the body. Hence slightly
higher Cl was observed for the pile-system breakwater.
While most of the previous studies on heave motion floating breakwater as
compared to restrained body reported that a slightly better performance was found in the
heave motion model, the contrary was found in the STEPFLOAT breakwater system.
This phenomenon could be attributed to the relatively light mass of the floating
breakwater model. With its light structure characteristic, lesser toss energy is needed to
induce the heave motion of the body. As a result, there is no significant energy lost due
to the induced heave motion by the toss energy. For this reason, little difference was
observed for the Cl curves of the restrained case and the pile-system STEPFLOAT.
While wave reflection remaining low and energy lost has no considerable difference as
compared to the restrained case, a result of slightly higher Ct for the pile-system
STEPFLOAT was obtained.
From the Ct trend lines for both cases of restrained body and pile-system
breakwater, as projected in Figures 5.22 and 5.23, it is clearly seen that the difference of
both Ct curves is somewhat little, i.e. the Ct difference ranging from 0.03 - 0.11 for 2row system and 0.02 - 0.08 for 3-row system. Therefore, as compensation for smaller
wave reflection which is more preferable for the pile-supported STEPFLOAT system,
especially for the safety of navigational vessels, a slightly higher Ct as compared to the
restrained case has to be accepted as tolerance.
146
Figures 5.24 through 5.27 show the results comparison of 2-row STEPFLOAT
system between the restrained case and vertical-pile system, with respect to the four
dimensionless variables, i.e. B/L, D/L, H/L and d/L. Similar results comparison of 3-row
system are shown in Figures 5.28 through 5.31. Regardless of type of mooring methods,
in general, all plots of Ct show decreasing trends as the dimensionless variables increase.
The decreasing trends of Ct with respect to the respective dimensionless variables have
been reported in the previous sections. However, the values of Ct for the case of vertical
pile-system STEPFLOAT are slightly higher than the restrained case.
5.3.2 Performance Evaluation of Pile-System STEPFLOAT in terms of System
Arrangements
5.3.2.1 Two-row vs Three-row
Figure 5.32 compares the pile-system STEPFLOAT breakwater for 2-row and 3row systems. The plots relate T with Ct and the difference of Ct between 2-row and 3row STEPFLOAT, 'Ct [2-3] = Ct 2-row - Ct 3-row. The floating structure can be made wide
enough in comparison to the wave that it is designed for so as to allow all of the
breaking energy to expend itself on the surface of the structure, leaving little more than a
rush of water off its surface to the leeside of the structure. Logically, it is expected that
the wider the width of the floating breakwater, the better the performance it will provide.
In general, the plots of Ct against T, as demonstrated in Figure 5.32, show that 3row system achieved lower Ct as compared to the 2-row STEPFLOAT, for the tested
wave period before 1.0 sec. However, as the wave period becomes greater than 1.0 sec,
147
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.05
0.10
0.15
0.20
0.25
0.30
B/L
Figure 5.24 : Ct vs B/L - Comparison of 2-row STEPFLOAT between restrained
case and vertical-pile system
1.0
0.8
0.6
Ct
vertical piles
0.4
0.2
restrained case
0.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D/L
Figure 5.25 : Ct vs D/L - Comparison of 2-row STEPFLOAT between restrained
case and vertical-pile system
148
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.00
0.02
0.04
0.06
0.08
0.10
H/L
Figure 5.26 : Ct vs H/L - Comparison of 2-row STEPFLOAT between restrained
case and vertical-pile system
1.0
0.8
0.6
Ct
vertical piles
0.4
0.2
0.0
0.15
restrained case
0.25
0.35
0.45
0.55
0.65
d/L
Figure 5.27 : Ct vs d/L - Comparison of 2-row STEPFLOAT between restrained
case and vertical-pile system
149
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
B/L
Figure 5.28 : Ct vs B/L - Comparison of 3-row STEPFLOAT between restrained
case and vertical-pile system
1.0
0.8
Ct
0.6
vertical piles
0.4
restrained case
0.2
0.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D/L
Figure 5.29 : Ct vs D/L - Comparison of 3-row STEPFLOAT between restrained
case and vertical-pile system
150
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.00
0.02
0.04
0.06
0.08
0.10
H/L
Figure 5.30 : Ct vs H/L - Comparison of 3-row STEPFLOAT between restrained
case and vertical-pile system
1.0
0.8
0.6
Ct
vertical piles
0.4
restrained case
0.2
0.0
0.15
0.25
0.35
0.45
0.55
0.65
d/L
Figure 5.31 : Ct vs d/L - Comparison of 3-row STEPFLOAT between restrained
case and vertical-pile system
151
1.0
2-row
Ct & 'Ct[2-3]
0.8
3-row
0.6
0.4
Ct
0.2
0.0
'Ct [2-3]
-0.2
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
T (sec)
Figure 5.32 : Ct & 'Ct[2-3] vs T - Performance comparison between 2-row and 3-row
STEPFLOAT systems
it was observed that the data points of Ct for both 2-row and 3-row systems do not show
any clear distinction. This might be attributed to the fact that relatively longer incoming
wave cannot sense the presence of an object effectively in its path which is much smaller
than its wave length. The plots of 'Ct [2-3] against T show that for T < 1.0 sec, most of the
data points occupy the region above the horizontal band of 0 and a 50% increase in
breakwater width causes a corresponding maximum decrease in the transmission
coefficient by approximately 17%. As expected, the STEPFLOAT system consisting of a
higher number of rows would give better wave attenuation characteristics. For T > 1.0
sec, the data points fall randomly in the horizontal band around 0.
Figure 5.33 shows the Ct curves against B/L for 2-row and 3-row systems. In
general, all the results show the amount of transmitted energy decreasing with the
increase of the B/L regardless of the number of rows. The threshold level of Ct = 0.5 is
achieved for 2-row system for B/L • 0.1478. For the 3-row system, this level is only
attained when B/L • 0.2068. According to Tobiasson and Kollmeyer (1991), for an
152
1.0
0.8
2-row
3-row
Ct
0.6
0.4
0.2
0.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
B/L
Figure 5.33 : Ct vs B/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
object to affect an incident wave, that object must be at least as large across as onequarter of the incident wave’s length. However, in the case of the STEPFLOAT, the
values of B/L are both smaller than 0.25 (or ¼) for the 2-row and 3-row systems in order
for the incident wave to sense the presence of the structure. For this, a relatively smaller
structure width is required to attenuate wave effectively, thus reducing the cost of
material and the necessary space of a specific site to construct the STEPFLOAT system.
Figures 5.34 through 5.36 show the plots of Ct against D/L, H/L and d/L,
respectively. All Ct plots show decreasing trends as the D/L, H/L and d/L increase. In
general, 3-row STEPFLOAT system gives lower Ct than the 2-row system for D/L >
0.036 and d/L > 0.273 as shown in Figures 5.34 and 5.36, respectively. The diminishing
values of Ct with increasing of H/L are seen in Figure 5.35. The overall performance
indicates that flat waves (lower values of H/L) are transmitted with ease whereas steeper
waves (greater values of H/L) are arrested effectively. It is also found that the data points
of Ct for 2-row and 3-row systems overlap each other for the given range of H/L.
153
1.0
0.8
Ct
0.6
2-row
0.4
3-row
0.2
0.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D/L
Figure 5.34 : Ct vs D/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
1.0
2-row
0.8
3-row
Ct
0.6
0.4
0.2
0.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
H/L
Figure 5.35 : Ct vs H/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
154
1.0
0.8
Ct
0.6
2-row
0.4
3-row
0.2
0.0
0.15
0.25
0.35
0.45
0.55
0.65
d/L
Figure 5.36 : Ct vs d/L - Performance comparison between 2-row and 3-row
STEPFLOAT systems
5.3.2.2 Three-row vs G = b
Figure 5.37 indicates the influence of pontoon spacing of G = b on the
transmission coefficient with respect to the wave period, as compared to the 3-row
STEPFLOAT system. The coefficient of transmission exhibits lower Ct with shorter
wave period or shorter wave length for both 3-row and G = b systems and tends to reach
a minimum of about Ct = 0.2 at the highest frequency generated in the laboratory. It is
noted that, in general, the G = b system performs slightly more effective in attenuating
wave energy than the 3-row system. From the plots of 'Ct [3-b] against T as shown in
Figure 5.37, it was observed that a decrease in Ct by 17% is obtained at T = 1.01 sec
while a maximum decrease in Ct by 18% is achieved at T = 1.26 sec. For T < 0.90 sec,
the data points fall randomly in the horizontal band around 0, indicating that the G = b
system with a pontoon spacing of b gives no significant difference in attenuating
155
1.0
3-row
Ct & 'Ct[3-b]
0.8
G=b
0.6
0.4
Ct
0.2
0.0
-0.2
'Ct [3-b]
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
T (sec)
Figure 5.37 : Ct & 'Ct[3-b] vs T - Performance comparison between 3-row and G = b
STEPFLOAT systems
relatively shorter incident waves as compared to the 3-row system. For the tested range
of wave period, lower Ct curve was found for G = b system, especially in the longer
wave period condition. This is because in relatively longer waves (lower frequency), the
spacing of b allows the two pontoons to act as a continuous structure functioning like a
single unit of 3-row system, spanning a significant part of the wave length while the
trapped middle section, i.e. the empty section between the two pontoons, enhances the
wave energy dissipation through the formation of turbulence and eddies.
The variation of Ct for the 3-row and G = b systems as a function of the B/L is
presented in Figure 5.38. For the G = b system, only when the B/L • 0.1955 is the
threshold level of Ct = 0.5 attained. In comparison to the 3-row system, the G = b system
generally provides lower Ct, especially when the B/L is smaller. Figure 5.39 shows a
decreasing trend of Ct as the D/L increases for the two cases. Similar trends are observed
for the plots of Ct against H/L and d/L as shown in Figures 5.40 and 5.41, respectively.
156
1.0
0.8
3-row
Ct
0.6
0.4
G=b
0.2
0.0
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
B/L
Figure 5.38 : Ct vs B/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
1.0
0.8
3-row
Ct
0.6
0.4
G=b
0.2
0.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D/L
Figure 5.39 : Ct vs D/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
157
1.0
0.8
3-row
Ct
0.6
0.4
G=b
0.2
0.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
H/L
Figure 5.40 : Ct vs H/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
1.0
0.8
3-row
Ct
0.6
0.4
G=b
0.2
0.0
0.15
0.25
0.35
0.45
0.55
0.65
d/L
Figure 5.41 : Ct vs d/L - Performance comparison between 3-row and G = b
STEPFLOAT systems
158
5.3.2.3 G = 0 vs G = b vs G = 2b
Figure 5.42 shows the variations of Ct as a function of the wave period for the
twin-pontoon STEPFLOAT breakwater system with the pontoon spacing, G = 0 (which
is a 2-row system), G = b and G = 2b, indicating the influence of pontoon spacing. The
G = 0 system generally provides higher Ct if compared to the G = b and G = 2b systems.
The difference of Ct between G = 0 and G = b STEPFLOAT systems, 'Ct [0-b], as shown
in Figure 5.42, indicates that a maximum decrease in Ct of 15% was achieved by the G =
b system, with most of the data points scatter above the horizontal band of 0.
For T < 0.8373 sec, it can be seen from the curves of Ct that the G = 2b system
performs slightly better with lower Ct than the G = b system. The plots of the difference
of Ct between G = 2b and G = b STEPFLOAT systems, 'Ct [2b-b], as shown in Figure
5.42, exhibits a maximum decrease in Ct of 12% by the G = 2b system for T < 0.8373
sec. However, for T > 0.8373 sec, the G = b system gives better and more effective wave
attenuation than the G = 2b system. The 'Ct [2b-b], as shown in Figure 5.42, demonstrates
that a maximum of 13 % decrease in Ct by the G = b system was observed.
It is noted that at low frequencies when T > 0.8373 sec, the system with smaller
spacing, i.e. G = b, leads to lower values of Ct, while at higher frequencies when T <
0.8373 sec, the larger pontoon spacing, i.e. G = 2b, results in lower Ct values. This is
because in relatively longer waves (lower frequency) the smaller spacing allows the two
pontoons to act as a continuous structure functioning like a single unit, spanning a
considerable portion of wave length, whereas in shorter waves (higher frequency)
pontoons with a larger spacing tend to act independently as two separated single pontoon
breakwaters in series, hence enhance the wave energy dissipation mechanism.
159
0.8
0.7
0.6
Ct
0.5
0.4
0.3
G=0
G=b
0.2
G = 2b
0.1
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
'Ct[0-b]
0.2
0.0
-0.2
'Ct[2b-b]
0.2
0.0
-0.2
T (sec)
Figure 5.42 : Ct, 'Ct[0-b] & 'Ct[2b-b] vs T - Performance comparison between G = 0,
G = b and G = 2b STEPFLOAT systems
160
Figure 5.43 shows the variations of Ct as a function of B/L for G = 0, G = b and
G = 2b systems. For the G = 2b system, the threshold level of Ct = 0.5 is only achieved
when B/L • 0.2713 while for the G = 0 and G = b systems, this level is attained when
B/L > 0.1478 and B/L > 0.1955, respectively, as reported in earlier section. Figures 5.44
through 5.46 show decreasing trend when Ct values were plotted against D/L, H/L and
d/L, respectively, for the three systems. Figure 5.47 shows the plots of Ct against G/L
with a decreasing trend of Ct as G/L increases.
1.0
G=0
0.8
G=b
G = 2b
Ct
0.6
0.4
0.2
0.0
0.05
0.15
0.25
0.35
0.45
0.55
B/L
Figure 5.43 : Ct vs B/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
161
1.0
G=0
0.8
G=b
G = 2b
Ct
0.6
0.4
0.2
0.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
D/L
Figure 5.44 : Ct vs D/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
1.0
G=0
0.8
G=b
G = 2b
Ct
0.6
0.4
0.2
0.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
H/L
Figure 5.45 : Ct vs H/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
162
1.0
G=0
0.8
G=b
G = 2b
Ct
0.6
0.4
0.2
0.0
0.15
0.25
0.35
0.45
0.55
0.65
d/L
Figure 5.46 : Ct vs d/L - Performance comparison between G = 0, G = b and G = 2b
STEPFLOAT systems
1.0
0.8
Ct
0.6
G = 2b
0.4
G=b
0.2
0.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
G/L
Figure 5.47 : Ct vs G/L - Performance comparison between G = b and G = 2b
STEPFLOAT systems
163
5.4 Comparison on the Performance between the STEPFLOAT and Previous
Floating Breakwater Studies
Figure 5.48 shows a comparison between the results of the present experimental
results of Ct of 2-row, 3-row, G = b and G = 2b STEPFLOAT systems with the work of
other authors for different types of floating breakwaters. The Ct curves of various
existing types of floating breakwaters are extracted and superimposed into Figure 5.48,
with B/L ranging from 0 to 1. All Ct trendlines show reasonably good agreement in terms
of the trend of the curve. The figure shows that the suggested STEPFLOAT breakwater
system is more efficient compared to the other results. However, an exact comparison
cannot be made due to the different experimental criteria used in the laboratory by
different investigators.
Without taking account of the floating breakwater’s draft, tested wave steepness
and water depth, the STEPFLOAT system generally has excellent wave dampening
ability over most of the previous floating breakwaters. The graphs in Figure 5.48 reveal
the significant influence of certain geometric characteristic of floating structures on
wave attenuation. The Ct decreases proportionally with the increase of the B/L ratio
revealing the significant influence of the width of structures on wave attenuation.
For instance, in order to reduce a one meter long incident wave height by 50%,
the STEPFLOAT breakwater system would need to be constructed at least 0.15 m, 0.21
m, 0.20 m and 0.27 m wide for the 2-row, 3-row, G = b and G = 2b systems,
respectively. The single row of SSFBW system for D/d = 0.242 would need 0.23 m wide
to provide such degree of protection while A-frame floating breakwater requires
breakwater width of 0.62 m. The required width for the rectangular cylinder to bring
down the one meter incident wave height by half is 0.43 m, which is higher than that of
the box-type floating breakwater with B = 0.30 m. However, the Y-frame with pipes is
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
[H]
[M]
0.1
[N]
[G]
0.2
[I]
0.3
[P]
[E]
0.4
[O]
[J]
[C]
[B]
B/L
0.5
[Q]
[L]
[K]
0.6
0.7
[F]
[D]
0.8
0.9
[A]
1.0
[N]
[O]
[P]
[Q]
STEPFLOAT:
Present Study - 2-Row System
Present Study - 3-Row System
Present Study - G = b System
Present Study - G = 2b System
Log Raft [Riche & Nece, 1974]
Goodyear [Harms, 1979]
Wave-Maze [Stitt & Noble, 1963]
Twin Plate [Kumar et al. , 2001]
Wave-Guard [Harms, 1979]
A-Frame [Brebner & Ofuya, 1968]
Y-Frame - No Pipes [Mani, 1991]
Y-Frame - With Pipes [Mani, 1991]
Box [Nece et al. , 1988]
Rectangular Cylinder
[Yamamoto et al. , 1982]
[K] Tethered Float [Harms, 1980]
[L] 2-Tier Floating Pipe
[Purushotham et al. , 2001]
[M] 1-Row SSFBW with D/d = 0.24
[Teh, 2002]
[A]
[B]
[C]
[D]
[E]
[F]
[G]
[H]
[I]
[J]
LEGEND:
Figure 5.48 : Comparison of floating breakwaters efficiency between the STEPFLOAT and those from previous studies
Ct
164
165
capable to offer a minimum of 50% reduction of wave height with the required
breakwater width of only 0.15 m, which is comparable to the one provided by the 2-row
STEPFLOAT system. Nonetheless, it has to be noted that the draft of the Y-frame
extended by the attachment of a row of pipes at the bottom of the trapezoidal float has
enhanced the wave attenuation capability of the floating breakwater system, thus smaller
breakwater width is required. For the rest of the floating breakwaters, their width would
need to be built larger than 0.62 m to attain the 50% wave height reduction.
Teh (2002) reported that, for the range of 0 < B/L < 1.0, some of the floating
breakwaters, mainly shallow-draft structures such as the log raft, the tethered float, the
Goodyear floating tire breakwater, the Wave-Maze, the Wave-Guard, the floating plates
and the floating pipes, are found less effective in attenuating the wave with even greater
value of B/L. As a result, the sea surface is occupied by a large surface area of these
structures in order to achieve a lower Ct. This is unfavourable as the spread may cause
difficulties in installation and handling as well as taking up too much space.
CHAPTER 6
PARAMETRIC ANALYSIS AND EMPIRICAL RELATIONSHIPS
6.1
Introduction
In this Chapter, Ct, Cr and Cl are calculated and presented for the laboratory
experiments on the vertical pile-system STEPFLOAT floating breakwater with different
system arrangements with respect to the specific dimensionless variables.
6.2
Parametric Analysis and Empirical Relationships
Parametric analysis was used to study the influence of independent variables on
the transmission coefficient. This preliminary examination describes qualitatively the
general trends of the Ct, Cr and Cl with respect to the individual independent variables
and investigates quantitatively the empirical relationship of the Ct associated with the
167
trend as a function of a specific independent variable, for all independent variables that
are under consideration. Several empirical relationships are investigated in this section.
Included are the four general dimensionless structural geometrical variables (relative
width,
B
D
and relative draft, ) and the dimensionless hydraulic variables (wave
L
L
steepness,
H
d
and relative depth, ). An additional dimensionless structural
L
L
geometrical variable, namely relative gap size,
G
is included for the case of the G = b
L
and G = 2b systems.
The second order polynomial and exponential trend lines for the plots of Ct from
the least squares regression analysis for the entire data set are shown. The equations for
these least squares fit lines are given by
Ct
a 0 a1 x a 2 x 2
(6.1)
Ct
ce bx
(6.2)
where a0, a1 and a2 are constants for the second order polynomial trend line [Equation
(6.1)] while c and b are constants for the exponential fit line [Equation (6.2)], and e is
the base of the natural logarithm. x is a dummy variable representing the independent
nondimensional variable. However, for the plots of Ct in this chapter, only trend lines
with higher R2 are shown in the figures.
168
6.2.1 Influence of Relative Width,
B
L
6.2.1.1 Two-row System
The literature has indicated that the relative width,
B
has great influence on
L
wave transmission of floating breakwaters and has been used as one of the most
representative parameters in result presentation by most of the researchers in their
studies on floating breakwaters. Figure 6.1 shows the relationship between B/L and the
measured Ct, Cr and Cl for the two-row system with D/d = 0.133. Based on the fit line of
Ct, it is found that the B/L is a strong governing parameter to Ct. The results indicate that
Ct increases with decrease in B/L, whereas Cl increases with the increase in B/L. No
appreciable variation can be found in Cr with increasing B/L. The values of Cr stay
within the range from 0.10 to 0.43, with most of the data points scatter closely around
0.2.
The scatter in this data is considered small and the least squares regression
analysis for Ct indicates a correspondingly high R2 = 0.8621 (for the exponential fit line).
The regression coefficients for the second order polynomial and the constants by the
exponential trend lines of Ct are also listed in Tables 6.1 and 6.2, respectively. Thus, this
parameter does a good job predicting the STEPFLOAT performance.
169
1.0
0.8
Ct , Cr & Cl
Cl
0.6
Ct = 0.9436e
-4.2975(B/L)
2
Ct
R = 0.8621
0.4
0.2
0.0
0.05
Cr
0.10
0.15
0.20
0.25
0.30
B/L
Figure 6.1 : Measured Ct, Cr & Cl versus B/L of 2-row system with D/d = 0.133
Table 6.1 : Summary of regression analysis parameters for the 2-row vertical
pile-system STEPFLOAT breakwater (second order polynomial)
Independent Variable x
Name
Symbol
B/L
Relative width
D/L
Relative draft
H/L
Wave steepness
d/L
Relative depth
Dependent variable = Ct
2
R
0.8447
0.8447
0.5494
0.8447
Regression Coefficients
a1
a2
a0
0.8706
-2.8526
2.6432
0.8706
-9.9840
32.3795
0.6959
-7.3947
39.5794
0.8706
-1.3312
0.5756
Table 6.2 : Summary of regression analysis parameters for the 2-row vertical
pile-system STEPFLOAT breakwater (exponential)
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
Dependent variable = Ct
Constants
2
R
0.8621
0.8621
0.4983
0.8621
c
0.9436
0.9436
0.6512
0.9436
b
-4.2975
-15.0413
-8.2953
-2.0055
170
6.2.1.2 Three-row System
From the results of the three-row system in Figure 6.2, it is noticed that the
transmitted energy decreases with the increase of B/L while the energy lost has the
inverse trend. The trend between Ct and B/L suggests that relative width did have an
effect on Ct. Most of the data points of Cr are exhibited within the range from 0.16 to
0.39. The Cr curve seems to increase slowly with the increase in B/L. The regression
analysis of Ct gave a higher R2 = 0.9124 (for the exponential trend line). Again, the
regression analysis coefficients for the second order polynomial and the constants by the
exponential fit line of Ct are listed in Tables 6.3 and 6.4, respectively.
6.2.1.3 G = b System
A trend between Ct and B/L was observed with the G = b system, as can be seen
in Figure 6.3. For the given range of B/L, the trend line of Ct decreases as the B/L
increases, whereas the Cl fit line increases with the increase in B/L. Similar to the
previous two systems, there is no appreciable variation can be found in Cr with
increasing B/L for G = b system except that the curve starts to increase at the larger end
of B/L. In general, the Cr data points stay around 0.2. The second order polynomial
regression coefficients for the Ct curve are listed in Table 6.5 while the constants of the
exponential fit line of Ct are shown in Table 6.6. The polynomial and exponential fit
lines of Ct gave the R2 = 0.9431 and 0.9238, respectively.
171
1.0
Ct , Cr & Cl
0.8
0.6
Cl
Ct = 1.1501e
Ct
-4.0282(B/L)
2
R = 0.9124
0.4
Cr
0.2
0.0
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
B/L
Figure 6.2 : Measured Ct, Cr & Cl versus B/L of 3-row system with D/d = 0.133
Table 6.3 : Summary of regression analysis parameters for the 3-row vertical
pile-system STEPFLOAT breakwater (second order polynomial)
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
2
R
0.8961
0.8961
0.7436
0.8961
Regression Coefficients
a0
a1
a2
1.0267
-3.0677
2.6852
1.0267
-15.5942
69.3861
0.8144
-10.7992
54.2211
1.0267
-2.0792
1.2335
Dependent variable = Ct
Table 6.4 : Summary of regression analysis parameters for the 3-row vertical
pile-system STEPFLOAT breakwater (exponential)
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
Dependent variable = Ct
Constants
R2
0.9124
0.9124
0.6526
0.9124
c
1.1501
1.1501
0.7864
1.1501
b
-4.0282
-20.4766
-12.9918
-2.7302
172
1.0
Ct, Cr & Cl
0.8
Cl
2
0.6
Ct = 1.1864(B/L) - 2.0914(B/L) + 0.8691
2
Ct
R = 0.9431
0.4
Cr
0.2
0.0
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
B/L
Figure 6.3 : Measured Ct, Cr & Cl versus B/L of G = b system with D/d = 0.133
Table 6.5 : Summary of regression analysis parameters for the G = b vertical
pile-system STEPFLOAT breakwater (second order polynomial)
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
Relative gap size
G/L
Dependent variable = Ct
2
R
0.9431
0.9431
0.4877
0.9431
0.9431
Regression Coefficients
a0
a1
a2
0.8691
-2.0914
1.1864
0.8691
-10.6313
30.6577
0.7041
-9.3395
58.9867
0.8691
-1.4175
0.5450
0.8691
-6.3788
11.0368
Table 6.6 : Summary of regression analysis parameters for the G = b vertical
pile-system STEPFLOAT breakwater (exponential)
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
Relative gap size
G/L
Dependent variable = Ct
Constants
R2
0.9238
0.9238
0.3870
0.9238
0.9238
c
1.0468
1.0468
0.6339
1.0468
1.0468
b
-3.7801
-19.2153
-9.7316
-2.5620
-11.5292
173
6.2.1.4 G = 2b System
Figure 6.4 shows the results of Ct, Cr and Cl versus B/L for G = 2b system.
Again, as previous discussed system, a similar trend of Ct versus B/L was observed with
the G = 2b system, suggesting that B/L did have an effect on Ct. Therefore, a statistical
analysis will be performed in the following chapter to see if the influence of B/L is
statistically significant. As expected, Cl curve increases as the B/L increases. A trend
between Cr and B/L was observed with the G = 2b system (a concave shape of results
with the nadir between 0.30 and 0.35). The regression analysis of Ct gave a high R2 =
0.9422 (for second order polynomial curve) and its regression coefficients are listed in
Table 6.7. The constants by the exponential fit line of Ct are listed in Table 6.8.
1.0
Ct, Cr & Cl
0.8
0.6
Cl
2
Ct = 0.0085(B/L) - 1.3235(B/L) + 0.8755
Ct
2
R = 0.9422
0.4
0.2
Cr
0.0
0.10
0.20
0.30
0.40
0.50
0.60
B/L
Figure 6.4 : Measured Ct, Cr & Cl versus B/L of G = 2b system with D/d = 0.133
174
Table 6.7 : Summary of regression analysis parameters for the G = 2b vertical
pile-system STEPFLOAT breakwater (second order polynomial)
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
Relative gap size
G/L
Regression Coefficients
a0
a1
a2
0.8755
-1.3235
0.0085
0.8755
-8.9335
0.3868
0.7192
-7.6584
27.2829
0.8755
-1.1911
0.0069
0.8755
-2.6800
0.0348
2
R
0.9422
0.9422
0.5955
0.9422
0.9422
Dependent variable = Ct
Table 6.8 : Summary of regression analysis parameters for the G = 2b vertical
pile-system STEPFLOAT breakwater (exponential)
Constants
Independent Variable x
Name
Symbol
Relative width
B/L
Relative draft
D/L
Wave steepness
H/L
Relative depth
d/L
Relative gap size
G/L
Dependent variable = Ct
6.2.2 Influence of Relative Draft,
2
R
0.9169
0.9169
0.5165
0.9169
0.9169
c
1.2962
1.2962
0.7343
1.2962
1.2962
b
-3.5118
-23.7048
-12.8463
-3.1606
-7.1114
D
L
6.2.2.1 Two-row System
The effect of draft for the 2-row STEPFLOAT breakwater is demonstrated in
terms of draft to wave length ratio, D/L in Figure 6.5. A decreasing trend of Ct curve is
175
observed as D/L increases, whereas Cl increases with an increase in D/L. The variation
of Cr with D/L is not appreciable. Most of the data points are scattered at 0.1 < Cr < 0.3.
From the data points of Ct, it is found that the variation of draft has a profound effect on
wave attenuation. The regression analysis of Ct indicates a correspondingly high R2 =
0.8621. The second order polynomial regression coefficients and the constants of
exponential relationship are listed in Tables 6.1 and 6.2, respectively.
1.0
Ct, Cr & Cl
0.8
0.6
Cl
Ct
-15.041(D/L)
Ct = 0.9436e
2
R = 0.8621
0.4
Cr
0.2
0.0
0.01
0.03
0.05
0.07
0.09
D/L
Figure 6.5 : Measured Ct, Cr & Cl versus D/L of 2-row system with D/d = 0.133
6.2.2.2 Three-row System
Figure 6.6 relates D/L with respective Ct, Cr and Cl for the three-row system. It
proved to be a good indicator, as there was a very defined trend showing that as D/L
increased, Ct decreased. The trend line of Cl increases slowly as the D/L increases. Most
of the data points of Cr, however, are scattered fairly uniform in the range of 0.16 - 0.39,
176
giving a very mild curve or nearly linear horizontal line. The R2 = 0.8961 and 0.9124 for
the second order polynomial and exponential curves of Ct, respectively and related
regression analysis coefficients or constants are again shown in Tables 6.3 and 6.4.
1.0
0.8
Ct, Cr & Cl
Cl
0.6
-20.4766(D/L)
Ct
Ct = 1.1501e
2
R = 0.9124
0.4
0.2
0.0
0.01
Cr
0.03
0.05
0.07
0.09
D/L
Figure 6.6 : Measured Ct, Cr & Cl versus D/L of 3-row system with D/d = 0.133
6.2.2.3 G = b System
The results of Ct, Cr and Cl versus D/L for the G = b system are shown in Figure
6.7. Similar to the previous discussed systems, it is found that there is a huge variation of
Ct with increasing D/L. The curve of Cl increases slowly as the D/L increases and the
data points are generally above Cl = 0.7. The data points of Cr are scattered at 0.1 < Cr <
0.4. High values of Cl and low values of Cr indicate that most of the incident wave
energy is dissipated instead of reflected. The regression analysis gave R2 = 0.9431 and
0.9238 for the second order polynomial and exponential curves, respectively. The
177
corresponding regression analysis coefficients or constants are listed in Tables 6.5 and
6.6.
1.0
Ct, Cr & Cl
0.8
0.6
Cl
2
Ct = 30.658(D/L) - 10.631(D/L) + 0.8691
2
Ct
R = 0.9431
0.4
0.2
Cr
0.0
0.01
0.03
0.05
0.07
0.09
D/L
Figure 6.7 : Measured Ct, Cr & Cl versus D/L of G = b system with D/d = 0.133
6.2.2.4 G = 2b System
A trend between Ct and D/L, as shown in Figure 6.8, was observed for the G = 2b
system, suggesting that D/L had a strong influence on Ct. As expected, the curve of Cl
lies at the upper level region of the Figure 6.8 with most of the data points above Cl =
0.7. Again, the data points of Cr are generally scattered in the lower region at 0.1 < Cr <
0.4 with a slight concave shape of results with the nadir around D/L = 0.5. The second
order regression analysis of Ct gave an R2 = 0.9422 and its corresponding coefficients
are listed in Table 6.7. The constants of exponential curve for the same data points of Ct
are listed in Table 6.8.
178
1.0
Ct, Cr & Cl
0.8
0.6
Cl
2
Ct = 0.3868(D/L) - 8.9335(D/L) + 0.8755
2
Ct
R = 0.9422
0.4
Cr
0.2
0.0
0.01
0.03
0.05
0.07
0.09
D/L
Figure 6.8 : Measured Ct, Cr & Cl versus D/L of G = 2b system with D/d = 0.133
6.2.3 Influence of Wave Steepness,
H
L
6.2.3.1 Two-row System
Figure 6.9 shows the Ct, Cr and Cl versus H/L for the two-row system. There is a
trend showing that as wave steepness increased, Ct decreased. When steeper waves pass
over or beneath the floating breakwater, they create larger eddies at the edges of the
structure due to a larger elevation head difference than flatter waves, resulting in greater
energy dissipation and thus lower transmission. Again, the Cl fit line occupies the upper
level of the chart with mostly all the data points above Cl = 0.7. As for Cr, the data points
are scattered around Cr = 0.2. The R2 = 0.5494 for the second order polynomial curve
and its corresponding regression analysis coefficients are again listed in Table 6.1. The
constants of exponential equation are listed in Table 6.2. In summary, although the R2
179
values for both regression types are slightly lower, this parameter of wave steepness has
some validity in predicting the STEPFLOAT performance. The scatter in the data and
low correlation may be due to the limited data set of wave heights and wave lengths.
1.0
Cl
0.8
Ct, Cr & Cl
2
Ct = 39.579(H/L) - 7.3947(H/L) + 0.6959
0.6
2
R = 0.5494
0.4
Ct
0.2
Cr
0.0
0.00
0.02
0.04
0.06
0.08
0.10
H/L
Figure 6.9 : Measured Ct, Cr & Cl versus H/L of 2-row system with D/d = 0.133
6.2.3.2 Three-row System
The data points of Ct, Cr and Cl are respectively plotted against H/L in Figure
6.10. The Ct curve exhibits a trend with its data points scatter at 0.20 < Ct < 0.76, for the
given range of H/L. Cl and Cr data points are respectively scattered at 0.62 < Cl < 0.96
and 0.16 < Cr < 0.39. It was observed that the data points of Cr scatter uniformly over
the H/L. The second order polynomial regression analysis gave a higher R2 = 0.7436,
compared to the 2-row system. The coefficients or constants of both the second order
polynomial and exponential equations are listed in Tables 6.3 and 6.4.
180
1.0
Cl
Ct, Cr & Cl
0.8
2
Ct = 54.221(H/L) - 10.799(H/L) + 0.8144
0.6
2
R = 0.7436
0.4
Ct
0.2
Cr
0.0
0.00
0.02
0.04
0.06
0.08
0.10
H/L
Figure 6.10 : Measured Ct, Cr & Cl versus H/L of 3-row system with D/d = 0.133
6.2.3.3 G = b System
A similar trend to that described earlier was found for the G = b system. Figure
6.11 demonstrates that the system altered mildly the magnitude of Ct but did not change
the trend of decreasing Ct with increasing wave steepness. Again, as seen in Figure 6.11,
most of the incident wave energy is attenuated through the mechanism of wave
dissipation with high values of Cl while Cr remains its uniform trend with most of the
data points scatter around Cr = 0.2. The R2 = 0.4877 and 0.3870 for the second order
polynomial and exponential fit lines, respectively. Tables 6.5 and 6.6 show their
respective regression analysis constants.
181
1.0
Cl
Ct, Cr & Cl
0.8
2
Ct = 58.987(H/L) - 9.3395(H/L) + 0.7041
0.6
2
R = 0.4877
0.4
Ct
0.2
Cr
0.0
0.00
0.02
0.04
0.06
0.08
0.10
H/L
Figure 6.11 : Measured Ct, Cr & Cl versus H/L of G = b system with D/d = 0.133
6.2.3.4 G = 2b System
Similar to the other system arrangements, the G = 2b system has a decreased Ct
trend line with increased wave steepness, showing that Ct values are dependent upon
H/L, as can be seen in Figure 6.12. Again, the fit lines of Cl and Cr demonstrate similar
trend as previously discussed systems. The second order polynomial regression analysis
gave an R2 = 0.5955. Related regression coefficients are listed in Tables 6.7 and 6.8 for
the polynomial and exponential equations.
182
1.0
Ct, Cr & Cl
0.8
Cl
2
Ct = 27.283(H/L) - 7.6584(H/L) + 0.7192
0.6
0.4
2
R = 0.5955
Ct
0.2
Cr
0.0
0.00
0.02
0.04
0.06
0.08
0.10
H/L
Figure 6.12 : Measured Ct, Cr & Cl versus H/L of G = 2b system with D/d = 0.133
6.2.4 Influence of Relative Depth,
d
L
6.2.4.1 Two-row System
Figure 6.13 shows the relationship between d/L and measured Ct, Cr and Cl. The
data is plotted in the similar pattern as in the previous graphs. From the curve of Ct, d/L
proved to be a good indicator, as there was a defined trend showing that as d/L
increased, Ct decreased. It shows that STEPFLOAT in relatively deeper water gives
better performance with lower values of Ct. Again, it was observed that the Cl fit line and
Cr curve occupy the upper and lower parts of the graph above Cl = 0.7 and below Cr =
0.3, respectively. The regression analysis of Ct indicates a correspondingly high R2 =
0.8621 (for the exponential fit line). The regression coefficients are respectively listed in
Tables 6.1 and 6.2 for the second order polynomial and exponential regression.
183
1.0
Cl
Ct, Cr & Cl
0.8
Ct = 0.9436e-2.0055(d/L)
0.6
R2 = 0.8621
0.4
Ct
0.2
0.0
0.10
Cr
0.20
0.30
0.40
0.50
0.60
0.70
d/L
Figure 6.13 : Measured Ct, Cr & Cl versus d/L of 2-row system with D/d = 0.133
6.2.4.2 Three-row System
The results of measured Ct, Cr and Cl versus d/L for the three-row system are
shown in Figure 6.14. A trend between Ct and d/L was observed, suggesting that d/L has
an effect on Ct. Cl increases as d/L increases. The data points of Cr are scattered between
0.1 and 0.4. The R2 = 0.8961 and 0.9124 for the second order polynomial and
exponential fit lines, respectively and their corresponding regression coefficients are
listed in Tables 6.3 and 6.4.
184
1.0
Ct, Cr & Cl
0.8
0.6
Cl
Ct
-2.7302(d/L)
Ct = 1.1501e
2
R = 0.9124
0.4
Cr
0.2
0.0
0.10
0.20
0.30
0.40
0.50
0.60
0.70
d/L
Figure 6.14 : Measured Ct, Cr & Cl versus d/L of 3-row system with D/d = 0.133
6.2.4.3 G = b System
From the results of the G = b system in Figure 6.15, it is noticed that the
transmitted energy decreases with the increase of d/L while the energy lost has the
inverse trend. The trend between Ct and d/L suggests that relative water depth did have
an effect on Ct. Most of the data points of Cr are exhibited within the range from 0.09 to
0.37. The Cr curve seems to increase slowly with the increase in d/L. The regression
analysis of Ct gave an R2 = 0.9431 (for the second order polynomial trend line). Again,
the regression analysis coefficients for the second order polynomial and the constants by
the exponential fit line of Ct are listed in Tables 6.5 and 6.6, respectively.
185
1.0
Ct, Cr & Cl
0.8
0.6
Cl
Ct
2
Ct = 0.545(d/L) - 1.4175(d/L) + 0.8691
2
R = 0.9431
0.4
Cr
0.2
0.0
0.10
0.20
0.30
0.40
0.50
0.60
0.70
d/L
Figure 6.15 : Measured Ct, Cr & Cl versus d/L of G = b system with D/d = 0.133
6.2.4.4 G = 2b System
A similar trend to that described earlier was found for the G = 2b system. Figure
6.16 demonstrates that the system altered the magnitude of Ct but did not change the
trend of decreasing Ct with increasing wave steepness. Again, as seen in Figure 6.16,
most of the incident wave energy is attenuated through the mechanism of wave
dissipation with high values of Cl while Cr demonstrates a slight concave shape of
results with the nadir around d/L = 0.35. The data points of Cr are generally scattered
between 0.1 and 0.4. The R2 = 0.9422 and 0.9169 for the second order polynomial and
exponential fit lines, respectively. Tables 6.7 and 6.8 show their respective regression
analysis constants.
186
1.0
Ct, Cr & Cl
0.8
0.6
Cl
Ct
2
Ct = 0.0069(d/L) - 1.1911(d/L) + 0.8755
2
R = 0.9422
0.4
0.2
0.0
0.10
Cr
0.20
0.30
0.40
0.50
0.60
0.70
d/L
Figure 6.16 : Measured Ct, Cr & Cl versus d/L of G = 2b system with D/d = 0.133
6.2.5 Influence of Relative Gap Size,
G
L
6.2.5.1 G = b System
A defined trend of Ct curve was visible signifying that the relative gap size or
spacing between the rows of modules was an important factor affecting Ct, as can be
noted in Figure 6.17. Cl and Cr curves stay above Cl = 0.7 and below Cr = 0.4,
respectively, indicating that the system performs effectively as a better wave energy
dissipator rather than as a reflector. The regression analysis of Ct gave an R2 = 0.9431
(for the second order polynomial fit line). The regression coefficients or constants for the
polynomial and exponential equations are listed in Tables 6.5 and 6.6, respectively.
187
1.0
Ct, Cr & Cl
0.8
0.6
Cl
Ct
2
Ct = 11.037(G/L) - 6.3788(G/L) + 0.8691
2
R = 0.9431
0.4
0.2
Cr
0.0
0.03
0.05
0.07
0.09
0.11
0.13
0.15
G/L
Figure 6.17 : Measured Ct, Cr & Cl versus G/L of G = b system with D/d = 0.133
6.2.5.2 G = 2b System
The plot given in Figure 6.18 shows the measured Ct, Cr and Cl versus G/L for
the G = 2b system. The curve of Ct highlights the decrease in Ct with an increase in G/L.
Strong trend regarding the influence of the G/L on Ct is apparent. A similar trend for Cl
and Cr to that described in the G = b system was found for the G = 2b system. The
regression analysis of Ct indicates a correspondingly R2 = 0.9422 (for the second order
polynomial curve). The second order polynomial regression coefficients and the
constants of exponential relationship are listed in Tables 6.7 and 6.8, respectively
188
1.0
Ct, Cr & Cl
0.8
0.6
Cl
Ct
2
Ct = 0.0348(G/L) - 2.68(G/L) + 0.8755
2
R = 0.9422
0.4
Cr
0.2
0.0
0.05
0.10
0.15
0.20
0.25
0.30
G/L
Figure 6.18 : Measured Ct, Cr & Cl versus G/L of G = 2b system with D/d = 0.133
CHAPTER 7
MULTIPLE LINEAR REGRESSION ANALYSIS AND DIAGNOSTICS
7.1
Introduction
This chapter discusses the statistical analysis performed on the experimental test
results. The objective of the statistical analysis is to develop a series of probabilistic
models or empirical equations that predict the STEPFLOAT breakwater efficiency for
the 2-row, 3-row, G = b and G = 2b systems, subject to a variety of input conditions,
which include the structural geometrical characteristics (breakwater width, draft and
spacing between pontoons) and the hydraulic characteristics (wave height, wave length
and water depth). The statistical analysis, specifically the Multiple Linear Regression
Analysis, was performed using the SPSS Version 12.0 for Windows (SPSS Inc., 2004).
Statistical software is essential for analyzing data. It channels a user’s energy into
thinking about a problem instead of being preoccupied with computational details. SPSS
Version 12.0 for Windows is a desktop statistical software specifically catered towards
scientific research.
190
7.2
Multiple Linear Regression Analysis
Multiple linear regression analysis is a powerful and frequently used tool for
modeling relationships among variables (Norušis, 2000). In multiple regression, the
empirical relationships that relate the dependent variable, i.e. Ct to two or more predictor
terms as depicted in Equations (7.1) and (7.2), imply that each of the predictor terms
have a linear relationship with the dependent variable. In order to predict the efficiency
of the STEPFLOAT system in terms of Ct, a myriad of possible predictors were
considered and determined in dimensional analysis which in turn resulted in the
formation of non-dimensional relationships. Several dimensionless variables were
identified as being significant following completion of the dimensional analysis and
parametric analysis.
Ct
B0 B1
B
D
H
d
B 2 B3
B4
L
L
L
L
for 2-row & 3-row systems
(7.1)
B0 B1
G
B
D
H
d
B2 B3
B4 B5
L
L
L
L
L
for G = b & G = 2b systems
(7.2)
or
Ct
The first step in developing an empirical model to predict the Ct is to select the
variables to include in the equation. As Ct is a dimensionless dependent variable, it is
desirable for the components within the equation to be dimensionless. In order to nondimensionalize all the structural and hydraulic variables, the dimensional variables were
divided by wave length to form dimensionless variables for the analyses of the efficiency
of the STEPFLOAT system. The parametric analysis showed that the dimensionless
variables, i.e. relative width, relative draft, wave steepness, relative water depth and
relative gap were important factors affecting the performance of the STEPFLOAT
system.
191
The form of the equation is the next consideration. The simplest equation is
linear which has the basic form as shown in Equations (7.1) and (7.2). When hydraulic
variables such as wave height and wave length are present in the dimensionless
independent variables, the relationships are not normally linear. For this reason of nonlinearity, higher order equations were considered and several criteria have to be satisfied.
The data were transformed by utilizing intrinsically linear functions. The most widely
used intrinsically linear functions are the natural logarithmic and power models. The
transformation for natural logarithmic analysis involves taking the natural logarithm of
the independent variables, then proceeding with the regression while for the power
model, it involves taking the logs of both the dependent and independent variables
before the regression analysis.
Basic linear, natural logarithmic and power relationships were investigated and
their suitability was judged by the following criteria:
(i)
The relationship between the dependent and the independent variables is
linear.
(ii)
The magnitude of the squared multiple correlation coefficient, R2, which is a
general measure of how well the equation fit the data.
(iii)
The ability of the equation to incorporate all the desired significant variables
while minimizing the number of statistically fitted parameters.
(iv)
Residuals should be normally distributed showing that there is no bias in the
equation.
Based on the desired criteria as well as the results of the multiple regression analyses on
the three different forms of equations, the basic linear relationship would be the most
preferable as it gives better model. Therefore, detailed discussion on the multiple
regression analysis will be limited to the derivation of basic linear empirical
relationships.
192
According to Norušis (2000), the enter and stepwise methods are sufficient for
most purposes. When conducting the regression, the author utilized the SPSS Version
12.0 for Windows statistical software and conducted both the enter and stepwise
analyses using the SPSS Linear Regression procedure. In the enter method, all
independent variables are entered into the equation as a group while the stepwise method
involves the selection of independent variables proceeds by steps and are evaluated
according to the selection criteria for removal (probability of F to remove • 0.10) and
entry (probability of F to enter ” 0.05).
The multiple linear regression equation, as seen in Equation (7.1), contains a
constant and four partial regression coefficients (B1 through B4) - one for each of the four
B
B
independent variables, for 2-row and 3-row STEPFLOAT systems. The regression
equation, for G = b and G = 2b systems, contains a constant and five partial regression
coefficients (B1 through B5) as shown in Equation (7.2). The least squares method is
B
B
used to estimate the values of the coefficients in which the coefficients that result in the
smallest sum of squared differences between the observed and predicted values of the Ct
are selected.
7.2.1
Examination of the Variables
Before estimating the partial regression coefficients, the independent variables
(B/L, D/L, H/L, d/L and G/L) have to be assured that they are linearly related to the
dependent variable, Ct. Figures 7.1 through 7.4 show the matrix scatterplots of the Ct and
the independent variables for 2-row, 3-row, G = b and G = 2b, respectively. The top row
of each matrix shows the relationships between Ct and the independent variables. The
relationship between Ct and all the independent variables appears to be more or less
d/L
H/L
D/L
B/L
Ct
193
Ct
B/L
D/L
H/L
d/L
d/L
H/L
D/L
B/L
Ct
Figure 7.1 : Scatterplot matrix of the Ct and the 4 independent variables for 2-row
Ct
B/L
D/L
H/L
d/L
Figure 7.2 : Scatterplot matrix of the Ct and the 4 independent variables for 3-row
G/L
d/L
H/L
D/L
B/L
Ct
194
Ct
B/L
D/L
H/L
d/L
G/L
G/L
d/L
H/L
D/L
B/L
Ct
Figure 7.3 : Scatterplot matrix of the Ct and the 5 independent variables for G = b
Ct
B/L
D/L
H/L
d/L
G/L
Figure 7.4 : Scatterplot matrix of the Ct and the 5 independent variables for G = 2b
195
linear. Since all of the independent variables have a linear relationship with the Ct, it
makes sense to compute the multiple linear regression equations using the independent
variables without any transformations.
However, it was observed from the matrix scatterplots that there are strong
dependencies among the independent variables except the wave steepness, indicating
that one variable is almost a linear combination of the other independent variables with
the tolerance value close to zero as shown in the collinearity statistics. If any of the
tolerances (the strength of the linear relationships among the independent variables) are
small, multicollinearity may cause problems such as the coefficients in the wrong sign,
high values of standard error of the regression coefficients, etc. This may have serious
effect on the estimates of the regression coefficients and the general applicability of the
estimated model.
If the independent variables are identified to be very highly related, a regression
model that contains all of them may not be able to be estimated. One of the possible
remedial measures for solving the problem of multicollinearity is to delete certain
independent variables from the model, but this approach has the disadvantages of
discarding the important information contained in the deleted independent variables.
Considering that parametric analysis has demonstrated the significant influence of all the
independent variables on the performance of the STEPFLOAT system, it is decided that
instead of removing some of them from the model, two new composite parameters
BD/dL (for 2-row and 3-row systems) and BDG/dL2 (for G = b and G = 2b systems),
which incorporate the structural geometrical characteristics (B, D and G) as numerator
and hydraulic characteristics (d and L) as denominator, were examined. The new
dimensionless parameter BD/dL (hereafter referred to as BD number) is a combination of
three nondimensional parameters and is defined by Equation (7.3) while the composite
parameter BDG/dL2 (hereafter referred to as BDG number) is a combination of four
nondimensional parameters and is defined by Equation (7.4). Thus these two new
196
parameters incorporate many of the original factors (except H/L) affecting the
STEPFLOAT’s performance. H/L will be used as one of the regressor variables for the
multiple linear regression.
BD
dL
BDG
dL2
BDL
L L d
BDGL
L L Ld
(7.3)
(7.4)
Figures 7.5 through 7.8 show the results of the measured Ct versus BD number or
BDG number. It was observed that all curves of Ct give a decreasing trend as the BD
number or BDG number increases, thus both BD number and BDG number have
significant influence on Ct. Therefore, the new independent variable, i.e. BD number or
BDG number, and the wave steepness, H/L, are plotted together with Ct in the form of
matrix scatterplots for evaluation if the independent variables appear to be linearly
related to Ct. Figures 7.9 through 7.12 demonstrate the scatterplot matrixes of the Ct and
the independent variables of BD number or BDG number and H/L for 2-row, 3-row, G =
b and G = 2b systems, respectively.
It was observed from the top row of each scatterplot matrix that the BD number
or BDG number and H/L are more or less linearly related to the dependent variable Ct. It
was also noted from the scatterplot matrixes that the strength of the linear relationships
among the independent variables is not as strong as those discussed previously.
Presuming that the independent variables are not very highly related or multicollinearity
is absent, multiple linear regression analysis will be performed using the new composite
parameter of BD number or BDG number as well as H/L to estimate four different
regression models for 2-row, 3-row, G = b and G = 2b STEPFLOAT systems.
Collinearity statistics in the linear regression statistics will verify or confirm the
presence of multicollinearity.
197
1.0
-32.2314 (BD/dL)
Ct = 0.9436e
0.8
2
R = 0.8621
Ct
0.6
0.4
0.2
0.0
0.01
0.02
0.03
0.04
BD/dL
Figure 7.5 : Measured Ct versus BD/dL for 2-row STEPFLOAT system
1.0
-30.2114 (BD/dL)
Ct = 1.1501e
0.8
2
R = 0.9124
Ct
0.6
0.4
0.2
0.0
0.01
0.02
0.03
0.04
0.05
0.06
BD/dL
Figure 7.6 : Measured Ct versus BD/dL for 3-row STEPFLOAT system
198
1.0
-30.2114 (BDG/dL^2)
Ct = 1.1501e
0.8
2
R = 0.9124
Ct
0.6
0.4
0.2
0.0
0.000
0.002
0.004
0.006
0.008
2
BDG/dL
Figure 7.7 : Measured Ct versus BDG/dL2 for G = b STEPFLOAT system
1.0
-30.2114 (BDG/dL^2)
Ct = 1.1501e
0.8
2
R = 0.9124
Ct
0.6
0.4
0.2
0.0
0.000
0.005
0.010
0.015
0.020
2
BDG/dL
Figure 7.8 : Measured Ct versus BDG/dL2 for G = 2b STEPFLOAT system
H/L
BD/dL
Ct
199
Ct
BD/dL
H/L
H/L
BD/dL
Ct
Figure 7.9 : Scatterplot matrix of the Ct and the 2 independent variables for 2-row
Ct
BD/dL
H/L
Figure 7.10 : Scatterplot matrix of the Ct and the 2 independent variables for 3-row
H/L
BDG/dL2
Ct
200
Ct
BDG/dL2
H/L
H/L
BDG/dL2
Ct
Figure 7.11 : Scatterplot matrix of the Ct and the 2 independent variables for G = b
Ct
BDG/dL2
H/L
Figure 7.12: Scatterplot matrix of the Ct and the 2 independent variables for G = 2b
201
7.2.2
Multiple Linear Regression Models of Ct
A multiple linear regression was performed on the results within the data sets for
each system. After an exhaustive analysis of the basic linear relationship as well as the
natural logarithmic and power models, the basic linear relationship was adopted. The
reasoning being aptness of the model, as the others did not adequately describe the
behaviour of the wave attenuation process. The basic linear relationship generally
predicts the dependent variable Ct reasonably well and results in a “good” regression
model. The proposed wave transmission equation is a proportional estimation between
statistical validity and practical implication. The partial regression coefficients were
developed for the 2-row and 3-row systems by taking the BD number (BD/dL) and wave
steepness (H/L) and regressing these variables against Ct. For the G = b and G = 2b
systems, the variable G was included in the BD number to form BDG number
(BDG/dL2), therefore BDG number replaced the BD number analyzed for the 2-row and
3-row equations. The empirical relationships that relate the Ct to the BD number or BDG
number, as well as the H/L are depicted in Equations (7.5) and (7.6).
Ct
B0 B1
H
BD
B2
L
dL
for 2-row & 3-row systems
(7.5)
Ct
B0 B1
BDG
H
B2
2
L
dL
for G = b & G = 2b systems
(7.6)
Multiple linear regression method by SPSS inputs all the independent variables
in a single step. While all variables are entered into the equation as a group by the enter
method, the stepwise method choose variables one at a time for entry into the equation
or removal from it, based on the Stepping Method Criteria using probability of F with
the specified entry probability value < 0.05 and the removal probability value > 0.10, for
all cases. SPSS returned the results from the multiple linear regression, which included
the fitted parameter, R2 and the standard error of the estimate.
202
7.2.2.1 Two-row Equation
An empirical formula for Ct was derived to predict wave attenuation capability of
a 2-row STEPFLOAT system. The resulting linear mathematical relationship is shown in
Equation (7.7). Both enter and stepwise methods yield similar results as attached in
Appendix A1. It should be stressed that Equation (7.7) should only be used within the
variable range given in Table 7.1.
Ct
0.792 12.190
BD H
dL L
(7.7)
Table 7.1 : Variable range for 2-row empirical model
Variable
BD/dL
H/L
B/L
D/L
d/L
Range of variable
0.0121 - 0.0368
0.0058 - 0.0811
0.0904 - 0.2757
0.0258 - 0.0788
0.1937 - 0.5907
R is the correlation coefficient between the observed value of the Ct and the
predicted value based on the regression model. The observed value of 0.93 is quite large,
indicating that the linear regression model predicts well. The linear empirical model also
produced the coefficient of multiple determination, R2 of 0.86, which indicates that the
model accounts for about 86% of the observed variability in the Ct response (refer to the
SPSS output in Appendix A1). Based on the results shown in the model summary by the
stepwise method in Appendix A1, adjusted R2 increased from 0.837 for the model 1
(which considers only BD/dL in the equation) to 0.854 for the model 2 [which is the
model as in Equation (7.7)]. The adjusted R2 statistic is an easy way to guard against
overfitting, that is including regressors that are not really useful. The increase of adjusted
R2 indicates that the addition of the H/L into the model reduced the error or residual
203
mean square. In addition, the standard error for model 2 is reduced to 0.048185 from
0.050817 (the standard error for model 1). Therefore, it could be concluded that adding
the H/L to the model does result in a meaningful reduction in unexplained variability in
the Ct.
The analysis-of-variance or ANOVA table in Appendix A1 is used to test several
equivalent null hypotheses: that there is no linear relationship between the Ct and the
independent variables, that all of the partial regression coefficients are zero, and that the
value of multiple R2 is zero. The test of the null hypothesis is based on the ratio of the
regression mean square to the residual mean square. The ratio of the two mean squares,
labeled F, is 129.259. Since the observed significance level is less than 0.05, the null
hypothesis is rejected and it can be concluded that Ct is linearly related to either BD
number or H/L, or both. At least of one of the Ct regression coefficients is not zero.
However, it was noted that this does not necessarily imply that the equation found is an
appropriate model for predicting Ct as a function of BD number and H/L. Further tests of
model adequacy are required before this model is comfortably used in practice.
The coefficients for the independent variables are listed in the column labeled B
in Appendix A1. Using these coefficients, the estimated regression equation can be
written as Equation (7.7). In the multiple regression equation, the partial regression
coefficient for a variable indicates how much the value of the Ct changes when the value
of that independent variable increases by one and the value of the other independent
variable do not change. The negative coefficients show that the predicted value of the Ct
decreases when the values of the BD number and the H/L increase. The standard errors
are a useful measure of the precision of estimation for the regression coefficients. The
SPSS output in Appendix A1 reports that the standard errors for B0, B1 and B2 are 0.022,
B
B
B
1.224 and 0.414. The estimated intercept and regression coefficients are considerably
larger than the magnitude of their respective standard errors. This implies good precision
of estimation. The importance of the two independent variables relative to the given
204
model is determined based on the beta weights, which make partial regression
coefficients somewhat more comparable. In the output, these beta coefficients are in the
column labeled Beta. It was observed that BD/dL, with the beta coefficient of -0.787,
appears to be a more significant variable affecting the Ct value if compared to the H/L
with a smaller beta coefficient of -0.191. However, the values of the beta coefficients
still depend on the other independent variables in the model, so they do not reflect in any
absolute sense the importance of the individual independent variables (Norušis, 2000).
The significance of the individual regression coefficient can be determined based
on the output of the t statistic (by dividing the estimated coefficient by its standard error)
and its observed significance level. This is a partial test as the regression coefficient
depends on all the other regressor variables that are in the model. Since the significance
level for all partial regression coefficients is less than 0.05, it indicates that both the BD
number and H/L contribute significantly to the model.
Appendix A1 also shows the matrix of Pearson correlation coefficients for all of
the variables in the model. Pearson correlation matrix demonstrates the strength of the
linear relationship between pairs of variables. The first part of the table contains the
observed correlation coefficients for each pair of variables. The second part of the table
contains the one-tailed observed significance levels and the third part of the table shows
the sample size on which the correlation is based. Looking at the extreme upper-right
corner, it was observed that the correlation coefficient between Ct and BD number equals
-0.917 while the correlation coefficient between Ct and H/L equals -0.728. Since the
observed significance levels are less than 0.05 or 0.01 alpha level, there is linear
relationship between Ct and BD number and H/L.
In order to determine the strength of linear relationship between the Ct and an
independent variable, while “controlling” or keeping constant the effects of other
205
independent variables, the partial correlation analysis is required. The partial correlation
coefficient is the correlation between two variables when the linear effects of other
variables are removed. Appendix A1 shows the partial correlation coefficient between Ct
and BD number when the linear effect of H/L is eliminated. The partial correlation
coefficient is -0.838 with the observed significance level less than 0.05, thus linear
relationship between Ct and BD number exists. The partial correlation coefficient
between Ct and H/L, with BD number as the control variable, is -0.349. The
corresponding significance level is 0.02, which is less than 0.05. Hence, linear
relationship exists between Ct and H/L.
From the collinearity statistics, it was observed that the tolerance value is 0.533 >
0.20 (as a rule of thumb, if tolerance is less than 0.20, a problem with multicollinearity is
indicated) or the variance-inflation-factor, VIF = 1.875 < 4 (VIF values above 4 suggest
a multicollinearity problem). Since both tolerance and VIF values fulfilled the
mentioned criteria, multicollinearity does not exist.
7.2.2.2 Three-row Equation
The full empirical equation for Ct for 3-row STEPFLOAT system, resulting from
the regression analysis, is given as Equation (7.8). Again, both enter and stepwise
methods yield similar results as attached in Appendix A2. It should be stressed that
Equation(7.8) should only be used within the variable range given in Table 7.2.
Ct
0.863 9.061
BD
H
2.332
dL
L
(7.8)
206
Table 7.2 : Variable range for 3-row empirical model
Variable
BD/dL
H/L
B/L
D/L
d/L
Range of variable
0.0175 - 0.0534
0.0094 - 0.0831
0.1313 - 0.4004
0.0258 - 0.0788
0.1937 - 0.5907
This linear empirical model of 3-row system provided a superior estimation than
the 2-row one. The correlation coefficient, R = 0.97 is rather large, implying that the
linear regression model predicts well. The coefficient of multiple determination, R2 =
0.94, indicating that 94% of the observed variability in Ct is “explained” by the two
independent variables, i.e. BD number and H/L (refer to the SPSS output in Appendix
A2). Based on the results shown in the model summary by the stepwise method in
Appendix A2, adjusted R2 increased from 0.882 for the model 1 (which considers only
BD/dL in the equation) to 0.938 for the model 2 [which considers BD/dL and H/L in the
model as in Equation (7.8)]. The increase of adjusted R2 indicates that with the
additional variable of the H/L to the model, it reduced the error or residual mean square.
It was also observed that the standard error for model 2 is reduced to 0.0387161 from
0.0535415 (the standard error for model 1). Therefore, it could be concluded that adding
the H/L to the model does result in a reduction in unexplained variability in the Ct.
The ANOVA table in Appendix A2 shows that the F value is 334.835 and the
corresponding significance level is less than 0.05, thus Ct is said to be linearly related to
either BD number or H/L, or both. The partial regression coefficients are shown in the
table of coefficients in Appendix A2. The negative coefficients show that the predicted
value of the Ct decreases when the values of the BD number and the H/L increase. The
SPSS output in Appendix A2 reports that the standard errors for B0, B1 and B2 are 0.018,
B
B
B
0.723 and 0.368. The estimated intercept (B0 = 0.863) and regression coefficients (B1 =
B
B
207
-9.061 and B2 = -2.332) are significantly larger than the magnitude of their respective
B
standard errors. This implies good precision of estimation for the regression coefficients.
The beta coefficients show that BD/dL, with the beta coefficient of -0.687,
appears to be a more influential variable as compared to the H/L with a smaller beta
coefficient of -0.348. The significance of the individual regression coefficient is
determined based on the output of the t statistic with t values equal to -12.535 and -6.343
for BD number and H/L, respectively. Since the significance level for both partial
regression coefficients is less than 0.05, it indicates that both the BD number and H/L
contribute significantly to the model.
The matrix of Pearson correlation coefficients for all of the variables in the
model shows that the correlation coefficient between Ct and BD number equals -0.940
while the correlation coefficient between Ct and H/L equals -0.849. Since the observed
significance levels are less than 0.05, there is linear relationship between Ct and BD
number and H/L. Appendix A2 also shows the partial correlation coefficient between Ct
and BD number when the linear effect of H/L is eliminated. The partial correlation
coefficient is -0.888 with the observed significance level less than 0.05. The partial
correlation coefficient between Ct and H/L, with BD number as the control variable, is
-0.699. The corresponding significance level is less than 0.05. Hence, linear relationship
exists between Ct and BD number as well as between Ct and H/L. From the collinearity
statistics, it was observed that the tolerance value is 0.468 > 0.20 and the VIF = 2.137 <
4. Therefore, multicollinearity does not exist.
208
7.2.2.3 G = b Equation
The full equations for the G = b STEPFLOAT system, resulting from the
regression analysis using the enter and stepwise methods, are given as Equations (7.9)
and (7.10), respectively. All independent variables are entered into the Equation (7.9) as
a group by the enter method while the stepwise method has excluded the variable of H/L
in the model as shown in the Excluded Variables table in Appendix A3. The statistics for
Model 1 as in the Excluded Variables table show that the variable of H/L has observed
significance level of 0.241, which is greater than 0.05 (the criterion for adding a
variable), so it is not eligible to enter the model. Table 7.3 gives the summary of the
applicable range for use of these equations for the G = b STEPFLOAT system.
Ct
0.630 58.952
BDG
H
0.439
2
L
dL
by the enter method
Ct
0.620 62.127
BDG
dL2
by the stepwise method
(7.9)
(7.10)
Table 7.3 : Variable range for G = b empirical model
Variable
BDG/dL
H/L
B/L
D/L
G/L
d/L
2
Range of variable
0.00075
0.0082
0.1313
0.0258
0.0430
0.1937
-
0.00701
0.0779
0.4004
0.0788
0.1313
0.5907
The correlation coefficient, R for both Equations (7.9) and (7.10) are 0.955 and
0.954, respectively. Generally, these imply that both the linear regression models predict
well. The coefficient of multiple determination, R2 for both models are similar, i.e. R2 =
209
0.91, indicating that 91% of the observed variability in Ct is “explained” by the
independent variable(s), i.e. BDG number and H/L or BDG number only (refer to the
SPSS output in Appendix A3). Based on the results shown in the model summary in
Appendix A3, the standard error of the estimate for both equations were also found to be
almost analogous, i.e. 0.0406260 [for Equation (7.9)] and 0.0408208 [for Equation
(7.10)].
The ANOVA tables for both the enter and stepwise methods in Appendix A3
show that the F values are 218.088 and 430.624, respectively. The corresponding
significance level for both models is less than 0.05, thus Ct is said to be linearly related
to either BDG number or H/L, or both. The partial regression coefficient(s) are shown in
the table of coefficients in Appendix A3.
The SPSS output in Appendix A3 reports that by using the enter method, the
standard errors for B0, B1 and B2 are 0.014, 4.001 and 0.370. The estimated intercept B0 =
B
B
B
B
0.630 and regression coefficient B1 = -58.952 are significantly larger than the magnitude
B
of their respective standard errors. This implies good precision of estimation for the
estimated intercept and regression coefficient B1. However, the standard error for B2 is
B
B
0.370, which is high and somewhat close to the magnitude of B2 = -0.439. This shows
B
that the estimated partial regression coefficient B2 has slightly lower precision of
B
estimation. It was also observed that the corresponding t statistic equals to -1.189 with
the significance level of 0.241 (this means that the finding has only a 75.9% chance of
being true and has less significant contribution to the model). Careful must be given
when extrapolating beyond the region containing the original observations. It is very
possible that a model that fits well in the region of the original data will no longer fit
well outside of that region (Montgomery and Runger, 2003). Therefore, it is
recommended that in predicting the Ct value, Equation (7.9) has to be applied carefully
where the applicable range of variable, as in Table 7.3, for use of the equation has to be
strictly followed.
210
Nevertheless, the stepwise method has excluded the variable of H/L in the model,
giving the standard errors for B0 and B1 as 0.011 and 2.994. The estimated intercept B0 =
B
B
B
0.620 and regression coefficient B1 = -62.127 are considerably larger than the magnitude
B
of their respective standard errors. This gives reasonable precision of estimation. The
significance of the regression coefficient is determined based on the t statistic with t
values equal to -20.751 for BDG number and its significance level is less than 0.05. This
indicates that the BDG number contributes significantly to the model. However, it
should be noted that the model with the exclusion of independent variable H/L, which
was previously found to be significantly influential to the Ct value, might be biased, i.e.
it does not represent the true underlying model. Therefore, it is again recommended that
in estimating the Ct value, Equation (7.10) has to be applied carefully within the
applicable range of variable, as in Table 7.3.
According to Montgomery and Runger (2003), a major criticism of variable
selection methods such as stepwise regression is that the analyst may conclude there is
one ‘best’ regression. Generally, this is not the case, because several equally good
regression models can often be used. The final model obtained from any model-building
procedures should be subjected to the usual adequacy checks or diagnostics, such as
residual analysis, which will be performed in the later sections. Therefore, based on the
statistical results, Equations (7.9) and (7.10) are equally good in predicting the
performance of the G = b STEPFLOAT system under the condition that the data points
lie within the joint region of original area among regressor variables but not outside the
region that is spanned by the original observations. Predicting the value of new
observations or estimating the performance of the floating breakwater at a point within
the ranges of regressor variables but outside the joint region of original area is
considered an extrapolation of the original regression model.
The matrix of Pearson correlation coefficients for all of the variables shows that
the correlation coefficient between Ct and BDG number equals -0.954 while the
211
correlation coefficient between Ct and H/L equals -0.677. Since the observed
significance levels are less than 0.05, there is linear relationship between Ct and BDG
number and H/L. Appendix A3 also shows the partial correlation coefficient between Ct
and BDG number when the linear effect of H/L is eliminated. The partial correlation
coefficient is -0.915 with the observed significance level less than 0.05. The partial
correlation coefficient between Ct and H/L, with BDG number as the control variable, is
-0.180. The corresponding significance level is 0.241, which is greater than 0.05. Hence,
strong linear relationship exists between Ct and BDG number but low linear correlation
was found between Ct and H/L. From the collinearity statistics, it was observed that the
tolerance values are 0.555 (with VIF = 1.803) and 1.000 (with VIF = 1.000) for the
Equations (7.9) and (7.10), respectively. Note that the tolerance values are greater than
0.20 and the VIF values are smaller than 4. Therefore, multicollinearity does not exist.
7.2.2.4 G = 2b Equation
The empirical equation for Ct for G = 2b STEPFLOAT system is given as
Equation (7.11). Again, both enter and stepwise methods produce similar results as
attached in Appendix A4. Equation (7.11) should only be used within the variable range
given in Table 7.4.
Ct
0.692 24.395
BDG
H
1.198
2
L
dL
(7.11)
The correlation coefficient, R = 0.97 is rather large, implying that the linear
regression model predicts well. The coefficient of multiple determination, R2 = 0.94,
indicating that 94% of the observed variability in Ct is “explained” by the two
212
Table 7.4 : Variable range for G = 2b empirical model
Range of variable
Variable
BDG/dL
H/L
B/L
D/L
G/L
d/L
2
0.0020
0.0079
0.1743
0.0258
0.0861
0.1937
-
0.0186
0.0840
0.5317
0.0788
0.2626
0.5907
independent variables, i.e. BDG number and H/L (refer to the SPSS output in Appendix
A4). Based on the results shown in the model summary by the stepwise method in
Appendix A4, adjusted R2 increased from 0.924 for the model 1 (which considers only
BDG/dL2 in the equation) to 0.939 for the model 2 [which considers BDG/dL2 and H/L
in the model as in Equation (7.11)]. The increase of adjusted R2 indicates that with the
additional variable of the H/L to the model, it reduced the error or residual mean square.
It was also observed that the standard error for model 2 is reduced to 0.0393489 from
0.0440984 (the standard error for model 1). Therefore, it could be concluded that adding
the H/L to the model does result in a reduction in unexplained variability in the Ct.
The ANOVA table in Appendix A4 shows that the F value is 340.721 and the
corresponding significance level is less than 0.05, thus Ct is said to be linearly related to
either BDG number or H/L, or both. The partial regression coefficients are shown in the
table of coefficients in Appendix A4. The negative coefficients show that the predicted
value of the Ct decreases when the values of the BDG number and the H/L increase. The
SPSS output in Appendix A4 reports that the standard errors for B0, B1 and B2 are 0.013,
B
B
B
1.528 and 0.346. The estimated intercept (B0 = 0.692) and regression coefficients (B1 =
B
B
-24.395 and B2 = -1.198) are significantly larger than the magnitude of their respective
B
standard errors. This implies good precision of estimation for the regression coefficients.
213
The beta coefficients show that BDG/dL2, with the beta coefficient of -0.835,
appears to be a more influential variable as compared to the H/L with a smaller beta
coefficient of -0.181. The significance of the individual regression coefficient is
determined based on the output of the t statistic with t values equal to -15.969 and -3.465
for BDG number and H/L, respectively. Since the significance level for both partial
regression coefficients is less than 0.05, it indicates that both the BDG number and H/L
contribute significantly to the model.
The matrix of Pearson correlation coefficients for all of the variables in the
model shows that the correlation coefficient between Ct and BDG number equals -0.962
while the correlation coefficient between Ct and H/L equals -0.768. Since the observed
significance levels are less than 0.05, there is linear relationship between Ct and BDG
number and H/L. Appendix A4 also shows the partial correlation coefficient between Ct
and BDG number when the linear effect of H/L is eliminated. The partial correlation
coefficient is -0.927 with the observed significance level less than 0.05. The partial
correlation coefficient between Ct and H/L, with BDG number as the control variable, is
-0.472. The corresponding significance level is less than 0.05. Hence, linear relationship
exists between Ct and BDG number as well as between Ct and H/L. From the collinearity
statistics, it was observed that the tolerance value is 0.506 > 0.20 and the VIF = 1.976 <
4. Therefore, multicollinearity does not exist.
7.2.3 Multiple Regression Diagnostics
In this section of multiple regression diagnostics, examination using residuals
and other diagnostics techniques for adequacy checks and to identify data points that are
in some way unusual are performed.
214
7.2.3.1 Two-row Model
Figure 7.13 graphically shows the scatterplot of predicted and observed values of
Ct. The data points are reasonably evenly distributed on either side of the fitted 45°
straight line, which is a line of perfect agreement. In general, the agreement between
experimental and predicted results is rather good with the coefficient of determination,
R2 = 0.8602. Table 7.5 compares the predicted and actual Ct values for the 2-row system
and shows that the maximum discrepancy between the predicted and observed Ct was
19.56% while the average absolute prediction error was approximately 8.19%.
2
R = 0.8602
1.0
0.9
0.8
Predicted Ct
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Observed Ct
Figure 7.13 : Predicted and observed Ct for 2-row system
215
Table 7.5 : Comparison of predicted and observed Ct for 2-row system
Predicted Ct
Observed Ct
Case Name
Test 1-2
0.6396
0.6211
Test 3-4
0.6255
0.5564
Test 5-6
0.6122
0.6456
Test 7-8
0.5930
0.6168
Test 9-10
0.5657
0.5736
Test 11-12
0.5463
0.5973
Test 13-14
0.5252
0.5779
Test 15-16
0.4969
0.4509
Test 17-18
0.4638
0.3731
Test 19-20
0.4339
0.4137
Test 21-22
0.4065
0.4109
Test 23-24
0.3763
0.3295
Test 25-26
0.3452
0.3270
Test 27-28
0.3211
0.3356
Test 29-30
0.2995
0.2988
Test 31-32
0.6257
0.6940
Test 33-34
0.6091
0.5238
Test 35-36
0.6019
0.6832
Test 37-38
0.5792
0.5812
Test 39-40
0.5405
0.4630
Test 41-42
0.5200
0.5317
Test 43-44
0.4994
0.5298
Test 45-46
0.4617
0.4011
Test 47-48
0.4211
0.4026
Test 49-50
0.3978
0.4536
Test 51-52
0.3615
0.3083
Test 53-54
0.3388
0.3996
Test 55-56
0.3125
0.3178
Test 57-58
0.2875
0.2481
Test 59-60
0.2750
0.2633
Test 61-62
0.6179
0.6625
Test 63-64
0.6041
0.6110
Test 65-66
0.5937
0.6550
Test 67-68
0.5635
0.4975
Test 69-70
0.5275
0.4456
Test 71-72
0.5036
0.4673
Test 73-74
0.4772
0.4533
Test 75-76
0.4522
0.5213
Test 77-78
0.4302
0.5013
Test 79-80
0.3903
0.4157
Test 81-82
0.3593
0.3810
Test 83-84
0.3362
0.3613
Test 85-86
0.3145
0.3305
Test 87-88
0.2904
0.2967
Test 89-90
0.2770
0.2904
Total
Number of data
Average absolute prediction error, %
% difference
2.8988
11.0447
-5.4557
-4.0006
-1.3860
-9.3306
-10.0365
9.2604
19.5650
4.6430
-1.0925
12.4263
5.2843
-4.4928
0.2136
-10.9167
13.9996
-13.5047
-0.3508
14.3314
-2.2535
-6.0893
13.1236
4.3854
-14.0401
14.7121
-17.9501
-1.7167
13.6898
4.2393
-7.2111
-1.1341
-10.3352
11.7169
15.5160
7.2018
4.9985
-15.2957
-16.5148
-6.5140
-6.0394
-7.4700
-5.0886
-2.1668
-4.8249
Absolute % difference
2.8988
11.0447
5.4557
4.0006
1.3860
9.3306
10.0365
9.2604
19.5650
4.6430
1.0925
12.4263
5.2843
4.4928
0.2136
10.9167
13.9996
13.5047
0.3508
14.3314
2.2535
6.0893
13.1236
4.3854
14.0401
14.7121
17.9501
1.7167
13.6898
4.2393
7.2111
1.1341
10.3352
11.7169
15.5160
7.2018
4.9985
15.2957
16.5148
6.5140
6.0394
7.4700
5.0886
2.1668
4.8249
368.4617
45
8.1880
216
The variability of the predicted value of Ct is not constant for all points but
depends on the value of the independent variable. Cases with values of the independent
variable close to the mean value have smaller variability for the predicted value of Ct.
The Studentized residual, by dividing the observed residual by an estimate of the
standard deviation of the residual at that point, takes into account the differences in
variability from point to point (Norušis, 2000). In order to see the impact of a case on the
computation of the regression statistics, Studentized deleted residual (i.e. Studentized
residual for a case when the case is excluded from the computation of the regression
statistics) is recommended by Norušis (2000). When there are departures from the
regression assumptions or any violations against regression suitability criteria or if there
are unusual and influential data points, it would be easily identified using Studentized
deleted residuals. The residuals should show no pattern when plotted against the
predicted values.
In Figure 7.14, the Studentized deleted residuals (Studentized deleted residuals
will be referred as simply residuals throughout the rest of this chapter) are plotted
against the predicted Ct to further verify the aptness of the model. Most of the residuals
fall in a horizontal band around 0 with the residuals for predicted Ct below 0.45 have
somewhat less spread than the residuals for the larger predicted values. The lesser the
scatter in the plots, the better the fit of the model. If there is trend in the residuals either
upward or downward the aptness of the model has to be questioned. In general, the
residuals appear to be randomly scattered around the horizontal line through 0 with no
obvious trend and there are no outliers observed, thus Equation (7.7) is valid.
217
3.0
Studentized Deleted Residual
2.0
1.0
0.0
-1.0
-2.0
-3.0
0.2
0.3
0.4
0.5
0.6
0.7
Unstandardized Predicted Ct
Figure 7.14 : Studentized deleted residuals versus predicted Ct for 2-row system
7.2.3.2 Three-row Model
The observed and predicted Ct values are plotted against each other in Figure
7.15. The data points are reasonably randomly distributed on either side of a 45° line,
which is shown for easier observation of a 1:1 correlation between predicted and
observed Ct. The agreement between observed and predicted Ct is good with an R2 =
0.9410. The model gives a general equation that passes through the points but individual
data points can still be over or under predicted by as much as 23.45% with an average
absolute prediction error of approximately 6.72%.
218
2
R = 0.9410
1
0.9
0.8
Predicted Ct
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Observed Ct
Figure 7.15 : Predicted and observed Ct for 3-row system
Figure 7.16 shows the scatterplot of the residuals against the predicted Ct. In
general, the residuals appear to be randomly distributed around the horizontal band
around 0. It was observed that there is no obvious trend in the plots, thus Equation (7.8)
is valid.
219
3.0
Studentized Deleted Residual
2.0
1.0
0.0
-1.0
-2.0
-3.0
0.2
0.3
0.4
0.5
0.6
0.7
Unstandardized Predicted Ct
Figure 7.16 : Studentized deleted residuals versus predicted Ct for 3-row system
7.2.3.3 G = b Model
Figures 7.17 and 7.18 show the comparison of predicted and observed values of
Ct for G = b system based on the Equations (7.9) and (7.10), respectively. Again, data
points are evenly scattered on either side of the fitted 45° straight line for both models.
The agreement between experimental and predicted results is good with the coefficient
of determination, R2 = 0.9122 and 0.9092 for the model based on Equations (7.9) and
(7.10), respectively. The average absolute prediction error was approximately 7.95% [for
Equation (7.9)] and 8.21% [for Equation (7.10)].
220
2
R = 0.9122
1.0
0.9
0.8
Predicted Ct
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Observed Ct
Figure 7.17 : Predicted and observed Ct for G = b system [Equation (7.9)]
2
R = 0.9092
1.0
0.9
0.8
Predicted Ct
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Observed Ct
Figure 7.18 : Predicted and observed Ct for G = b system [Equation (7.10)]
221
In Figures 7.19 and 7.20, the residuals are plotted against the predicted Ct. Most
of the residuals fall in a horizontal band around 0 with the residuals for predicted Ct
below 0.40 have somewhat less spread than the residuals for the larger predicted values.
In general, the residuals appear to be randomly scattered around the horizontal line
through 0 with no obvious trend, thus Equations (7.9) and (7.10) are valid.
3.0
Studentized Deleted Residual
2.0
1.0
0.0
-1.0
-2.0
-3.0
0.1
0.2
0.3
0.4
0.5
0.6
Unstandardized Predicted Ct
Figure 7.19 : Studentized deleted residuals versus predicted Ct for G = b system
[Equation (7.9)]
222
3.0
Studentized Deleted Residual
2.0
1.0
0.0
-1.0
-2.0
-3.0
0.1
0.2
0.3
0.4
0.5
0.6
Unstandardized Predicted Ct
Figure 7.20 : Studentized deleted residuals versus predicted Ct for G = b system
[Equation (7.10)]
7.2.3.4 G = 2b Model
Figure 7.21 shows the comparison of predicted and observed values of Ct for G =
2b system. As other models, it was observed that data points are randomly scattered on
either side of the fitted 45° straight line. The agreement between experimental and
predicted results is good with the R2 = 0.9419. The average absolute prediction error was
calculated to be approximately 6.70%.
223
In Figure 7.22, the residuals are plotted against the predicted Ct. Most of the
residuals fall in a horizontal band around 0 with the residuals for predicted Ct below 0.40
have somewhat less spread than the residuals for the larger predicted values. The
residuals generally appear to be randomly distributed around the horizontal line through
0 with no obvious trend, thus Equation (7.11) is valid.
2
R = 0.9419
1.0
0.9
0.8
Predicted Ct
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Observed Ct
Figure 7.21 : Predicted and observed Ct for G = 2b system
224
Studentized Deleted Residual
4.0
2.0
0.0
-2.0
-4.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Unstandardized Predicted Ct
Figure 7.22 : Studentized deleted residuals versus predicted Ct for G = 2b system
CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS
8.1
Summary and Conclusions
Current environmental constraints and financial restrictions on development of
marinas, harbours, aquaculture facilities, recreational beaches and other coastal facilities
dictate that alternatives to traditional coastal protection structures such as rubble-mound
and caisson breakwaters are essential to the futu re of coastal engineering. As a result, the
composite stepped-slope floating breakwater (STEPFLOAT) system has been developed
based on the concept which utilizes mainly dissipation to reduce wave energy and
therefore attenuating incident waves to an acceptable level.
The purpose of this research was to conduct a series of laboratory experiments to
evaluate and predict the hydraulic performance of the STEPFLOAT breakwater system
using vertical piles as mooring method. The STEPFLOAT is a floating break
water with
stepped-slope features at the upper layer and multiple sharp edges all over the
polyhedron that serve to intercept waves by dissipating the wave energy as the waves
impinges on the surface of the structure, thus providing a sheltered region from waves.
226
The mooring system using vertical piles was suggested in order to overcome the problem
of roll and sway motions, which will in turn generate secondary waves in the leeside of
the floating breakwater, the di sadvantage of mooring lines in region with large tidal
range and to improve the performance of the STEPFLOAT system as a pier or
pedestrian walkway in small craft harbours a nd marinas. Invention disclosure and patent
application on the STEPFLOAT break
water system have been submitted to the Research
Management Centre of Universiti Teknologi Malaysia for further action.
The hydraulic performance of a floating breakwater can be quantified by its
ability to reduce waves in the sheltered region and this may be quantified by the wave
transmission coefficient Ct. Several STEPFLOAT system arrangements were tested in
the unidirectional wave flume at the COEI Laboratory in 2004 to investigate the
influence of engineering design parameters on the coefficient of transmission which is
useful and essential for future design purposes. The physical model scale was 1:10.
W
ave transmission was measured and evaluated. The ability of the suggested pilesystem floating breakwater to attenuate in cident regular waves was determined by
comparing its results with the results of chain moored breakwater sy stem and restrained
case. A summary of Ct, Cr and Cl for the STEPFLOAT breakwater in terms of various
system arrangements and mooring systems is tabulated in Table 8.1. Finally the
following summary of the major findings and important conclusions of the present study
were drawn:
(a)
The chain-moored 2-row STEPFLOAT system tends to become fully suspended
and taut and impart snap loads as the waves impinge upon the floating
breakwater. This has subsequently induced sway and roll motions which in turn
generate secondary waves behind the structure. Hence unsatisfactory
performance with higher Ct or a minimum of Ct = 0.6 was achieved.
227
Table 8.1 : Summary of Ct, Cr and Cl for the STEPFLOAT in terms of various
conditions
System arrangements
Operational
system arrangements and mooring systems
d
D/d
Chain-moored
53 cm
0.104
T
0.68 - 1.05 sec
d/L
0.32 - 0.73
C t : 0.60 - 0.97
Mooring systems
Restrained case
45 cm
0.133
0.69 - 1.11 sec (for 2-row)
0.70 - 1.33 sec (for 3-row)
0.25 - 0.60 (for 2-row)
0.19 - 0.59 (for 3-row)
C t : 0.17 - 0.56
2-row
C r : 0.17 - 0.41
C r : 0.28 - 0.58
C r : 0.10 - 0.43
C l : 0.10 - 0.72
C l : 0.72 - 0.93
C l : 0.63 - 0.94
C t : 0.11 - 0.69
C t : 0.20 - 0.72
C r : 0.21 - 0.65
C r : 0.16 - 0.39
C l : 0.43 - 0.92
C l : 0.62 - 0.96
3-row
n.a.
Pile-supported
45 cm
0.133
0.70 - 1.33 sec
0.19 - 0.59
C t : 0.25 - 0.69
C t : 0.17 - 0.67
G=b
n.a.
n.a.
C r : 0.09 - 0.37
C l : 0.71 - 0.94
C t : 0.16 - 0.69
G = 2b
n.a.
n.a.
C r : 0.11 - 0.41
C l : 0.62 - 0.94
Note: n.a. = not available
(b)
The suggested pile-system STEPFLOAT, for the case of 2-row system, has
significant wave attenuation capability as compared to the chain-moored system.
A [Ct]red ranging from 33.42% to 39.30% was obtained.
(c)
The restrained case of the STEPFLOAT system was investigated, as it is
restricted from motion, in order to find the transmission and reflection
coefficients of the body without motion so that these could be used as a reference
to determine the effect of heave and limited roll motions of the pile-system
STEPFLOAT on wave attenuation. The suggested vertical pile-system for 2-row
and 3-row STEPFLOAT showed the same behaviour as the restrained structure
but with slightly higher values for Ct (approximately 10% difference). Most of
the incident wave energy was reflected backoffshore for the case of restrained
228
body if compared to lower wave reflection by the vertical pile-system
STEPFLOAT with a difference of about 20% between the two cases. Hence the
initial objective of developing the STEPFLOAT break
water as a wave energy
dissipator rather than a wave energy reflector was achieved.
(d)
The pile-supported STEPFLOAT break
water system with 3-row, G = b and G =
2b arrangements were found to be effective in attenuating incident wave heights
up to 80% for wave period of less than 1.33 sec and were also hydrodynamically
stable due to wave-structure interaction. Ct was found to decrease with increasing
B/L, D/L, H/L, d/L and G/L.
(e)
The experimental results proved that there were significant amount of energy loss
in the incident wave energy. Observation of the experiments showed that part of
the wave energy loss is dissipated in the form of eddies created around the sharp
edges of the floating body. These eddies occupy an area on the seaward side of
the structure which is larger than on the leeward side. Furthermore, the position
of these eddies changes with the wave surface elevation around the
STEPFLOAT. The STEPFLOAT is structurally different from other existing
floating breakwaters, thus exhibits fundamentally different functional
characteristics as a break
water. The S TEPFLOAT intercepts and dissipates most
of the incident wave energy rather than redirecting it to the seaward region of the
sea. hWile most of the existing floa ting break
waters act as wave energy
reflectors, the STEPFLOAT mak
es a better wave energy dissipator.
(f)
Experimental results showed that 3-row system achieved slightly lower Ct as
compared to the 2-row system for the tested wave period below 1.0 sec. There is
no clear distinction shown for results with wave period greater than 1.0 sec.
(g)
In comparison with 3-row system, the G = b system generally performed slightly
more effective in attenuating wave energy. For T < 0.90 sec, the G = b system
with a pontoon spacing of b gives no significant differences in attenuating
relatively shorter incident waves as compared to the 3-row system.
(h)
The spacing between the breakwater pontoons is an important parameter that
brings in additional waterline beam for a given individual or combined module
width. The G = 0 (or 2-row) system generally provides higher Ct if compared to
229
the G = b and G = 2b systems. For T > 0.8373 sec, the G = b system gives better
and more effective wave attenuation than G = 2b system while the contrary was
achieved for T < 0.8373 sec. In relatively lower frequency (or longer waves), the
smaller spacing allows the two pontoons to act as a continuous structure
functioning like a single unit, spanning a considerable portion of wave length,
whereas in higher frequency (or shorter waves), a twin-pontoons section with
larger spacing tend to act independently as two separate single pontoon
breakwaters in series. Thus, for twin- pontoon section to achieve better wave
attenuation performance, the spacing should be approximately equal to the
pontoon width, for T > 0.8373 sec.
(i)
The relative width B/L is one of the dominant parameters in determining the
efficiency of the STEPFLOAT. As B/L increases, the degree of wave
transmission generally decreases. To be effective with an achievement of the
threshold level of Ct = 0.5, the 2-row, 3-row, G = b and G = 2b systems should
have a ratio of the overall breakwater width to wave length of at least 0.1478,
0.2068, 0.1955 and 0.2713, respectively.
(j)
A comparison on wave attenuation ability was made among the STEPFLOAT,
the SSFBW
, the rectangular pontoon a nd also results of various floating
breakwater designs. In general, the findings of the present study show reasonably
good agreement with the results from other researchers. The STEPFLOAT
generally has excellent wave attenuation ability over most of the previous
floating breakwaters. The range of B/L for the STEPFLOAT breakwater system
is relatively smaller than most of the other floating break
waters and this implies
that a relatively smaller structure width is sufficient for the STEPFLOAT to
achieve similar or better level of wave attenuation as achieved by other floating
breakwaters. Thus STEPFLOAT will potenti ally appear to be a competitively
cost-effective floating breakwater in the future market.
(k)
Several dimensionless variables, i.e. B/L, D/L, G/L, H/L and d/L were
investigated individually relative to measured Ct. Although the relationship
between Ct and all the independent variables appears to be more or less linear,
there are strong dependencies among the independent variables
230
(multicollinearity) except wave steepness. Hence, in order to yield a regression
model that contains all the independent variables, two new composite parameters
BD number (for 2-row and 3-row systems) and BDG number (for G = b and G =
2b systems), which incorporate the structural geometrical and hydraulic
characteristics, were introduced and examined. Both BD and BDG numbers
proved to have significant influence on Ct.
(l)
A series of empirical design models were developed for the 2-row, 3-row, G = b
and G = 2b pile-supported STEPFLOAT systems based on multiple linear
regression analysis. The predicted results are in reasonably good agreement with
the experimental values with the average absolute prediction error for all system
arrangements below 9%. The empirical multiple linear regression model
predictions were a good fit to the measured data for the range of experimental
conditions tested. Thus, the empirical equations for the pile-supported
STEPFLOAT with different system arrangements as shown in Table 8.2 can be
used as a preliminary tool to obtain related engineering design parameters.
Table 8.2 : Summary of Ct predictive equations
System
Method
Equations
2-row
Enter &Stepwise
Ct
3-row
Enter &Stepwise
Ct
Enter
Ct
Stepwise
Ct
Enter &Stepwise
Ct
G=b
G = 2b
BD H
dL L
BD
H
0.863 9.061
2.332
dL
L
BDG
H
0.630 58.952
0.439
2
L
dL
BDG
0.620 62.127
dL2
BDG
H
0.692 24.395
1.198
2
L
dL
0.792 12.190
These equations are only applicable when the variables lie within the tested range
given in previous chapter.
231
8.2
Recommendations for Future Research
The present study considers the case of a composite stepped-slope floating
break
water system interacting with monochroma tic wave train. The workcarried out and
the study performed of the related literature indicates that there are several extensions of
the present study which may improve the overall understanding of the behavior and
performance of the STEPFLOAT system. Prospective extensions of the present study
include the following:
(a)
Promising data were obtained from the laboratory data sets. Although most of the
data has been analyzed, much remains to be done with this data. It was
impossible to investigate every potential relationship within the scope of this
thesis. Reflection is often a major concern and design criterion for engineers,
especially in areas of ship traffic. Maximizing energy dissipation may be great
importance, for both reflection and transmission purposes. Additional study
should be done with these data sets in comparison to workof other researchers,
particularly in the area of the reflection coefficient and loss coefficient as well as
the development of equations that focus on reflection and energy dissipation.
(b)
In the present study, the case of the STEPFLOAT break
water system interacting
with regular waves is considered, whereas it is of interest to consider the wavestructure interaction of the floating breakwater to random waves. This would be
useful to assess the extent to which the regular wave results of the present study
can be applied to random waves.
(c)
The present study considers the one-dimensional case of the STEPFLOAT
breakwater system subjected to normally incident regular waves. For future
research, study on the similar problem in a two-dimensional or three-dimensional
scale to investigate the wave diffraction around the floating breakwater is
recommended. Examination of the influence of oblique regular and irregular
waves on the STEPFLOAT breakwater of finite length may be considered.
232
(d)
The influence of the hydrodynamics load associated with oblique and normally
incident waves on the bending moments and torsion at breakwa ter section needs
to be assessed in this breakwater de sign. Study on the influence of impulsive
loads such as wave forces on the mooring lines or piles is recommended for
proper mooring design. The consequences of connections between floating
modular units and mooring structures failing should be considered as connections
have generally been unsatisfactory and are often under-designed for most of the
existing floating breakwater designs. Failures of either the breakwater structure
or moorings are relatively common. British Standards Institution (1999)
highlighted that the low success rate of floating break
waters is due in part to the
difficulty of predicting wave forces. Physical models should be used whenever
possible. Models will however only provide approximate estimates of wave
forces and the nature of wave loading. Allowance should be made for tests at the
installation stage and monitoring of long-term performance. This would be
helpful in the evaluation of extreme motions and forces.
(e)
The mechanics of motion of the STEPFLOAT are not thoroughly investigated in
the present study. A study on the motions of the STEPFLOAT in response to
wave is highly recommended.
(f)
During experiments, strong turbulence and eddies were observed. K
nowing the
effects of water particles turbulence around the floating break
water on bed scour
may be great importance for the understanding of scour problems posed by the
wave hydrodynamics and this may lead to the suggestions of proper mitigating
measures against the bed scour problems.
(g)
Stepped-slope feature of the STEPFLOAT are sufficiently good in intercepting
and dissipating wave energy effectively. hW ile the feature of the upper layer of
the floating breakwater remained, it is wo rthwhile to further investigate the
performance of the floating breakwater using different shapes for the bottom
layer of the floating breakw ater (such as the suggested shapes for the bottom
layer in Figure 8.1) in order to further enhance the performance and stability of
the structure. A hull shape bottom layer might also contribute to the wave
attenuation performance which worthwhile to have a trial in the laboratory.
233
Figure 8.1 : Cross-section of the suggested shapes for the bottom layer of the
STEPFLOAT
(h)
As a way to improve the efficiency of the STEPFLOAT system, an additional
indigenous design unit, i.e. wave screen or seaweed curtain as in Figure 3.8, is
recommended to be incorporated into the STEPFLOAT system as an option. The
artificial seaweeds will be fixed to the bottom of the STEPFLOAT and extended
into the water acting as a sheet of ‘grass’ keel or curtain to further reduce the
wave transmission as well as to minimize the oscillating currents beneath the
structure. The idea can also minimize the cross sectional area of the floating
breakwater immersed under water, thus reducing the construction costs.
(i)
The equations presented could be further expanded by conducting additional tests
investigating additional variables relating to the motions and mooring system.
(j)
Laboratory experimental data should be verified by data from the prototype tests
in the field. As there are more parameters that might probably affect the
efficiency of the STEPFLOAT in the field compared to the controlled conditions
in the laboratory, it is expected that there will be some discrepancies of results
between laboratory study and prototype test.
(k)
W
hile physical modeling provides the most
reliable estimates of the performance
of the STEPFLOAT, numerical modeling is useful in refining design of the
STEPFLOAT to an optimum degree prior to physical modeling test.
234
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APPENDIX A1
Results of the multiple linear regression analysis for a 2-row STEPFLOAT system
(a)
The Enter Method
b
Variables Entered/Removed
Variables
Variables
Model
Entered
Removed
a
1
.
HL, BDdL
a. All requested variables entered.
b. Dependent Variable: Ct
Method
Enter
b
Model Summary
Model
1
R
Adjusted R Std. Error of the
Square
Estimate
R Square
a
0.927
a. Predictors: (Constant), HL, BDdL
b. Dependent Variable: Ct
0.86
0.854
0.048185
b
ANOVA
Model
1
Sum of Squares
0.600
Regression
0.098
Residual
0.698
Total
a. Predictors: Constant, HL, BDdL
b. Dependent Variable: Ct
df
Mean Square
2
0.300
42
0.002
44
F
129.259
Sig.
a
0.000
a
Coefficients
Unstandardized
Coefficients
B
Std. Error
Model
0.792
0.022
1
Constant
-12.190
1.224
BDdL
-1.000
0.414
HL
a. Dependent Variable: Ct
Standardized
Coefficients
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Beta
Bound Bound Tolerance VIF
35.974 0.000
0.748
0.837
-0.787 -9.959 0.000 -14.660 -9.720
0.533 1.875
-0.191 -2.414 0.020 -1.836 -0.164
0.533 1.875
t
Sig.
240
(b)
The Stepwise Method
a
Variables Entered/Removed
Variables Variables
Model
Entered
Removed
1
BDdL
.
2
HL
.
a. Dependent Variable: Ct
Method
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
c
Model Summary
Model
1
Adjusted R Std. Error of the
R Square
Square
Estimate
R
a
0.917
0.841
b
2
0.927
0.860
a. Predictors: Constant, BDdL
b. Predictors: (Constant), BDdL, HL
c. Dependent Variable: Ct
ANOVA
Model
1
0.837
0.854
0.050817
0.048185
c
Sum of Squares
Regression
0.587
Residual
0.111
0.698
Total
Regression
0.600
2
Residual
0.098
0.698
Total
a. Predictors: Constant, BDdL
b. Predictors: Constant, BDdL, HL
c. Dependent Variable: Ct
df
1
43
44
2
42
44
Mean Square
0.587
0.003
F
227.188
Sig.
a
0.000
0.300
0.002
129.259
0.000
b
a
Coefficients
Unstandardized
Coefficients
B
Std. Error
Model
1 Constant
0.794
0.023
BDdL
-14.209
0.943
2 Constant
0.792
0.022
BDdL
-12.190
1.224
HL
-1.000
0.414
a. Dependent Variable: Ct
Standardized
Coefficients
t
Sig.
Beta
34.172
-0.917 -15.073
35.974
-0.787 -9.959
-0.191 -2.414
0.000
0.000
0.000
0.000
0.020
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Bound Bound Tolerance VIF
0.747 0.840
-16.110 -12.308
1.000 1.000
0.748 0.837
-14.660 -9.720
0.533 1.875
-1.836 -0.164
0.533 1.875
241
b
Excluded Variables
Collinearity Statistics
Minimum
Partial
VIF
Tolerance
Sig.
Correlation Tolerance
0.020
-0.349
0.533
1.875
0.533
Model
1
Beta In
t
a
HL
-0.191
-2.414
a. Predictors in the Model: Constant, BDdL
b. Dependent Variable: Ct
Pearson Correlations
BDdL
Ct
Pearson Correlation
Sig. (1-tailed)
N
Partial Correlation
Control Variables
HL
Ct
BDdL
Control Variables
BDdL
Ct
HL
Ct
BDdL
HL
Ct
BDdL
HL
Ct
BDdL
HL
HL
1.000
-0.917
-0.728
-0.917
1.000
0.683
-0.728
0.683
1.000
.
0.000
0.000
0.000
.
0.000
0.000
0.000
.
45
45
45
45
45
45
45
45
45
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.838
0.000
42
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.349
0.020
42
BDdL
-0.838
0.000
42
1
.
0
HL
-0.349
0.020
42
1
.
0
242
APPENDIX A2
Results of the multiple linear regression analysis for a 3-row STEPFLOAT system
(a)
The Enter Method
b
Variables Entered/Removed
Variables
Variables
Model
Entered
Removed
a
1
.
HL, BDdL
Method
Enter
a. All requested variables entered.
b. Dependent Variable: Ct
b
Model Summary
Model
1
R
R Square
a
0.941
0.970
a. Predictors: (Constant), HL, BDdL
b. Dependent Variable: Ct
Adjusted R Std. Error of
Square
the Estimate
0.938
0.0387161
b
ANOVA
Model
1
Sum of Squares
1.004
Regression
0.063
Residual
1.067
Total
a. Predictors: Constant, HL, BDdL
b. Dependent Variable: Ct
df
Mean Square
2
0.502
42
0.001
44
F
334.835
Sig.
a
0.000
a
Coefficients
Unstandardized Standardized
Coefficients
Coefficients
B Std. Error
Model
0.018
1
Constant 0.863
-9.061
0.723
BDdL
-2.332
0.368
HL
a. Dependent Variable: Ct
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Beta
Bound Bound Tolerance VIF
48.800 0.000
0.828
0.899
-0.687 -12.535 0.000 -10.520 -7.602
0.468 2.137
-0.348 -6.343 0.000 -3.073 -1.590
0.468 2.137
t
Sig.
243
(b)
The Stepwise Method
a
Variables Entered/Removed
Variables Variables
Model
Entered
Removed
1
BDdL
.
2
HL
.
Method
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
a. Dependent Variable: Ct
c
Model Summary
Model
1
Adjusted R Std. Error of the
R Square
Square
Estimate
R
a
0.884
0.940
b
2
0.970
0.941
a. Predictors: Constant, BDdL
b. Predictors: (Constant), BDdL, HL
c. Dependent Variable: Ct
0.882
0.938
0.0535415
0.0387161
c
ANOVA
Model
1
Sum of Squares
0.943
Regression
0.123
Residual
1.067
Total
1.004
2
Regression
0.063
Residual
1.067
Total
a. Predictors: Constant, BDdL
b. Predictors: Constant, BDdL, HL
c. Dependent Variable: Ct
df
1
43
44
2
42
44
Mean Square
0.943
0.003
F
329.118
Sig.
a
0.000
0.502
0.001
334.835
0.000
b
a
Coefficients
Unstandardized
Coefficients
B
Std. Error
Model
1 Constant
0.861
0.024
BDdL
-12.406
0.684
2 Constant
0.863
0.018
BDdL
-9.061
0.723
HL
-2.332
0.368
a. Dependent Variable: Ct
Standardized
Coefficients
t
Sig.
Beta
35.211
-0.940 -18.142
48.800
-0.687 -12.535
-0.348 -6.343
0.000
0.000
0.000
0.000
0.000
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Bound Bound Tolerance VIF
0.812 0.911
-13.785 -11.027
1.000 1.000
0.828 0.899
-10.520 -7.602
0.468 2.137
-3.073 -1.590
0.468 2.137
244
b
Excluded Variables
Collinearity Statistics
Minimum
Partial
VIF
Tolerance
Sig. Correlation Tolerance
0.000
-0.699
0.468
2.137
0.468
Model
1
Beta In
t
a
HL
-0.348 -6.343
a. Predictors in the Model: Constant, BDdL
b. Dependent Variable: Ct
Pearson Correlations
BDdL
Ct
Pearson Correlation
Sig. (1-tailed)
N
Partial Correlation
Control Variables
HL
Ct
BDdL
Control Variables
BDdL
Ct
HL
Ct
BDdL
HL
Ct
BDdL
HL
Ct
BDdL
HL
HL
1.000
-0.940
-0.849
-0.940
1.000
0.729
-0.849
0.729
1.000
.
0.000
0.000
0.000
.
0.000
0.000
0.000
.
45
45
45
45
45
45
45
45
45
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.888
0.000
42
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.699
0.000
42
BDdL
-0.888
0.000
42
1
.
0
HL
-0.699
0.000
42
1
.
0
245
APPENDIX A3
Results of the multiple linear regression analysis for a G = b STEPFLOAT system
(a)
The Enter Method
b
Variables Entered/Removed
Variables
Variables
Model
Entered
Removed
a
.
HL, BDGdL2
1
a. All requested variables entered.
b. Dependent Variable: Ct
Method
Enter
b
Model Summary
Model
1
R
Adjusted R Std. Error of
Square
the Estimate
R Square
a
0.912
0.955
a. Predictors: (Constant), HL, BDGdL2
b. Dependent Variable: Ct
0.908
0.0406260
b
ANOVA
Model
1
Sum of Squares
0.720
Regression
0.069
Residual
0.789
Total
a. Predictors: Constant, HL, BDGdL2
b. Dependent Variable: Ct
df
Mean Square
2
0.360
42
0.002
44
F
218.088
Sig.
a
0.000
a
Coefficients
Unstandardized
Coefficients
B
Std. Error
Model
0.630
0.014
1
Constant
4.001
BDGdL2 -58.952
-0.439
0.370
HL
a. Dependent Variable: Ct
Standardized
Coefficients
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Beta
Bound Bound Tolerance VIF
44.024 0.000
0.601
0.659
-0.905 -14.734 0.000 -67.026 -50.878
0.555 1.803
-0.073 -1.189 0.241 -1.185
0.306
0.555 1.803
t
Sig.
246
(b)
The Stepwise Method
a
Variables Entered/Removed
Variables Variables
Model
Entered
Removed
1
BDGdL2
.
a. Dependent Variable: Ct
Method
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
b
Model Summary
Model
1
R
R Square
Adjusted R Std. Error of the
Square
Estimate
a
0.909
0.954
a. Predictors: Constant, BDGdL2
b. Dependent Variable: Ct
0.907
0.0408208
b
ANOVA
Model
1
Sum of Squares
Regression
Residual
Total
a. Predictors: Constant, BDGdL2
b. Dependent Variable: Ct
df
0.718
0.072
0.789
Mean Square
1
43
44
F
0.718
0.002
Sig.
430.624
0.000
a
a
Coefficients
Standardized
Coefficients
Unstandardized
Coefficients
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Beta
Bound Bound Tolerance VIF
55.232 0.000
0.597 0.642
-0.954 -20.751 0.000 -68.164 -56.089
1.000 1.000
B
Std. Error
Model
1 Constant
0.620
0.011
BDGdL2 -62.127
2.994
a. Dependent Variable: Ct
t
Sig.
b
Excluded Variables
Model
1
Beta In
HL
t
a
Sig.
-0.073 -1.189
0.241
a. Predictors in the Model: Constant, BDGdL2
b. Dependent Variable: Ct
Collinearity Statistics
Minimum
Partial
Tolerance
VIF
Tolerance
Correlation
-0.180
0.555
1.803
0.555
247
Pearson Correlations
BDGdL2
Ct
Pearson Correlation
Sig. (1-tailed)
N
Ct
BDGdL2
HL
Ct
BDGdL2
HL
Ct
BDGdL2
HL
Partial Correlation
Control Variables
HL
Ct
BDGdL2
Control Variables
BDGdL2
Ct
HL
HL
1.000
-0.954
-0.677
-0.954
1.000
0.667
-0.677
0.667
1.000
.
0.000
0.000
0.000
.
0.000
0.000
0.000
.
45
45
45
45
45
45
45
45
45
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.915
0.000
42
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.180
0.241
42
BDGdL2
-0.915
0.000
42
1
.
0
HL
-0.180
0.241
42
1
.
0
248
APPENDIX A4
Results of the multiple linear regression analysis for a G = 2b STEPFLOAT system
(a)
The Enter Method
b
Variables Entered/Removed
Variables
Variables
Model
Entered
Removed
HL,
.
1
a. All requested variables entered.
b. Dependent Variable: Ct
Method
Enter
b
Model Summary
Model
1
R
Adjusted R Std. Error of
Square
the Estimate
R Square
a
0.942
0.971
a. Predictors: (Constant), HL, BDGdL2
b. Dependent Variable: Ct
0.939
0.0393489
b
ANOVA
Model
1
Sum of Squares
1.055
Regression
0.065
Residual
1.120
Total
a. Predictors: Constant, HL, BDGdL2
b. Dependent Variable: Ct
df
Mean Square
2
0.528
42
0.002
44
F
340.721
Sig.
a
0.000
a
Coefficients
Unstandardized
Coefficients
B
Std. Error
Model
0.692
0.013
1 Constant
1.528
BDGdL2 -24.395
-1.198
0.346
HL
a. Dependent Variable: Ct
Standardized
Coefficients
95%
Collinearity
Confidence
Statistics
Lower Upper
Beta
Bound Bound Tolerance VIF
52.436 0.000 0.665 0.719
-0.835 -15.969 0.000 -27.478 -21.312
0.506 1.976
-0.181 -3.465 0.001 -1.896 -0.500
0.506 1.976
t
Sig.
249
(b)
The Stepwise Method
a
Variables Entered/Removed
Variables Variables
Model
Entered
Removed
1
BDGdL2
.
HL
a. Dependent Variable: Ct
.
Method
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
Stepwise (Criteria: Probability-of-F-to-enter <= .050,
Probability-of-F-to-remove >= .100).
c
Model Summary
Model
1
Adjusted R Std. Error of the
R Square
Square
Estimate
R
a
0.925
0.962
b
2
0.971
0.942
a. Predictors: Constant, BDGdL2
b. Predictors: Constant, BDGdL2, HL
c. Dependent Variable: Ct
0.924
0.939
0.0440984
0.0393489
c
ANOVA
Model
1
Sum of Squares
1.037
Regression
0.084
Residual
1.120
Total
1.055
2
Regression
0.065
Residual
1.120
Total
a. Predictors: Constant, BDGdL2
b. Predictors: Constant, BDGdL2, HL
c. Dependent Variable: Ct
df
1
43
44
2
42
44
Mean Square
1.037
0.002
F
532.999
Sig.
a
0.000
0.528
0.002
340.721
0.000
b
a
Coefficients
Unstandardized
Coefficients
B
Std. Error
Model
1 Constant
0.666
0.012
BDGdL2 -28.116
1.218
2 Constant
0.692
0.013
BDGdL2 -24.395
1.528
HL
-1.198
0.346
a. Dependent Variable: Ct
Standardized
Coefficients
t
Sig.
Beta
54.944
-0.962 -23.087
52.436
-0.835 -15.969
-0.181 -3.465
0.000
0.000
0.000
0.000
0.001
95% Confidence
Collinearity
Interval for B
Statistics
Lower Upper
Bound Bound Tolerance VIF
0.641 0.690
-30.572 -25.660
1.000 1.000
0.665 0.719
-27.478 -21.312
0.506 1.976
-1.896 -0.500
0.506 1.976
250
b
Excluded Variables
Collinearity Statistics
Minimum
Partial
VIF
Tolerance
Model
Beta In
t
Sig. Correlation Tolerance
a
1
HL
0.001
-0.472
0.506
1.976
0.506
-0.181 -3.465
a. Predictors in the Model: Constant, BDGdL2
b. Dependent Variable: Ct
Pearson Correlations
BDGdL2
Ct
Pearson Correlation
Sig. (1-tailed)
N
Ct
BDGdL2
HL
Ct
BDGdL2
HL
Ct
BDGdL2
HL
Partial Correlation
Control Variables
HL
Ct
BDGdL2
Control Variables
BDGdL2
Ct
HL
HL
1.000
-0.962
-0.768
-0.962
1.000
0.703
-0.768
0.703
1.000
.
0.000
0.000
0.000
.
0.000
0.000
0.000
.
45
45
45
45
45
45
45
45
45
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.927
0.000
42
Ct
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
1
.
0
-0.472
0.001
42
BDGdL2
-0.927
0.000
42
1
.
0
HL
-0.472
0.001
42
1
.
0