Development of simplified formulas to determine …...Development of simplified formulas to...

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Development of simplified formulas to determine wave induced loads on inland vessels operated in stretches of

water within the range of navigation IN(0.6 ≤ x ≤ 2)

Gian Carlo Matheus Torres

Master Thesis

presented in partial fulfillment of the requirements for the double degree:

“Advanced Master in Naval Architecture” conferred by University of Liege "Master of Sciences in Applied Mechanics, specialization in Hydrodynamics,

Energetics and Propulsion” conferred by Ecole Centrale de Nantes

developed at West Pomeranian University of Technology, Szczecin in the framework of the

“EMSHIP” Erasmus Mundus Master Course

in “Integrated Advanced Ship Design”

Ref. 159652-1-2009-1-BE-ERA MUNDUS-EMMC

Supervisors:

- Dr. Maciej Taczała, West Pomeranian University of Technology, Szczecin

- Dr. Nzengu Wa Nzengu. Bureau Veritas Inland Navigation. Antwerp.

Reviewer: - Prof. Robert Bronsart, University of Rostock

Szczecin, February 2017

P 2 Gian Carlo Matheus Torres

Master Thesis developed at West Pomeranian University of Technology, Szczecin

ABSTRACT

It is of major importance for design purposes that long-term effects coming from the sea are accurately

predicted by classification societies, which must not be exceeded during the lifetime operation of a

given vessel. It is encountered that Inland-Navigation Bureau Veritas rules (NR247) do not assess the

seakeeping behaviour and do not predict sea loads adequately. To deal with this, a development of

empirical-formulas was carried to account for inland-vessel responses when operated in stretches of

water within the range of navigation of 0.6 m ≤ Hs ≤ 2 m, based on linear potential-flow theory,

boundary element method and the 3D linear panel method theory. A frequency-domain linear analysis

for an impossed constant speed of 10 knots is taken into account. Using the Belgian Coastal Scatter

Diagram, it was found the hydrodynamic long-term response for a return period of 17 years, composed

of motions, accelerations, relative elevation, shear forces and bending moments; encountered in a set

of 46 inland vessels. Then, by mean of a regression process, a set of empirical equations was proposed

accounting for these effects. They were validated against an additional set of 13 direct-calculation

results, showing good agreement. Finally, to take into account discrepancies originated from linear-

theory assumptions, proposed empirical models are corrected by taking into account nonlinear

hydrodynamic effects.

Keywords: Inland Navigation, Seakeeping, Bending Moment, Potential Flow, Rule-formulas, Linear

Analysis, RAO, Long-term value, Statistics, Regression.

Development of simplified formulas to determine wave induced loads on inland vessels operated in

stretches of water within the range of navigation IN(0.6 ≤ x ≤ 2)

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“EMSHIP” Erasmus Mundus Master Course, period of study September 2015 – February 2017

To my fundamental pillar, Jesus Christ. To my Italian-Venezuelan Family, and my Belgian Family.

To my dear beloved fiancé: Camille.



DECLARATION OF AUTHORSHIP

I declare that this thesis and the work presented in it are my own and have been generated by me as

the result of my own original research.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With the exception of such

quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have made clear exactly what

was done by others and what I have contributed myself.

This thesis contains no material that has been submitted previously, in whole or in part, for the award

of any other academic degree or diploma.

I cede copyright of the thesis in favour of the West Pomeranian University of Technology.

Date: Signature



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ACKNOWLEDGEMENTS

This thesis was developed in the frame of the European Master Course in “Integrated Advanced Ship

Design” named “EMSHIP” for “European Education in Advanced Ship Design”, Ref.: 159652-1-

2009-1-BE-ERA MUNDUS-EMMC.



CONTENTS

ABSTRACT

LIST OF FIGURES 10

LIST OF TABLES 12

NOMENCLATURE 13

1. INTRODUCTION 14

1.1. General................................................................................................................................. 14

1.2. Objectives............................................................................................................................. 15

1.3. Steps..................................................................................................................................... 15

2. BUREAU-VERITAS APPROACH 16

2.1 Definition of Range of Navigation according BV-NR217 rules............................................. 16

2.1.2. Range of Navigation..................................................................................................... 16

2.2 Vessel Motion and Acceleration............................................................................................ 18

2.2.1 General considerations................................................................................................. 18

2.2.2 Vessel Motion and Acceleration.................................................................................... 18

2.2.3 Vessel relative motions................................................................................................. 21

2.3. Loading conditions............................................................................................................... 21

2.3.1. Lightship........................................................................................................................ 21

2.3.2 Fully loaded vessel 22

2.4. Issues encountered when assessing BV NR217 rules........................................................... 22

2.4.1 Rudakovic (2015) results.............................................................................................. 22

2.4.2. Rudakovic (2015) conclusion........................................................................................ 24

3. PROBLEM INSIGHT 25

3.1. HYDROSTAR..................................................................................................................... 25

3.1.1. Panel Method................................................................................................................. 25

3.2. Belgian coastal Scatter Diagram........................................................................................... 26

3.3. Conventions......................................................................................................................... 27

3.4. Return Period........................................................................................................................ 28

3.5. Range of Navigation to be accounted................................................................................... 29

3.6. Additional features to be considered..................................................................................... 29

3.7. Investigated vessel within the study...................................................................................... 30

4. THEORETICAL FRAMEWORK 33



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4.1. Scheme of study.................................................................................................................... 33

4.2. Statistical Representation of the Sea Surface........................................................................ 34

4.3. Regular gravity harmonic waves.......................................................................................... 35

4.3.1. Wave speed.................................................................................................................. 35

4.3.2 Wave Steepness............................................................................................................. 35

4.3.3. Influence of wave steepness......................................................................................... 36

4.3.4. Laplace condition......................................................................................................... 36

4.3.5. Wave potential and wave surface elevation.................................................................. 37

4.3.6. Boundary Conditions (Boundary Element Methods)................................................... 37

4.3.7. Dispersion Relation...................................................................................................... 41

4.3.8. Total Wave Energy....................................................................................................... 42

4.4. Irregular Waves.................................................................................................................... 42

4.4.1. Statistics....................................................................................................................... 42

4.4.2. Sea-state definition....................................................................................................... 44

4.4.3. Wave Energy Spectra........................................................................................... 45

4.4.4. Wave Height and Period definition from Wave Energy Spectra........................... 45

4.4.5. Rayleigh Distribution of wave peaks.................................................................... 46

4.4.6. JONSWAP Wave Spectra.................................................................................... 48

4.4.7. Directional Spreading.......................................................................................... 48

4.5. Linear Response to first order excitation.............................................................................. 49

4.5.1. Linear Potential Theory: Linear Mass-Spring System................................................. 49

4.5.2. Frames of reference..................................................................................................... 50

4.5.3. Motions of the ship....................................................................................................... 51

4.5.4. Plane of symmetry........................................................................................................ 52

4.5.5. Wave encounter frequency........................................................................................... 52

4.5.6. Loads Superposition..................................................................................................... 53

4.5.7. Boundary Conditions describing linear ship-wave interaction.................................... 54

4.5.8. Forces and Moments on the hull wetted surface S........................................................ 56

4.5.9. Hydrodynamic Loads................................................................................................... 57

4.5.10. Wave and Diffraction Loads....................................................................................... 59

4.5.11. Hydrostatic restoring loads........................................................................................ 60

4.5.12. Correction of the Spring Matrix due to free-surface effects........................................ 61

4.5.13. Linear Harmonic Ship Response to regular wave excitations.................................... 63



4.5.14. Coupled Equations of Motion..................................................................................... 63

4.5.15. Relative Wave Elevation............................................................................................ 64

4.5.16. Total Inertia matrix of the ship................................................................................... 64

4.5.17. Response Amplitude Operator................................................................................... 65

4.6. Response in Irregular Waves................................................................................................ 65

4.6.1. Response Amplitude and Period of a regular-response spectrum................................ 66

4.6.2. Maximum short-term linear response........................................................................... 67

4.6.3 Long Term Statistics..................................................................................................... 67

4.7. Higher order analysis............................................................................................................ 68

4.7.1. Source of higher order effects in the Belgian Coastal Scatter Diagram....................... 69

4.7.2. Additional Effects to be taken into account.................................................................. 70

4.7.3 General conclusions. Additional effects to consider in linear analysis.......................... 78

5. DIRECT CALCULATIONS USING HYDROSTAR CODE 80

5.1. Methodology........................................................................................................................ 80

5.2. ARGOS................................................................................................................................ 81

5.2.2. Hydrostatic Particulars................................................................................................ 81

5.3. Hydrostar reference system.................................................................................................. 82

5.4. Mesh Generation.................................................................................................................. 82

5.4.1. Input to .hsmsh............................................................................................................. 82

5.4.2. Output of .hsmsh file.................................................................................................... 83

5.5. Additional Hydrostatic computation: Loading conditions.................................................. 86

5.5.1. hstat input..................................................................................................................... 86

5.5.2. hstat output................................................................................................................... 88

5.6. Diffraction radiation computation........................................................................................ 90

5.6.1. Radiation solution........................................................................................................ 90

5.6.2. The diffraction solution................................................................................................ 91

5.6.3. hsrdf input.................................................................................................................... 91

5.6.4. hsrdf output.................................................................................................................. 91

5.7. Motion Computation............................................................................................................ 91

5.7.1. hsmcn input.................................................................................................................. 92

5.7.2. hsmcn output................................................................................................................ 92

5.8.1. hsprs input.................................................................................................................... 93

5.8.2. hsprs output.................................................................................................................. 93



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5.9. Construction of the transfer functions............................................................................. 94

5.10. Long term value for a given response................................................................................. 97

5.10.1. hspec input................................................................................................................. 97

5.10.2. hspec output............................................................................................................... 97

6. EMPIRICAL FORMULAS 99

6.1. General Scheme.................................................................................................................... 99

6.2. Developing the problem....................................................................................................... 99

6.3. Characteristis of the long-term-response.............................................................................. 100

6.4. Methodology........................................................................................................................ 103

6.5. Proposed Equations................................................................................................................ 108

6.5.1. Sway Acceleration, in [m/s2]....................................................................................... 108

6.5.2. Surge Acceleration, in [m/s2]...................................................................................... 109

6.5.3. Heave Acceleration, in [m/s2]..................................................................................... 109

6.5.4. Pitch Amplitude, in [rad].............................................................................................. 110

6.5.5. Pitch Acceleration, in [rad/s2]..................................................................................... 110

6.5.6. Pitch Period, in [s]....................................................................................................... 110

6.5.7 Vessel relative motion.................................................................................................. 110

6.6 Nonlinear Correction............................................................................................................ 111

6.6.1 Heave Nonlinear Coefficient....................................................................................... 112

6.6.2 Pitch Nonlinear Coefficient......................................................................................... 112

6.6.3 Relative Motion Nonlinear Coefficient........................................................................ 112

6.6.4 Heave Nonlinear Coefficient....................................................................................... 112

6.6.5 Pitch Forward Speed Coefficient................................................................................. 112

6.6.6 Relative Motion Forward Speed Coefficient............................................................... 112

6.6.7 Heave Experimental Coefficient.................................................................................. 112

6.6.8 Total Corrections......................................................................................................... 113

7. CONCLUSIONS 114

8. REFERENCES 116

APPENDIX A. TENDIENCIES. 118

APPENDIX B. VALIDATIONS. 127



LIST OF FIGURES

Figure 1. Direct Calculation compared to NR 217 for 85% of the lifetime of the vessel.................... 23

Figure 2. Direct Calculation compared to NR 217 for 85% of the lifetime of the vessel.................... 23

Figure 3. 3-D Representation of the Hull Form of a Crude Oil Carrier.............................................. 26

Figure 4. Belgium seaway spectrum up to HS = 2.0 m...................................................................... 26

Figure 5: Waves direction βgeo......................................................................................................... 27

Figure 6: Azimuth α’......................................................................................................................... 27

Figure 7. Belgium coastal area where scatter diagram belongs......................................................... 28

Figure 8. A typical tank vessel for inland navigation......................................................................... 31

Figure 9. General arrangement plan of a typical IN Tanker............................................................... 32

Figure 10. A sum of many simple sinusoidal waves makes an Irregular Sea..................................... 34

Figure 11. Regular-wave reference system for regular-harmonic wave definitions.......................... 35

Figure 12. Wave steepness influence................................................................................................ 36

Figure 13. Sea Bed boundary condition............................................................................................ 38

Figure 14. Dynamic free surface boundary condition........................................................................ 39

Figure 15. Kinematic Boundary Condition....................................................................................... 40

Figure 16. Schematic of the ocean-surface elevation probability density function............................ 42

Figure 17. Schematic of a random ergodic process........................................................................... 44

Figure 18. Ocean wave elevation sampling at a specific location...................................................... 44

Figure 19. Rayleigh Distribution....................................................................................................... 47

Figure 20. Probability of exceeding.................................................................................................. 47

Figure 21. Wave Spectra with Directional Spreading........................................................................ 49

Figure 22. Block Diagram for a Linear System. Linear relation between Motions and Waves.......... 50

Figure 23. Coordinate Systems......................................................................................................... 51

Figure 24. Mode of motions of a ship. Steadily translating and body-bound reference systems........ 51

Figure 25. Frequency of Encounter.................................................................................................. 52

Figure 26. Schematic of a linear superposition of hydromechanical and exciting forces................... 53

Figure 27. Direction cosines for a normal vector n located on the wetted hull surface.................... 55

Figure 28. Green’s second theorem for a cylinder of surface S*........................................................ 56

Figure 29. Metacentric Height reduction caused by free surface effects in wall-sided tanks............. 62

Figure 30. Schematic of GZ-Curve, corrected for free surface effects within tanks.......................... 62



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Figure 31. Scheme of available approaches to study a ship response under sea influence................. 69

Figure 32. Belgium seaway spectrum up to HS = 2.0 m..................................................................... 70

Figure 33. Roll damping coefficient (in percentage of the critical damping)..................................... 71

Figure 34. Probability Distribution. Irregular wave in model test (Tp = 12 s, Hs = 11.5 m)............... 73

Figure 35. Heave motions of ship with/without forward speed......................................................... 74

Figure 36. Pitch motions of ship with/without forward speed........................................................... 75

Figure 37. Bending moment from waves. Ship with/without forward speed..................................... 76

Figure 38. 3-hour extreme value with different forward speeds........................................................ 77

Figure 39. RAO’s of Roll and Pitch of a Containership..................................................................... 78

Figure 40. Schematic of non-linear correction to be considered........................................................ 79

Figure 41. ARGOS modules............................................................................................................. 81

Figure 42. Hydrostar reference system.............................................................................................. 82

Figure 43. Coordinate system (Obl , Xbl , Ybl , Zbl) used to represent body-lines in .hul file........... 83

Figure 44. Body-lines of Vessel 43, at different sections................................................................... 84

Figure 45. Vessel 45 mesh. LOA = 135 m; B = 22.5 m. Nlongm; Ntrans.......................................... 85

Figure 46. Hschk module report........................................................................................................ 85

Figure 47. Loading distribution for Vessel 20 at maximum draught (fully loaded)........................... 87

Figure 48. Total length scheme of the cargo-midship section of Vessel 20....................................... 87

Figure 49. Fully-loaded conditions for Vessel 20. Hydrostatic difference with hslec module........... 89

Figure 50. Correction of hydrostatic values from input loading conditions. Vessel 20...................... 90

Figure 51. Vessel 43. Quadratic roll damping estimation using hsdmp............................................. 92

Figure 52. Proposed locations to assess relative wave elevation, wave loads and VBM.................... 94

Figure 53. Ship system of reference relative to wave angle incidence............................................... 95

Figure 54. Roll-acceleration transfer function for Vessel 20............................................................. 96

Figure 55. Relative wave elevation transfer function for Friendship, at CARGO_AP...................... 96

Figure 56. Heave acceleration long-term-value of Vessel 20............................................................ 98

Figure 57. Pitch acceleration long-term-value of Vessel 20.............................................................. 98

Figure 58. Vessel 17 IN vessel. Long-term values............................................................................ 100

Figure 59. Vessel 31 IN vessel. Long-term values............................................................................ 101

Figure 60. Deviation from Hs maximum assumed in Table 3............................................................ 102

Figure 61. Heave acceleration vs aB. Range of slope values is assessed........................................... 103

Figure 62. 11 x 18 matrix combining the proposed slope vector with ship’s features........................ 104

Figure 63. Surge acceleration vs combination: aB·B. Linear behaviour is verified........................... 105



LIST OF TABLES

Table 1. Values of wave height H. 17

Table 2. Characteristics of vessels used in Rudakovic (2015) research........................................... 22

Table 3. Range of Navigation under investigation........................................................................... 29

Table 4. Set of Inland Navigation vessels to be studied in the present work................................... 30

Table 5. Irregular sea state characteristics. Guo et al. (2016).......................................................... 72

Table 6. LNG tanker characteristics at maximum draft. Bingjie Guo et al. (2016)......................... 73

Table 7. Deviation linear method extreme values with respect to the nonlinear ones at V=0 kn.... 76

Table 8. Characteristics of Vessel 17 and Vessel 31. Inland Navigation Vessels............................ 101

Table 9. Characteristics of Vessel 17 and Vessel 31. Inland Navigation Vessels............................ 105



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NOMENCLATURE

Usual Designation Symbol

Rule length L

Length overall LOA

Draught T

Depth D

Displacement Δ

Density ρ

Concentrated loads P

Linearly distributed loads q

Surface distributed loads (pressure) p

Bending moment M

Stresses σ, τ

Vessel speed V



1. INTRODUCTION

1.1. General

Ocean surface waves cause periodic loads on all sorts of man-made structures in the sea. It does not

matter whether these structures are fixed or floating on the surface or deeper in the sea. Most of the

offshore structures including ships respond in a specific way to the wave-induced periodic loads. This

(dynamic) response includes accelerations, harmonic displacements and internal loads. Effects as

added resistance to the advance, reduced sustained speed (with associated longer travel times), increase

of fuel consumption and shipping water on deck, are consequence of the ship-waves interaction on the

same hand.

When designing a vessel, it is of interest for a naval architect to rely on an adequate ‘design response’,

namely, an extreme dynamic internal load or displacement with an associated (small) probability that

will be exceeded, which come from a previous dynamic analysis or directly from classification

societies rules. It serves as an adequate input to undertake FE model analysis, so structure strength and

intrinsic behaviour can be studied accordingly on a solid basis. Thus, it is of major importance that

loads coming from the sea are accurately predicted by classification societies and should not be

exceeded during the lifetime operation of a given vessel.

According to the Common Structural Rules for Bulk Carriers (IACS, 2006a) and the Common

Structural Rules for Double Hull Oil Tankers (IACS, 2006b), when assessing hull girder strength

against extreme loads that are overtaken only once during the ship’s lifetime, long-term loads should

be provided, based on: fully-loaded conditions at design draft and normal ballast loading conditions at

light draft. This study will be carried out taking into account only fully loaded conditions to create a

data base of 46-ships modelled as done in the work of S. Rudakovic (2015), using linear potential

flow theory on Hydrostar code (belonging to Bureau Veritas), which accounts for the boundary

element methods and 3D linear panel method theory. Belgian coastal Wave Spectrum, provided by the

institute of Oceanography located in Ostend, will be used in order to obtain a linear extreme response

value for each mode of motion and for internal loads of a given inland navigation vessel. It will serve

to assess the accuracy of the current BV NR217 rules and to propose a set of new empirical formulas

as an alternative to correct current rules. Finally, to compensate for errors of linear theory caused by

truncating the pressure distribution at the still-water level, proposed empirical models are corrected by

taking into account nonlinear hydrodynamic effects.



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1.2. Objectives

I. Obtaining long-term responses for a set of 46 vessels simulated under fully loaded conditions,

based on linear potential flow theory. It includes amplitudes, velocities and accelerations acting

on and about the centre of gravity; and the relative elevation, shear forces and bending moments

exerting influence on specific ship-hull locations.

II. Propose empirical formulas accounting for the main motions and accelerations parameters,

including relative motions.

III. Correct the empirical formulas taking into account nonlinear effects and forward speed effects.

1.3. Steps

1) Definition of the approach encountered in BV NR217 rules:

a) Range of Navigation.

b) Seakeeping behaviour: vessel motions, accelerations and relative motion.

c) Loading conditions.

d) Present the issues encountered and capacity of prediction of BV NR217 rules.

2) Problem insight

3) Present the theoretical framework of all phenomena involved in the seakeeping analysis

a) Sea-waves mechanics explained from potential flow theory.

b) Statistics of sea waves.

c) Hydromechanical ship response to wave induced excitation.

d) Statistics of the ship response: short and long-term statistics for a return period of 17 years.

4) Simulation methodology.

a) Meshing process with the help of ship bodylines and approved general arrangements.

b) Radiation/diffraction analysis, considering V = 10 knots.

c) Correction due to viscous effects on roll motion.

d) Calculations of motions, velocities and accelerations.

e) Definition of locations on the ship where loads and relative wave elevations are obtained.

f) Construction of first order transfer functions (RAO).

g) Response long-term values, when subjected to the Belgian Scatter Diagram conditions.

5) Regression process is undertaken to find relationship between results and ship’s main features.

6) Proposal of empirical formulas

7) Correction to empirical formulas due to nonlinear potential effects.



2. BUREAU-VERITAS APPROACH

Following it will be detailed all rules for Inland Navigation vessels that Bureau Veritas has, regarding

to the seakeeping mode of motion of a given vessel. Important parameters are defined as the Range of

Navigation definition according rules, wave height accounted within each range of navigation, bending

moment coming from the wave loads and motion and accelerations parameters. It is important to

correctly define the Range of Navigation of each vessel, as they direct influence the design procedure;

specifically on:

1. Local and global strength

2. Stability

3. Freeboard

4. Safety clearance

5. Equipment design

6. Steering system design.

Below are defined the important parameters to take into account for the present investigation:

2.1 Definition of Range of Navigation according BV-NR217 rules

2.1.1 Character IN (Inland Navigation)

It indicates the type of waters covered by BV Rules:

• all inland waterways;

• all restricted maritime stretches of water up to a significant wave height of 2 m;

• other waters showing comparable conditions.

Restricted maritime stretches of water Inland navigation vessels may operate in coastwise restricted

maritime stretches of water complying with the range of navigation specified above where allowed by

the competent National Authorities. Possible specific requirements of National Authorities for

operation in maritime stretches are to be complied with and take precedence on the present Rules in

case of conflict.



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2.1.2. Range of Navigation

Wave height H

The wave height H corresponding to the ranges of navigation are defined as shown in Table 1:

Table 1. Values of wave height H.

Range of Navigation Wave Height in [m]

IN(0) H = 0

IN(0.6) H = 0.6

IN(0.6 < x < 1.2) IN(0.6 < Hs < 1.2)

Maximum wave height, in m, for IN(0) and IN(0,6):

According to Part A, Chapter 1, Section 3, [11] of NR217 rules, the range of navigation IN(0) is

assigned to a vessel having a structure with scantlings deemed suitable to navigate on still and smooth

stretches of water. The range of navigation IN(0,6) is assigned to a vessel having a structure with

scantlings deemed suitable to navigate on stretches of water where there may be strong currents and a

certain roughness of the surface on which a maximum wave height of 0,6 m can develop.

Maximum significant wave height, in m, for IN(0,6 < x ≤ 2)

The significant wave height considered in the Rules corresponds to H1/3 which means the average of

33% of the total number of waves having the greater heights between wave trough and wave crest,

observed over a short period. In accordance to Part A, Chapter 1, Section 3, [11], the range IN(0,6

< x ≤ 2) is assigned to a vessel having structure scantlings and other design features deemed suitable

to navigate on stretches of water as estuaries, lakes and some restricted maritime stretches of water.

Navigation coefficient

The navigation coefficient is defined as:

𝑛 = 0.85 𝐻𝑠 (1)



Where:

𝑛 = navigation coefficient.

Hs = maximum significant wave height, in [m], as defined above.

Wave Bending Moment

In part Part B, Chapter 3, Section 2, Number 3 of Inland Navigation Rules, bending moment coming

from the waves is defined by taking into account the stream and water conditions in the navigation

zone is to be considered, except for range of navigation IN(0).

Range of navigation IN(0,6 < x ≤ 2)

For range of navigation IN(0,6 < x ≤ 2), the absolute value of the wave-induced bending moment

amidships is to be obtained, in kN.m, from the following formula:

MW = 0.021∙n∙C∙L2∙B∙(CB+0.7) (2)

2.2 Vessel Motion and Acceleration

In part Part B, Chapter 3, Section 3 of Inland Navigation Rules, seakeeping parameters according to

a vessel response depending on the wave height encountered in the proper wave scatter diagram,

following expressions are defined.

2.2.1 General considerations

Vessel motions and accelerations are assumed to be periodic. The motion amplitudes, defined by the

formulas of this section, are half of the crest to through amplitudes.

2.2.2 Vessel Motion and Acceleration

Wave parameter, in m:

hW = 11.44− | L− 250

110 |

3

(3)



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Motion and acceleration parameter:

aB = 0.33n (0.04V

√L+1.1

hW

L) (4)

Vessel absolute motions and accelerations

Surge. The surge acceleration aSU is to be taken equal to:

aSU = 0.5 𝑚 𝑠2⁄ (5)

Sway. The sway acceleration aSW, in m/s2, is obtained from the formula:

aSW = 7.6aB (6)

Heave. The heave acceleration aH is obtained, in m/s2, appliying the formula:

aH = 9.81aB (7)

Roll.

Roll amplitude AR:

AR = aB√E (8)

Where:

AR = roll amplitude, in [rad];

E = Parameter defined as:

E = 1.39GM

δ2

B (9)

GM = distance, in m, from the vessel’s centre of gravity to the transverse metacentre, for

the loading considered; when GM is not known, the following values may be assumed:

full load: GM = 0,07 B

lightship: GM = 0,18 B



Roll period:

TR = 2.2δ

√GM (10)

Where:

TR = roll period, in [s],

δ = roll radius of gyration, in m, for the loading considered; when δ is not known, the

following value may be assumed, in full load and lightship conditions: δ = 0,35B.

Roll Acceleration:

αR = AR (2𝜋

𝑇𝑅)2

(11)

Where:

αR = roll acceleration, in [m2/s].

Pitch. The pitch amplitude AP, period TP and acceleration αP are obtained from below

formulas

AP = 0.328aB (1.32−hW

L) (

0.6

CB

)0.75

(12)

Where:

AP = pitch amplitude, in [rad];

Pitch period:

TP = 0.575√L (13)

Where:

TR = roll period, in [s],

δ = roll radius of gyration, in m, for the loading considered; when δ is not known, the

following value may be assumed, in full load and lightship conditions: δ = 0,35B.

Pitch Acceleration:

αP = AP (2π

TP

)2

(14)



21


Where:

αR = roll acceleration, in [m2/s].

Yaw. The yaw acceleration αY is obtained, in rad/s2, from the following formula:

αY = 15.5aB

L (15)

2.2.3 Vessel relative motions

The vessel relative motion (relative surface elevation) h1 is to be taken equal to:

• for H ≤ 0.6:

h1 = 0.6 n (16)

• for H > 0.6:

h1 = 0.08∙n∙C∙(CB + 0.7) (17)

2.3. Loading conditions

The most relevant load cases from the large number of possible wave situations are taken into account

by choosing so-called dominant load parameters. They are specified by IACS and Bureau Veritas

Inland Navigation to expedite the analysis. For tankers, cargo and containerships, following loading

conditions must be considered:

2.3.1. Lightship

The vessel is assumed empty, without supplies nor ballast. For self-propelled cargo vessels and tank

vessels, the light standard loading conditions are: supplies: 100%, ballast tanks: 50%.



2.3.2 Fully loaded vessel

The vessel is considered to be homogeneously loaded at its maximum draught, without supplies nor

ballast. For self-propelled cargo vessels and tank vessels, the vessel is considered to be homogeneously

loaded at its maximum draught with 10% of supplies (without ballast). Only these conditions are taken

into account for the present study.

2.4. Issues encountered when assessing BV NR217 rules

Stefan Rudakovic carried out a research (Bureau Veritas, Antwerp, September 2015), regarding

direct calculations on a data base of 13 inland navigation vessels using Hydrostar software to obtain

the wave induced response extreme-value within range of navigation IN(0.6 ≤ x ≤ 2).

Table 2. Characteristics of vessels used in Rudakovic (2015) research.

Vessel's

Reference

L B Tmax Δ(Tmax) CB GM

[m] [m] [m] [ton] [-] [m]

1 65.95 10.50 3.45 2001.36 0.82 1.34

2 83.05 9.56 3.60 2730.14 0.93 0.31

3 82.95 9.40 3.07 2228.24 0.91 0.57

4 84.05 10.95 2.80 2300.89 0.87 1.28

5 84.05 11.40 4.30 3643.27 0.86 1.26

6 108.2 11.45 2.60 2862.11 0.87 3.24

7 108.2 11.40 2.46 2699.97 0.87 3.34

8 118.4 11.40 4.30 5389.68 0.91 0.51

9 122 11.40 4.50 5863.35 0.91 1.49

10 131.6 14.50 3.60 6447.47 0.92 3.43

11 133 14.50 4.00 7241.54 0.92 2.00

12 133 11.45 2.68 3820.68 0.91 3.31

13 133 11.45 3.33 4818.32 0.93 1.87

2.4.1 Rudakovic (2015) results

An adequate prediction of the vertical bending moment is appreciated in the figures presented in the

Fig. 1. Also, Rudakovic (2015) mentioned that “when it is necessary to have a reference value of the

accelerations, Bureau Veritas NR 217 does not predict accurately these values. It represent an issue

when given an adequate value for the machinery located within the vessel and electronic devices. No

adequate design is possible only based on these rules” (see Fig. 2).



23


Figure 1. Direct Calculation compared to NR 217 for 85% of the lifetime of the vessel (20 years) on waves.

Vertical Bending Moment for different vessels (from left to right): considering significant wave height of

Hs=0.6 m and 1.2 m.

Figure 2. Direct Calculation compared to NR 217 for 85% of the lifetime of the vessel (20 years) on waves.

Acceleration in (from above left to bottom right): Heave, Roll, Pitch and Surge.



2.4.2. Rudakovic (2015) conclusion

Following conclusions were derived:

“Results from this calculation can be used to modify and improve current inland navigation rules

BV NR 217”.

“Imposed speed at V = 10 kn for all the cases simulated is justified that in rough weather, forward

speed is involuntarily or voluntarily reduced, and in that case, using only this speed, results are on

a safer side”.

“Results from NR 217 rules when compared to direct calculation results for motions and

accelerations shows significant difference, in most cases underestimating real values. Furthermore,

in most cases, NR 217 formulas have adequate assumptions of forms of equations, but coefficients

should be modified”.

“Considered cases are only for fully loaded vessels”.



25


3. PROBLEM INSIGHT

The problem arises as NR217 formulas are not able to correctly predict the values which they are made

for. The work of Rudakovic (Bureau Veritas, 2015) will be extended in order to broad the data-base

of ships available in order to have enough data to undertake a proposal of empirical formulas that

allows the prediction of the seakeeping behaviour of any inland-navigation vessel.

Inland navigation vessels are scheduled to navigate in rivers, estuaries and coasts at maximum distance

of 5 NM. Big amount of these kind of vessels spend their time of operation sailing by the coast, were

higher waves are encountered. Therefore, in order to predict accurately the behaviour of each vessel

in open-seas close to the coast, a new set of formulas should be developed from direct calculation

extreme-value results, which are obtained using the Hydrostar code provided by Bureau Veritas and

the Belgian Coastal Scatter Diagram.

3.1. HYDROSTAR

HYDROSTAR is used to perform frequency-domain simulations of a rigid ship in extreme waves. The

numerical method is based on potential flow theory. It can be used to calculate global responses and

local loads on ship hulls at any forward speed. It solves the linear 3D radiation/diffraction problem by

the Rankine Panel method by taking into account these forward speed effects.

3.1.1. Panel Method

The use of 3D Rankine Panel Method theory as hull surface boundary condition is implemented in the

present research to calculate seakeeping behaviour from a given ship-sea interaction.

According to J.M.J. Journée et al, (2001), the panel method is a numerical code suitable for

calculations of the (potential) flow around a given hull, based on the principle of Green’s integral

theorem. It is possible to transform a three-dimensional linear homogeneous differential equation into

a two-dimensional integral equation. In this way, the three-dimensional Laplace (potential) equation

can be transformed to a surface integral equation, known as Green’s identity. The integral equation

represents a distribution of sources (or sinks) and dipoles on the surface. To solve the integral equation

numerically, the surface of the body is divided in a number of n x n panels, as shown for a crude oil

carrier below:



Figure 3. 3-D Representation of the Hull Form of a Crude Oil Carrier.

The body surface is divided in N panels small enough to assume that the sources and doublets strength

and the fluid pressure is constant over each element.

3.2. Belgian coastal Scatter Diagram

Figure 4. Belgium seaway spectrum up to HS = 2.0 m.

A wave climate is modelled as a succession of short-term stationary sea states each one having a

duration of 3 hours. To carry out the calculations, Rudakovic (2015) made used of the Belgian Coastal



27


Scatter Diagram (given by the Institute of Oceanography in Ostend, Belgium), up to HS = 2.0 m, as

shown above. This scatter diagram is adopted for the present study on the same hand. Notice the fact

that the maximum wave height allowed for Inland vessels to operate (Hs = 2.0 m) has a narrow band

of peak frequencies in these conditions: [0.79 – 1.05] rad/s.

3.3. Conventions

Each sea state belonging to the Belgian Scatter Diagram is described by its significant wave height Hs

in [m], peak period (Tp) in s. and direction βgeo in degrees, in the geographical reference following

denoted:

Figure 5: Waves direction βgeo.

The azimuth of the vessel is specified from north (clockwise), as in Fig. 6:

Figure 6: Azimuth α’.



3.4. Return Period

Long-term predictions are undertaken to study the extreme loads that are exceeded only once during

the ship’s lifetime. These extreme loads, generally based on a probability of exceedance of about 10−8,

correspond to the Recommendation No. 34 of IACS (2001) for a return period of at least 20 years for

a life period of more than 30 years. This recommendation generally serves as an accepted standard of

wave statistics to predict long-term (extreme) loads for operation in unrestricted waters over the service

life of the ship. It is based on wave statistics for the North Atlantic scatter diagram.

In the present research, by imposing that all the inland navigation vessels will have 20 years of life, a

85% of their life (17 years) is assumed to be spent in operation; sailing within the range of navigation

IN[0.6 < x < 2.0]. It represents a very conservative assumption to determine an adequate extreme value

for a given parameter under evaluation. Return period of 17 years was used in Rudakovic (2015) work

as well and it is adopted for present investigation.

In operation conditions, it is assumed that a given ship sails 50% of the lifetime in Azimuth 70º and

the other 50% of the lifetime in Azimuth 250º, as shown below:

Figure 7. Belgium coastal area where scatter diagram belongs. A vessel’s azimuth is specified.



29


3.5. Range of Navigation to be accounted

All vessels database to be developed under loading conditions specified in Sec. 3.7 will contain results

for 15 types of navigation limit, each one represented by a significant wave height Hs within the range

0.6 m ≤ x ≤ 2 m ( IN(0.6 < x ≤ 2) ) as defined in Part B, Chapter 3, Section 1, [5] of BV NR217

rules (Section 2 of present report); having a step of 0.1 meter.

Table 3. Range of Navigation under investigation.

Physical

Notation

BV NR17

Notation

Hs = 0.6 IN(0.6)

Hs = 0.7 IN(0.7)

Hs = 0.8 IN(0.8)

Hs = 0.9 IN(0.9)

Hs = 1.0 IN(1.0)

Hs = 1.1 IN(1.1)

Hs = 1.2 IN(1.2)

Hs = 1.3 IN(1.3)

Hs = 1.4 IN(1.4)

Hs = 1.5 IN(1.5)

Hs = 1.6 IN(1.6)

Hs = 1.7 IN(1.7)

Hs = 1.8 IN(1.8)

Hs = 1.9 IN(1.9)

Hs = 2.0 IN(2.0)

3.6. Additional features to be considered

To model the forward speed at 10 knots, it was used only the correction of the encounter

frequency, without taking into account perturbations on the sea surface due to the advance of

the ship (Neumann-Kelvin problem).

5 % of the critical roll damping was taken to account for the nonlinear damping coefficient.

As it will be modelled each ship navigating 100% in sea water, a density 𝜌 = 1.025 ton/m3.



Loading condition is imposed for maximum draught.

A constant waterdepth of h = 15 m is assumed.

A constant forward velocity V = 10 knots is imposed for all the ships to be modelled, as done

by Rudakovic (2015).

3.7. Investigated vessel within the study

A database composed of 40 Tanker vessels of different type, 4 container, 1 cargo vessel and 1

bulkcarrier, is used for the present study; all of them complying with the BV Rules for Classification

of Inland Navigation Vessels NR 217. Main characteristics of these vessel are given in the table below.

Table 4. Set of Inland Navigation vessels to be studied in the present work.

Vessel's

Reference

Type of LOA L B Tmax D Δ(Tmax) CB GM

Cargo Carrier [m] [m] [m] [m] [m] [ton] [-] [m]

1 ADN Tanker Type N 55.42 53.50 11.50 3.30 4.60 1766.88 0.82 1.77

2 ADN Tanker Type N 55.00 53.20 9.50 2.80 4.42 1223.00 0.82 1.09

3 ADN Tanker Type C 85.99 84.59 9.48 3.09 4.85 2236.00 0.87 0.61

4 ADN Tanker Type N 105.00 103.20 10.45 3.22 3.80 3142.77 0.87 1.82

5 ADN Tanker Type N 109.96 107.86 10.95 3.32 3.50 3992.00 0.97 1.43

6 ADN Tanker Type C 135.00 132.00 11.40 4.00 5.34 5800.50 0.92 1.48

7 ADN Tanker Type C 86.00 83.75 9.45 2.60 4.25 1904.00 0.88 0.92

8 ADN Tanker Type C 110.00 106.85 11.45 3.30 5.02 3813.32 0.90 0.77

9 ADN Tanker Type C 110.00 107.85 11.45 3.60 5.32 3992.00 0.86 0.71

10 ADN Tanker Type N 110.00 108.35 10.45 3.19 4.60 3277.92 0.87 0.39

11 ADN Tanker Type C 100.00 98.00 11.45 3.30 5.00 3263.00 0.84 0.58

12 ADN Tanker Type G 108.50 106.25 11.35 2.50 5.20 2687.00 0.85 1.85

13 ADN Tanker Type C 110.00 106.55 13.50 4.20 5.32 5590.44 0.87 2.34

14 Container 86.00 84.50 14.15 4.41 5.00 5148.00 0.94 1.52

15 ADN Tanker Type N 109.80 108.18 11.40 2.85 4.00 3097.00 0.85 2.19

16 ADN Tanker Type N 109.98 108.78 11.45 3.40 4.30 3843.00 0.88 1.92

17 ADN Tanker Type C 110.00 107.95 11.41 3.80 5.32 4297.35 0.88 0.48

18 ADN Tanker Type C 110.00 108.25 11.40 3.40 5.62 3874.00 0.89 1.06

19 ADN Tanker Type C 110.00 107.95 11.45 3.60 5.32 4095.00 0.88 0.64

20 ADN Tanker Type C 135.00 133.25 15.00 4.30 5.39 8099.11 0.91 2.99

21 Container 134.97 132.92 11.40 3.49 3.50 4949.00 0.90 1.61

22 ADN Tanker Type N 35.00 33.70 6.40 2.20 3.55 405.00 0.80 0.69

23 ADN Tanker Type C 110.00 107.84 13.50 3.94 5.32 5257.00 0.88 2.60

24 ADN Tanker Type C 110.00 108.37 11.40 3.75 5.40 4301.53 0.89 0.61

25 ADN Tanker Type C 110.00 108.00 11.45 4.32 5.57 4837.00 0.87 1.28



31


Figure 8. A typical tank vessel for inland navigation.

26 ADN Tanker Type C 110.00 108.41 11.45 3.40 5.30 3910.70 0.89 1.84

27 ADN Tanker Type G 95.04 92.30 11.40 2.70 5.70 2664.00 0.89 2.33

28 ADN Tanker Type N 110.00 108.50 11.45 3.50 4.99 4053.00 0.90 0.80

29 ADN Tanker Type C 110.00 108.00 11.45 3.20 4.67 3550.01 0.86 0.98

30 ADN Tanker Type C 135.00 133.65 16.80 4.63 5.75 9513.00 0.88 3.72

31 ADN Tanker Type C 85.96 84.47 9.64 3.16 4.70 2313.25 0.86 0.40

32 ADN Tanker Type C 110.00 107.75 13.50 4.12 5.32 5466.00 0.87 2.94

33 ADN Tanker Type C 130.00 128.10 11.45 4.11 5.70 5680.64 0.91 0.94

34 ADN Tanker Type C 85.00 83.63 10.50 3.46 5.10 2768.00 0.87 1.51

35 ADN Tanker Type N 85.95 84.03 9.00 2.55 5.75 1883.00 0.93 1.58

36 ADN Tanker Type N 109.92 107.57 10.45 3.33 4.00 3456.60 0.88 1.55

37 ADN Tanker Type C 110.00 107.20 11.45 3.76 5.40 4297.42 0.89 0.68

38 Bulkcarrier 38.50 38.00 5.06 2.54 2.60 460.00 0.91 0.56

39 Cargo Vessel 51.20 49.45 11.40 3.25 5.55 1681.08 0.86 0.34

40 ADN Tanker Type C 124.99 123.24 11.45 3.22 5.30 4265.33 0.90 2.02

41 Container 135.00 133.00 11.50 3.21 3.90 4545.11 0.89 0.55

42 ADN Tanker Type C 109.99 107.99 13.50 4.20 5.32 5583.43 0.87 2.55

43 ADN Tanker Type C 135.00 132.75 11.40 4.15 5.90 5977.57 0.91 0.45

44 ADN Tanker Type G 106.00 103.60 11.36 2.80 5.32 3040.39 0.88 2.46

45 ADN Tanker Type C 134.95 131.86 22.80 5.20 6.36 14728.90 0.90 7.02

46 Container 135.00 133.90 11.40 3.71 3.90 5448.00 0.93 2.20



Figure 9. General arrangement plan of a typical IN Tanker.

One of the main feature Inland vessels have is that they are very slender with a high Block Coefficient

value. For the loading conditions to be studied, it is noted that Inland vessels have a Block Coefficient

(CB) ≥ 0.82 in all the cases, even for containerships. 14 ships even have a CB ≥ 0.90. It is expected

that, unlike sea-going vessels (specially sea-going containerships), CB will have small (or negligible)

influence on motions and accelerations of the ship. Nevertheless, it is also assessed.



33


4. THEORETICAL FRAMEWORK

4.1. Scheme of study

Now that all input parameters have been defined for the present study, a theoretical approach is

presented below in order to understand the basis of the linear potential code to be adopted:

First, the characteristics of a harmonic regular wave are defined based on the Boundary

Element Theory for linear potential flow, where a potential of first and second order together

with a first and second order representation of the sea surface is accounted, in order to better

understand the assumptions to be taken and the source of the nonlinear corrections to be

implemented on proposed formulas;

Then, a statistical representation of the sea surface is built based on series of first-order n-

regular-waves; in consequence, theory of statistics for random processes is introduced and

spectral representation of the sea surface is put in terms of basic statistical concepts.

Once all statistical characteristics of a given sea state have been defined, a ship behaviour under wave

excitations is studied, based on:

First order response to a unit-amplitude regular wave excitation. Boundary conditions, defined

from Boundary Element Method theory and Green’s Second theorem, are defined for the hull

wetted surface. Sources of all the potentials influencing the response of the ship are studied.

The equations of motion of a given vessel under unit-amplitude wave excitation, is built;

Next, a statistical representation of the response is presented, which is linked to the approach

adopted for statistical representation of the sea surface;

After, a short term response for a given sea state will be defined,

And finally, a long-term response expression will be presented, which accounts for the

maximum response amplitude that can be encountered in all of the sea states contained in a

Scatter Diagram (as given in Fig. 4) during a determined return period (17 years for the present

study).



4.2. Statistical Representation of the Sea Surface

The surface of the ocean, that is, the pattern of sur face elevation, is highly irregular and totally random

(nonrepeating) even under relatively calm conditions. The first major contributions on this topic were

made by Pierson (1952) and Pierson et al. (1955), who proposed that the completely irregular and

nonrepeating pattern of the ocean surface 𝜁 is represented as the addition of an infinite number of

regular sinusoidal waves, of all frequencies:

ζ(x, t) = limN→∞

∑An cos(− knx − ωnt + θn)

N

n=1

(18)

where:

ζ(x, t) = wave surface elevation, measured from the mean water surface in [m].

𝐴𝑛 = amplitude of a n-wave, measured from the mean water surface in [m].

𝑛 = 1,2,3,…,N. couter.

λ= wave length, the horizontal distance between successive crests or troughs in [m].

k = 2π / λ = wave number in [rad/m].

T = wave period, the time between two successive crests to pass a fixed point on the x-axis or

the time between a crest to travel a distance equal to one wavelength, in [s].

ω = 2π/T = wave frequency, in [rad/s].

θ = phase angle, in [rad].

This allows the ocean surface to be described mathematically, and it also permits the use of statistical

methods to predict the maximum wave loads in a ship’s lifetime. Therefore, a need to study the water

regular waves beforehand arise.

Figure 10. A sum of many simple sinusoidal waves makes an Irregular Sea.



35


4.3. Regular gravity harmonic waves

Harmonic regular waves are seen from two points of view. Fig. 11a shows a wave profile as a function

of distance x along its propagation at fixed time. Fig. 11b is a time record of the water level observed

at one fixed location. Origin of the coordinate system is at the still water level with the positive z-axis

directed upward. The still water level is the average water level. The x-axis is positive in the direction

of wave propagation. The water depth, h, is measured between sea bed (z = -h) and still water level.

The highest point of the wave is its crest and the lowest point on its surface represents its trough.

Figure 11. Regular-wave reference system for regular-harmonic wave definitions.

The wave height H is the distance between wave trough level to the wave crest level, being H = 2A.

4.3.1. Wave speed

Wave speed or phase velocity, c; is finally given by:

c = λ

T = ω

k (19)

in which:

k : is the wave number in [rad/m] and

ω : is the circular wave frequency [rad/s].

4.3.2 Wave Steepness

The ratio of wave height to wave length is referred to the wave steepness, ε; as presented below:



ε = πH

λ= 2πA

λ (20)

4.3.3. Influence of wave steepness

In Fig. 12 is shown how wave steepness can affect the wave surface. 4 kind of waves are shown with

a constant wave length λ = 2π m ≈ 6.28 m while surface elevation changes. Fig. 12a. indicates that the

slope of the wave with amplitude s = 0.01 m is very small (ε = 0.01), giving place to an harmonic

sinusoidal behaviour. In Fig. 12b is shown the nondimensioned surface elevation. Wave crests higher

than wave amplitude and wave troughs lower than wave amplitude are encountered for

s = 0.1, 0.2 and 0.3 m, with a steeper non-sinusoidal behaviour. In conclusion, waves with s = ε > 0.01

have second order effects that cannot be neglected.

Figure 12. Wave steepness influence. 12a. Harmonic motions at different wave amplitude. 12b nondimensined

wave surface elevation.

4.3.4. Laplace condition

If the fluid is assumed to be incompressible, inviscid, irrotational (also without surface tension), the

continuity condition meets the condition div(V ) = 0. The flow resulting satisfies Laplace equation in

the x-z plane as shown below, where the reference defined in Fig. 11 is used.

𝜕2𝜙𝑤𝜕𝑥2

= 𝜕2𝜙𝑤𝜕𝑧2

= 0 (21)

Using the linear theory leads the harmonic displacements, velocities and accelerations of the water

particles, as well as the harmonic pressures; to have a linear relation with the wave surface elevation.

Navier stokes equations are simplified to describe potential flow theory, giving place to Bernoulli



37


equation for a no-stationary irrotational flow (with the velocity given in terms of its three components)

is in its general form:

∂𝜙𝑤∂t

+ 1

2(u2+ w2) +

p

ρ + gz = C* (22)

In where:

𝜙𝑤 : Potential describing the fluid behaviour in [m2/s]

(u, w) = 𝑣 : Velocity vector in [m/s];

p : Fluid pressure (absolute or relative) in [Pa];

ρ : Fluid density in [kg/m3].

If the flow is considered to have a small steepness, then linear theory can be applied and second order

effects such as (u, w) = 𝑣 can be neglected.

4.3.5. Wave potential and wave surface elevation

Both are schematically described as an influence of linear terms 𝜙𝑤(1)

and ζ(1)

respectively, plus the

influence of terms of higher order 𝜙𝑤(𝑛), 𝜁(𝑛); which are directly linked to the wave steepness value. If

(ε ≪ 1), higher order terms can be neglected.

𝜁 = 𝜁(1) + 𝜀𝜁(2) + 𝜀2𝜁(3) + ⋯ (23)

𝜙𝑤 = 𝜙𝑤(1)+ 𝜀𝜙𝑤

(2) + 𝜀2𝜙𝑤

(3)+ ⋯ (24)

4.3.6. Boundary Conditions (Boundary Element Methods)

Types of boundary conditions:

Nonlinear: Wave steepness is not sufficiently small. Higher order terms of Eq. 23 and 24

cannot be neglected.

Linear: Little perturbations with respect to the still water level occur, so wave slopes are very

small. Therefore, wave steepness becomes as well very small (ε ≪ 1). Terms with order equal

or higher than 2 are neglected; see schematic below:

𝜁 = 𝜁(1)+ O(𝜀) (25)



𝜙𝑤 = 𝜙𝑤(1) + O(𝜀) (26)

Sea Bed Boundary Condition

The vertical velocity of water particles at the sea bed is zero (no-leak condition), see Fig. 13. 1st and

2nd order terms of the potential are expressed below:

Figure 13. Sea Bed boundary condition.

1st order term:

𝜕𝜙𝑤(1)

𝜕𝑧 = 0 for z = − h (27)

2nd order term:

𝜕𝜙𝑤(2)

𝜕𝑧 = 0 for z = − h (28)

Free surface boundary condition

Using Taylor expansion, free surface boundary conditions are expanded at the still water level z = 0:

E(𝜙𝑤, 0, 𝑥, 𝑡) = ∑ζm

m!

𝜕𝑚

∂zm[𝐸(𝜙𝑤, 0, 𝑥, 𝑡)]

∞

m=0

(29)

For instance, the potential at the free surface can be expanded in a Taylor series to account for the

potential at 𝑧 = ζ, as following:

{𝜙𝑤(𝑥, 𝑧, 𝑡)}𝑧= ζ = {𝜙𝑤(𝑥, 𝑧, 𝑡)}𝑧= 0 + ζ {𝜕𝜙𝑤(𝑥, 𝑧, 𝑡)

𝜕𝑧}𝑧= 0

+ 𝜁2

2{𝜕2𝜙𝑤(𝑥, 𝑧, 𝑡)

𝜕𝑧2}𝑧= 0

+⋯ (30)

And derivation of the potential against time as presented in Bernoulli’s equation (see Eq. 30) to

describe free-surface hydrodynamics leads to:



39


{𝜕𝜙𝑤(𝑥, 𝑧, 𝑡)

𝜕𝑡}𝑧= ζ

= {𝜕𝜙𝑤(𝑥, 𝑧, 𝑡)

𝜕𝑡}𝑧= 0

+ ζ {𝜕2𝜙𝑤(𝑥, 𝑧, 𝑡)

𝜕𝑧𝜕𝑡}𝑧= 0

+ 𝜁2

2{𝜕3𝜙𝑤(𝑥, 𝑧, 𝑡)

𝜕𝑧2𝜕𝑡}𝑧= 0

… (31)

Dynamic free surface boundary condition (DFSBC)

The pressure, p, at the free surface of the fluid, z = ζ, is equal to the atmospheric pressure, p0.

Figure 14. Dynamic free surface boundary condition.

Having Bernoulli equation and taking into account that waves have a small steepness, it gives place to

the linear DFSBC:

𝜕𝜙𝑤(1)

𝜕𝑡 = − 𝑔𝜁(1) for z = 0 (32)

If wave steepness is not small enough, second order terms of velocity in Bernoulli equation have to be

taken into account. Potential second order DFSBC is then denoted as:

𝜕𝜙𝑤(2)

𝜕𝑡 + 𝑔𝜁(2) = − 𝜁(1)

∂2𝜙𝑤(1)

𝜕𝑡𝜕𝑧 −

𝜕

𝜕𝑧[|∇𝜙𝑤

(1)|2] for z = 0 (33)

With Laplace, sea bed boundary and dynamic free surface boundary conditions, it can be built an

expression for first and second order terms of the wave potential and wave surface elevation:

1st order terms:

𝜁(1)(𝑥, 𝑡) = Acos(kx − ωt) (34)

𝜙𝑤(1)(𝑥, 𝑧, 𝑡) =

Aω

ke-kzsin(kx − ωt) (35)



2nd order terms:

𝜁(2)(𝑥, 𝑡)= kA2(3 − tanh2(kh))

4tanh3(kh)cos(2(kx − ωt)) (36)

𝜙𝑤(2)(𝑥, 𝑧, 𝑡) =

3A2ω

8sinh4(kh)cosh(2k(z + h))sin(2(kx-ωt)) (37)

Total second order wave potential and wave surface elevation is obtained adding up 1st and 2nd order

terms:

𝜁(𝑥, 𝑡) = 𝜁(1) + 𝜁(2) (38)

𝜙𝑤(𝑥, 𝑧, 𝑡) = 𝜙𝑤(1) + 𝜙𝑤

(2) (39)

If the wave steepness ε = 2πA λ = kA⁄ is very small, the term kA2 in 𝜁(2)(Eq. 36) will be almost zero.

On the same hand, wave amplitude will be very small so the term A2ω in 𝜙𝑤

(2) (Eq. 37) is as well

almost zero. In conclusion, in both cases linear potential and linear surface elevation are adopted.

Kinematic Boundary Condition (KFSBC)

The vertical velocity of a water particle at the free surface of the fluid is identical to the vertical velocity

of that free surface itself (no-leak condition):

∂ζ

∂t = w (40)

Figure 15. Kinematic Boundary Condition.

Using Taylor expansion (Eq. 29) for the wave surface elevation from the calm water level (z = 0 m)

expanded to z = 𝜁; yields:



41


dz

dt = ∂ζ

∂t + ∂ζ

∂x

∂x

∂t = ∂ζ

∂t + u

∂ζ

∂x (41)

If wave steepness is small enough, the second term in the expression above is a product of two very

small values. This product becomes even smaller (second order) and can be ignored. Then, a

differentiation of the linear DFSBC (Eq. 32) with respect to time gives place to the KFSBC, known as

well as Cauchy-Poisson condition:

𝜕2𝜙𝑤(1)

𝜕𝑡2 + g

∂ζ

∂t = 0 for z = 0 (42)

If wave steepness is not so small, second order terms need to be taken into account, giving place to the

potential second order KFSBC:

𝜕2𝜙𝑤(2)

𝜕2𝑡 + g

𝜕𝜙𝑤(2)

𝜕𝑧 = − ζ(1)

∂

∂z(𝜕2𝜙𝑤

(1)

𝜕𝑡2 + g

𝜕𝜙𝑤(1)

𝜕𝑧) −

∂

∂z[|∇𝜙𝑤

(1)|2] for z = 0 (43)

4.3.7. Dispersion Relation

In order to establish a relationship between ω and k (or equivalently T and λ), a substitution of the

linear wave potential expression Eq. 35 into Eq. 42 leads to the dispersion relation for any arbitrary

water depth h:

ω2 = kg ∙ tanh (kh) (44) In deep water conditions ((tanh (kh) = 1), wave frequency (wave period) and wave number (wave

length) are related by a simpler form:

ω2 = kg or T2 = 2𝜋𝜆

g (45)

Second order term ζ(2) (Eq. 36) represent a wave which has the double of the frequency of the linear

term: (2(kx− ωt)). It is named ‘bound wave’ and it does not satisfy the dispersion relation presented

above. Nevertheless, dispersion relation is still valid for deep water conditions for the total second

order theory.



4.3.8. Total Wave Energy

The total energy, Ω, per unit area of water surface is defined as:

Ω = 1

2ρgA2 (46)

4.4. Irregular Waves

The sea surface is very confused; its image changes continuously with time without repeating itself.

Both the wave length between two successive crests or troughs and the vertical distance between a

crest and a trough vary continuously. When the sea time-history is available, as it is the case for this

study, a simpler analysis can be carried out to obtain statistical data from this record based on the fact

that it could be represented as a superposition of infinite regular waves with negligible steepness

influence. Therefore, only linear potential flow theory explained above can be subjected to the

following statistical anaylisis.

4.4.1. Statistics

When measuring sea surface elevation at a fixed location, a random value defined as wave height 𝐻

represents an outcome, among all possible outcomes. If all the possible values of the outcomes Hn (set

of all measured data) form a continuous distribution in space: -∞ < x < ∞, and each wave height value

is some portion of this space, then the probability to obtain a given value of H is simply the probability

that H lies within that portion of x. The degree of probability that a wave height Hi happens is described

by a wave surface-elevation probability density function, pH(x); as shown in Fig. 16 below:

Figure. 16. Schematic of the ocean-surface elevation probability density function.

Where the probability that H lies in an infinitesimal region [x , x + dx] is pH(x)dx; x represents the

value per se of H in [m].



43


Prob[x ≤ H ≤ x + dx] = pH(x)dx (47)

And the total area under the curve, defined as “zero-moment” of the distribution or simply 𝑚0, is equal

to unity:

Prob[ − ∞ ≤ H ≤ ∞] = 𝑚0 = ∫ pH(x)dx∞

−∞

= 1 (48)

To obtain E[H], namely the average or expected value of H, first moment 𝑚1 over the distribution

pH(x)dx is taken. This is referred as the mean value of all possible outcomes, which is the most direct

and most familiar type of average. The symbol μ* indicates mean value.

𝑚1 = μ* = E[H] = ∫ x pH(x)dx∞

−∞

(49)

The wave’s mean surface elevation 𝜁 is equals to E[H]. Following definitions of moments of higher

order k (𝑚k) are taken about the mean value μ of a distribution, and not from the origin of the range.

The moment arm becomes the deviation from the mean, (H − μ).

𝑚k = E[(H − μ)k] = ∫ (x − μ)k pH(x)dx

∞

−∞

(50)

In this sense is defined a second moment 𝑚2, which is a measure of the spread or dispersion of pH(x)

and is known as the variance, 𝜎2. It is defined accordingly as:

𝑚2 = 𝜎2 = E[(H − μ)2] = ∫ (x − μ)2 pH(x)dx (51)

∞

−∞

In which the measure of dispersion is usually taken as the positive square root of 𝜎2. This quantity 𝜎

is referred to as the standard deviation or root mean square of the deviation.

According to the definition of the moment arm (H− μ*), where μ* = �� accounting for wave’s mean

surface elevation; wave’s amplitude 𝐴𝑛 is defined as (𝐻 − ��)𝑛. By setting �� as the new surface

elevation reference fixed at ��= μ* = 0 m, namely the still water level, the frame of reference showed



in Fig. 11 for regular waves can be adopted now. All set of vertical displacement measurements will

be relative to the still water level.

4.4.2. Sea-state definition

Figure 17. Schematic of a random ergodic process.

Random processes, as the case of ocean waves, have a statistical behaviour such that when a time

average is taken from a single short time (typically 3-hours) sample, they are, in the limit, equal to the

same average measured from many samples when stopping all of them at certain time t1 , t2 or tn (see

Fig. 17 above). Such processes are known as ergodic processes. A measurement of a single sample

X(t) = 𝜁(t) is sufficiently typical to represent the entire process (see Fig. 18 below). In this way, a sea-

state is defined. With this property, the required averages are described by the moment arms 𝑚k.

Figure 18. Ocean wave elevation sampling at a specific location, measured every Δt during 3 h.



45


4.4.3. Wave Energy Spectra

The signal 𝜁(t) of the wave’s surface elevation at a specific location is irregular (Fig. 18 above) and it

is expressed via Fourier series analysis as the sum of a large number of sinusoidal regular waves, each

with its own frequency, amplitude and phase in the frequency domain (Eq. 52). These phases are

discarded and amplitudes An are obtained. Wave energy spectra is defined as following:

𝑆𝜁(𝜔𝑛) ∙ ∆𝜔 = ∑1

2𝐴𝑛

2(𝜔)

𝜔𝑛+∆𝜔

𝜔𝑛

(52)

Where:

∆𝜔 = a constant difference between two successive frequencies, in [rad/s];

𝑆𝜁(𝜔𝑛) = wave spectrum. Multiplied times 𝜌𝑔 leads to the energy per unit area of the waves

[kg m2/s2]. See Eq. 46.

From engineering point of view, the process described above happens on the contrary way. From a

given wave spectrum, series of sinusoidal regular waves are generated with an associated phase, which

only matters for purposes of regenerating the wave surface in terms of time 𝜁(t). It will not be as the

original due to random phase generation for each regular wave).

4.4.4. Wave Height and Period definition from Wave Energy Spectra

The moments of a sea state energy spectrum are given by:

𝑚𝑛𝜁 = σ𝜁2 =∫ 𝜔𝑛 ∙ 𝑆𝜁(𝜔) ∙ 𝑑𝜔

∞

0

; 𝑓𝑜𝑟 𝑛 = 1,2, … ,𝑁 (53)

Where:

𝑚0ζ = area under the response spectrum which is equal to its deviation;

𝑚1ζ = 1st moment of area (centroid of spectrum);

𝑚2ζ = 2nd moment of inertia, with (ωn − 𝜇∗)2 being the spectral radius of gyration.

The significant wave amplitude can be calculated from the wave spectrum. The significant amplitude

is defined to be the mean value of the highest one third part of the wave amplitudes, or equivalently:



A1/3 = 2 ∙ σζ = 2√𝑚0ζ (54)

The significant wave height is defined to be the mean value of the highest one third part of the wave

height; as defined in Sec. 2 in BV NR217 rules.

H1/3 = 𝐻𝑆 = 4 ∙ σζ = 4√𝑚0ζ (55)

in which σζ is the Root Mean Square value. A mean period, TP , can be found from the centroid of the

spectrum:

TP = 2π∙𝑚0ζ

𝑚1ζ (56)

and the average zero-crossing period, TZ, is found from the spectral radius of gyration:

TZ = 2π ∙ √𝑚0ζ

𝑚2ζ (57)

Both TP and TZ can be written in terms of 𝜔P and 𝜔Z as denoted in Eq. 18.

4.4.5. Rayleigh Distribution of wave peaks

For design purposes, it is more interesting to study wave peak values, A, rather than the full range of

values of 𝜁(t). Peak values of a random process are a special subgroup and, therefore, they have a

probability density function of their own. Rayleigh model propose a distribution for these peaks, in

terms of moments 𝑚𝑛𝜁.

𝑅𝑎𝜁(𝑡) =𝑡

𝑚0𝜁∙ 𝑒𝑥𝑝 {−

𝐴2

2 ∙ 𝑚0𝜁} (58)

In which:

𝑅𝑎𝜁(𝑡) = Rayleigh distribution of all the wave amplitudes, as seen in Fig. 19 below.

With the above specified distribution, the probability that a certain wave amplitude of value “a”

exceeds a chosen threshold value, a, is calculated using following expression:



47


With the above specified distribution, the probability that a certain wave amplitude of value “a”

exceeds a chosen threshold value, a, is calculated using following expression; (see Fig. 20):

𝑃𝑟𝑜𝑏[𝐴 > 𝑎] = ∫ 𝑅𝑎𝜁(𝑡) ∙ 𝑑𝑡∞

0

= 𝑒𝑥𝑝 {−𝑎2

2 ∙ 𝑚0𝜁} (59)

If the wave surface elevation is a Gaussian distribution, then the wave amplitude (A) statistics will

obey a Rayleigh distribution, as show in Fig. 19.

Figure 19. Rayleigh Distribution.

Figure 20. Probability of exceeding.



4.4.6. JONSWAP Wave Spectra

The JONSWAP formulation is based on an extensive wave measurement program known as the JOint

North Sea WAve Project. The spectrum represents wind-generated seas with fetch limitation. The

formulation is more general and englobes the spectrum of Pierson Moskowitz as a particular case. It

can be written as:

𝑆𝐽𝑆𝜁(𝜔) =1

80(0.065𝛾0.803 + 0.135) 𝐻𝑆

2𝜔𝑃4𝜔−5∙ 𝑒𝑥𝑝 [−

5

4(𝜔

𝜔𝑃𝐽)

−4

] ∙ 𝛾 ∙ 𝑒𝑥𝑝 [(−(𝜔 − 𝜔𝑃𝐽)

2

2𝜉2𝜔𝑃2 )

−4

] (60)

Where:

𝑆𝐽𝑆𝜁(𝜔) = JONSWAP formulation of a wave spectrum.

𝛾 = peak-enhancement factor, if equals to 1, JONSWAP formulation becomes identical to the

one adopted by Pierson-Moskowitz.

𝜉 = relative measure of the width of the peak;

𝜔PJ = JONSWAP spectrum mean frequency in [rad/s].

The wave climate, also defined as Scatter Diagram, is modelled as an ergodic succession of short-term

wave spectrums, where each short-term sea state is characterized by the two-parameter Pierson-

Moskowitz seaway spectrum (or JONSWAP Wave Spectra with 𝛾 =1) with a significant wave height

( 𝐻𝑆) and mean peak period period (TZ), as show in the Belgiun Coastal Scatter Diagram (Fig. 4).

4.4.7. Directional Spreading

A cosine-squared rule is often used to introduce directional spreading to the wave energy spectrum (to

the JONSWAP spectrum in this case). The unidirectional wave energy found in the previous section

is scaled as in the following formula:

𝑆𝐽𝑆𝜁(𝜔, 𝜇) = {2

𝜋∙ cos2(𝜇 − ��)} ∙ 𝑆𝐽𝑆𝜁(𝜔) ; 𝑓𝑜𝑟 𝑎 𝑟𝑎𝑛𝑔𝑒: −

𝜋

2≤ (𝜇 − ��) ≤ +

𝜋

2 (61)

Where to:

𝜇 = any wave direction, in [degree] or [rad].

�� = dominant wave direction, in [degree] or [rad].



49


Figure 21. Wave Spectra with Directional Spreading.

4.5. Linear Response to first order excitation

Now it is time to study how the linear response is, under determined irregular-sea conditions. It is

supposed that steady state conditions of ship response have been attained, as done by irregular waves

(see Sec. 4.4). Motion amplitudes and velocities are assumed to be small, so second order terms

belonging to the free surface condition and Bernoulli equation (see Eq. 22) are neglected

4.5.1. Linear Potential Theory: Linear Mass-Spring System

In the Fig. 22, the output of the system is the mode of motion of the floating structure (namely an

inland navigation vessel). At each frequency, a specific ratio between motion amplitude and wave

amplitude is intrinsically defined for a vessel, which is kept constant. In consequence, doubling the

input (wave) amplitude results in a doubled output amplitude, while the phase shifts between output

and input does not change. In consequence, motions, accelerations and loads will be analysed in the

so-called frequency domain for the present research. It can be obtained a very realistic mathematical

model by making use of a superposition of the frequency-dependent response r(ω) components at a

certain range of frequencies, giving place to an output r(t) as shown in Fig. 22.



Figure 22. Block Diagram for a Linear System. Linear relation between Motions and Waves.

4.5.2. Frames of reference

Three orthogonal coordinate systems are used to define the ship motions, see Fig. 23.

• An earth-bound coordinate system S(x0, y0, z0): Used to describe directionality of the waves. the

(S, x0, y0) - plane lies on the still water surface, the positive x0-axis is in the direction of the wave

propagation; it rotates at a horizontal angle μ relative to the translating axis system O(x, y, z). The

positive z0-axis is directed upwards.

• A body - bound coordinate system G(xb, yb, zb): This system is connected to the ship with its origin

at the ship’s centre of gravity, G; it remains invariant independently the mode of motion undergoing.

It moves at ship forward speed, V. The directions of the positive axes are: xb in the longitudinal forward

direction, yb in the lateral port side direction and zb upwards. If the ship is floating upright in still water,

the (G, xb, yb)-plane is parallel to the still water surface.

• A steadily translating coordinate system G(x, y, z): This system is used to describe the modes of

motion of a ship. It is moving forward with a constant speed V in x-direction. The (x, y)-plane lies in

the still water surface or the time-averaged position of the ship when subjected to wave effects. Its

origin is placed at the centre of gravity, G. The ship is supposed to carry out oscillations with respect

to this system of reference. The rule formulas shown in Sec. 2. (BV NR217) are based on this frame

of reference.



51


Figure 23. Coordinate Systems.

4.5.3. Motions of the ship

The motions of a vessel, just as for any other rigid body, can be split into three mutually perpendicular

translations of the centre of gravity G, and three rotations around G; see Fig. 24.

{ Translating modesabout G

} = {surgesway

heave } ∙ {

x y

z

} (62)

{ Rotating modesabout G

} = {roll pitch yaw

} ∙ {

φ

𝜃 ψ

} (63)

Figure 24. Mode of motions of a ship. Steadily translating and body-bound reference systems.



4.5.4. Plane of symmetry

Generally, a ship has a vertical-longitudinal plane of symmetry (center plane), so that its motions can

be split into symmetric and anti-symmetric components. Surge, heave and pitch represent symmetric

motions, meaning that a point at starboard has the same motion as the mirrored point at port side. The

remaining motions sway, roll and yaw are anti-symmetric motions.

4.5.5. Wave encounter frequency

When a vessel moves with a constant forward speed, the frequency at which it encounters the waves,

ωe, becomes important. At zero forward speed (V = 0 knots) or in beam waves (μ = 90 deg. or μ = 270

deg.) the frequencies ωe and ω come to be identical. Wave encounter frequency is determined as:

ωe = ω − kV cos μ= ω −ω2

gV cos μ (64)

Where:

ωe = frequency of encounter between ship and waves in [rad/s];

ω = wave frequency in [rad/s];

V = forward ship velocity in [m/s];

μ = angle of encounter, as defined in Fig. 25 below:

Figure 25. Frequency of Encounter.

Angle of encounter, μ, is redefined in terms of βgeo (see Fig. 5) for the calculation process.



53


Surface wave elevation is modified accounting for the angle μ at which a vessel encounter the waves:

𝜁ωe(𝑥, 𝑡, μ) = 𝐴 𝑐𝑜𝑠(ωet − kxb cos μ − kyb sin μ) (65)

Where:

𝜁ωe(𝑥, 𝑡, μ) = wave surface elevation for a particular frequency of encounter ωe, at a certain

location (xb , yb) on the ship seen from the steadily translating ship reference.

μ = angle of encounter, in [deg] or [rad].

4.5.6. Loads Superposition

Since the system is linear, the resulting loads arisen from wave-ship interactions are seen as the

superposition of two effects, see Fig. 26 below, for a ship advancing at given constant forward speed.

1. Hydromechanical forces and moments, induced by a linear radiation potential (𝜙𝑟) coming

from harmonic regular oscillations r(𝜔) of a ship hull moving in an undisturbed fluid surface

at a given frequency 𝜔 equals of the frequency of encounter ωe.

2. Wave exciting forces and moments, produced by an incident undisturbed lineal wave potential

(𝜙𝑤) and a linear diffraction potential (𝜙𝑑), coming from regular incident waves of unit

amplitude acting on a restrained ship hull at a frequency of encounter ωe.

Figure 26. Schematic of a linear superposition of hydromechanical and exciting forces and moments

This interaction between ship and waves generates a total linear fluid velocity potential, 𝜙𝑇:

𝜙𝑇(x,y,z,t) = 𝜙𝑟 + 𝜙𝑤+ 𝜙𝑑 (66)



Each of these velocity potentials in Eq. 66 has to meet certain requirements and boundary conditions

in the fluid, regarding to first order potential theory. Part of these were presented in Sec. 4.3. for the

first order wave potential 𝜙𝑤 (see Eqs. 27, 32 and 40). The additional boundary conditions associated

with the floating hull wetted surface which causes a diffracted and radiated linear potential, are

presented next.

4.5.7. Boundary Conditions describing linear ship-wave interaction

Kinematic Boundary Condition on the oscillating hull surface

The normal velocity vn(x,y,z;t) out of the oscillating hull wetted surface at a given location, arises from

the total linear potential 𝜙𝑇(x,y,z;t) acting in that location. It can be written as the addition of the

effects in all the 6 degree of freedom, each one put in terms of oscillatory velocities vj(t) and

generalized direction cosines fj(x,y,z) on the surface of the hull, S; as following:

∂𝜙𝑇(x,y,z;t)

∂n = vn(x,y,z;t) = ∑vj(t)∙fj(x,y,z) = (∑vj

6

j=1

) ∙(∑fj

6

j=1

) (67)

6

j=1

Direction cosines related to translations of the centre of gravity:

{ x y

z

}={

f1

f2

f3

} = {

𝑐𝑜𝑠(𝑛, 𝑥) 𝑐𝑜𝑠(𝑛, 𝑦)

𝑐𝑜𝑠(𝑛, 𝑧)} (68)

Direction cosines related to rotations about the centre of gravity:

{

φ

𝜃ψ

} = {

f4

f5

f6

} = {

𝑦 𝑐𝑜𝑠(𝑛, 𝑧) − 𝑧 𝑐𝑜𝑠(𝑛, 𝑦)

𝑧 𝑐𝑜𝑠(𝑛, 𝑥) − 𝑥 𝑐𝑜𝑠(𝑛, 𝑧)

𝑥 𝑐𝑜𝑠(𝑛, 𝑦) − 𝑦 𝑐𝑜𝑠(𝑛, 𝑥)} = {

𝑦𝑓3 − 𝑧𝑓2 𝑧𝑓1 − 𝑥𝑓3𝑥𝑓2 − 𝑦𝑓1

} (69)

Where normalized vectors 𝑓1, 𝑓2 and 𝑓3 are used to obtain 𝑓4, 𝑓5 and 𝑓6. With this information, normal

vector to the oscillating body is defined as:

∑ fj

3

j=1

= n = f1 �� + f2 �� + f3 �� (70)

While arm with respect of centre of gravity (G,x,y,z) is given by:



55


∑fj

6

j=4

= (r x n ) = f4 φ+ f5 𝜃 + f6 ψ (71)

Figure 27. Direction cosines for a normal vector �� located on the wetted hull surface.

Radiation Condition

As the distance, R, from the oscillating body becomes large, the radiation potential value 𝜙𝑟, tends to

zero:

𝑙𝑖𝑚𝑅→∞

𝜙𝑟 (72)

Green’s Second Theorem

Green’s second theorem transforms a large volume-integral into a surface-integral. Applying Green’s

second theorem to two separate velocity potentials Φj and Φk , leads to:

∭(Φj ∙ ( ∇2Φk) − Φk ∙ ( ∇

2Φj)) ∙ dV* = ∬(Φj

∂Φk∂n

− Φk∂Φj∂n) ∙dS*

S*V*

(73)



In Green’s theorem, S* is a closed surface with a volume V*, which is bounded by the wall of an

imaginary vertical circular cylinder with a very large radius R, the sea bottom at z = -h, the water

surface at z = 0 and the wetted surface of the floating body, S; see Fig. 28:

Figure 28. Green’s second theorem for a cylinder of surface S*.

If both of the above radiation potentials Φj and Φk fulfil the Laplace condition (∇2Φj = ∇2Φk= 0 ),

the left hand side of Eq. 73 becomes zero, and taking into account the boundary condition at the sea

bed (Eq. 27) and the radiation condition on the wall of the vertical imaginary cylinder in Fig. 28 (Eq.

72), the integral over the surface S* reduces to:

∬(Φj∂Φk∂n) ∙dS

S

=∬(Φk∂Φj∂n) ∙dS (74)

S

in which S becomes the wetted surface of the of the ship hull only.

4.5.8. Forces and Moments on the hull wetted surface S:

The hydrodynamic pressure acting on the hull wetted surface is obtained from the linearized Bernoulli

equation, using the known velocity potentials (see Eq. 66). Integration of this pressure over the wetted

hull surface, S (in the required direction) provides the hydrodynamic forces 𝐹 and moments �� (eq 75

and 76).

F = −∬(p ∙ n ) ∙ dS

S

(75)



57


M = −∬p ∙ (r x n ) ∙ dS

S

(76)

Where:

�� = outward normal vector on surface dS;

𝑟 = position vector of surface dS in the steadily translating coordinate system G(x, y, z);

p = pressure coming from Bernoulli linear equation, where 𝜙𝑤 now becomes 𝜙𝑇; in [Pa].

p = − ρ∂𝜙𝑇∂t − ρgz (77)

The hydromechanical forces 𝐹 and moments �� are split into four parts too:

F = Fr + Fw + Fd + Fs (78)

M = Mr + Mw + Md + Ms (79)

Where:

𝐹𝑟 , 𝑀𝑠 , caused by the radiated waves from the oscillating body in still water, described by 𝜙𝑟.

𝐹𝑤 , 𝑀𝑤 , induced by the approaching waves on the fixed body, described by 𝜙𝑤.

𝐹𝑑 , 𝑀𝑑 , brought by the diffracted waves, which are generated by the interaction between

approaching undisturbed waves and the ship. It is described by the linear potential 𝜙𝑑.

𝐹𝑠 , 𝑀𝑠 , appearing simply due to the hydrostatic buoyancy in still water.

4.5.9. Hydrodynamic Loads

Hydrodynamic forces and moments are split into a load in-phase with the harmonic acceleration and a

load in-phase with the harmonic velocity:

Xrk = ∑−akj Sj

6

j=1

− bkj Sj ; for k = 1,2,…,6 (80)

Where:

akj = hydrodynamic mass caused by the acceleration Sj in the direction j, generating a force in

direction k;

bkj= hydrodynamic damping from the velocity Sj in the direction j, causing a force in direction k.



��j = given harmonic velocity in direction j; in [m/s] or [rad/s].

Sj = given harmonic acceleration in direction j; in [m2/s] or [rad2/s].

The radiation force or moment in the direction k, written as Xrk associated with a linear potential Φk,

is caused by a forced harmonic oscillation of the body in the direction j linked to a potential Φj ; and

it is written as:

Xrk = ∑(ω2Sajakj

6

j=1

+iωSajbkj)e−iωt = ∑(−ω2Sajρ∫∫Φj

∂Φk∂n

∙dS

S

6

j=1

)e−iωt ; for k = 1,2,…,6

(81)

Where terms of hydrodynamic mass and damping matrix are defined as:

akj= − Re {ρ∬Φj∂Φk∂n

∙dS

𝑆

} ; bkj=− Im {ρω∬Φj∂Φk∂n

∙dS

𝑆

} ; for j = 1,2,…,6 and k = 1,2,…,6

(82)

When j = k, the force or moment is caused by a motion in that same direction. When j ≠ k, the force

in one direction results from the motion in another direction. It introduces what is called coupling

between the forces and moments (or motions) and leads to a 6 x 6 matrix.

Because of the symmetry of a ship, some coefficients are zero and the two matrices with hydrodynamic

coefficients for ship become:

6x6 Hydrodynamic mass matrix:

a =

(

a11 0 a13 0 a15 00 a22 0 a24 0 a26a31 0 a33 0 a35 00 a42 0 a44 0 a46a51 0 a53 0 a55 00 a62 0 a64 0 a66

)

(83)

6x6 Hydrodynamic damping matrix:

b =

(

b11 0 b13 0 b15 00 b22 0 b24 0 b26b31 0 b33 0 b35 00 b42 0 b44 0 b46b51 0 b53 0 b55 00 b62 0 b64 0 b66

)

(84)



59


Terms on the diagonals (such as a11 or b11 for example) are the primary coefficients relating properties

such as hydrodynamic mass or damping in one direction to the inertia forces in that same direction.

Off-diagonal terms (such as a13 or b13) represent hydrodynamic mass or damping only which is

associated with an inertia dependent force in one direction caused by a motion component in another.

4.5.10. Wave and Diffraction Loads

The wave-diffraction forces and moments in the direction k, Xwk ; are defined by a contribution of a

linear wave potential 𝜙𝑤 and a linear diffraction potential 𝜙𝑑, written in terms of a direction cosine in

k-direction (same direction of the linear radiation potential 𝜙𝑘 , see above). Following the Eq. 67, this

phenomena is expressed as following:

Xwk = − iωρe−iωt∬(Φw+Φd)fk∙dS

S

= − iωρe−iωt∬(Φw+Φd)∂Φk∂n

∙dS

S

for k=1,2,…,6 (85)

It is supposed that all linear wave potential, 𝜙𝑤, acting on the full wetted surface is diffracted as a

linear diffracting potential, 𝜙𝑑.

∂Φw∂n

= −∂Φd∂n

(86)

The radiation potential 𝜙𝑘 has to be determined for the constant forward speed case, taking an opposite

sign into account. So, the corresponding wave potential for deep water, has to be corrected as:

𝜙𝑤(𝑥, 𝑧, 𝑡) = Aω

ke-kzsin(ωt - kx 𝑐𝑜𝑠 𝜇 - ky 𝑠𝑖𝑛 𝜇 ) (87)

And the velocities of the water particles in the direction of the outward normal n on the surface of the

hull (see Fig. 27) is denoted as:

∂Φw∂n

= Φw∙k∙{f3 − i(f1 cos μ+f2 sin μ)} (88)

Then, the wave-diffraction forces and moments in the direction k, Xwk ; are written as:

Xwk = − iωρe−iωt (∬Φwfk∙dS − k∬Φw∙Φk∙{f3 − i(f1 cos μ+f2 sin μ)}∙dS

SS

) ; for k = 1,2,…,6

(89)



The first term in this expression accounting for wave loads is the so-called Froude-Krilov force or

moment, which is the wave load caused by the undisturbed incident wave. The second term is caused

by the wave disturbance due to the presence of the (fixed) hull surface, the so-called diffraction force.

Note the opposite sign between them, which follows the principle denoted in Eq. 86.

4.5.11. Hydrostatic restoring loads

The hydrostatic restoring force due to the oscillation of the body is written as the difference between

the hydrostatic force in still water and the constantly changing hydrostatic force due to oscillations in

the 6 degree of freedom. Motions in a given k-direction are able to generate forces and moments in a

j-direction.

𝑋ℎ𝑘 = ρg∬(∑𝑥��

6

𝑗=1

) ∙fk∙dS

S

; for j = 1,2,…,6 and k = 1,2,…,6 (90)

Where:

𝑥�� = the distance in j direction between centre of buoyancy position and new centre of gravity

position, due to motions in k direction; in [m] or [rad].

Only heave, roll and pitch motions (motions through the horizontal plane x,G,y in the steadily

translating reference) are influenced by these effects. The hydrostatic restoring force occurring from

Heave and Pitch motions will have influence on each other but roll hydrostatic restoring term will not

be influenced by any other motion. The hydrostatic restoring (spring) matrix is given below:

K =

(

0 0 0 0 0 00 0 0 0 0 00 0 K33 0 K35 00 0 0 K44 0 00 0 K53 0 K55 00 0 0 0 0 0

)

(91)

Where:

𝐾33 = 𝜌𝑔𝐴𝑤. Heave spring coefficient due to heave motion, in [kg/s2];

𝐾35 = 𝐾35 = −𝜌𝑔 ∫ 𝑥 ∙𝐴𝑤

𝑑𝑆. Heave spring coefficient due to pitch motion, in [kg/s2];



61


𝐾44 = 𝜌𝑔 ∫ 𝑦2 ∙𝐴𝑤

𝑑𝑆 = 𝜌𝑔𝛻𝐺𝑀𝑇 . Roll spring coefficient due to roll oscillations, in

[(m∙kg/rad∙s2)].

𝐾55 = 𝜌𝑔 ∫ 𝑥2 ∙𝐴𝑤

𝑑𝑆 = 𝜌𝑔𝛻𝐺𝑀𝐿 . Pitch spring coefficient due to pitch oscillations, in

[(m∙kg/rad∙s2)].

𝐴𝑤 = wetted area, in [m2].

𝐺𝑀𝑇 = transversal metacentric height, in [m].

𝐺𝑀𝐿 = longitudinal metacentric height, in [m].

It is assumed that hydrostatic restoring forces are linear by assuming small motions in heave, roll and

pitch mode of motions, so all 𝐾𝑗𝑘 coefficients are linearized depending only on linear values: GMT and

GML .

4.5.12. Correction of the Spring Matrix due to free-surface effects

Considering a tanker ship, which has large mid-ship section bounded by two vertical sidewalls

covering almost all their cross-section, if it is brought under a certain angle of heel ϕ (see Fig. 29),

then the righting stability lever arm is given by:

GZ = GNϕ ∙ sin ϕ = (GM + MNϕ

) ∙ sin ϕ (92)

Liquids in a bottom tank, a cargo tank or any other space, will affect both the static conditions where

inclinations of the ship-cross sections are described as heel angles, and dynamic conditions where sea

waves generate roll motions. If dynamic conditions are considered and sea waves exert influence in a

rolling vessel containing fluid in tanks, the righting stability lever arm will be reduced by GG’’ ∙ sin ϕ

(see Fig. 29):

GG’’ = ρ'∙i

ρ∙∇∙ (1 +

1

2tan2 ϕ) (93)

Where (all unities in SI):

GG’’ = free surface correction or the reduction of the metacentric height;

ρ′ = density of the liquid contained in the tank;

ρ = density of water (being salty or normal);



∇ = ship displacement for the considered loading condition;

i = transverse moment of inertia (second moment of area) of the liquid in the tank.

Figure 29. Metacentric Height reduction caused by free surface effects in wall-sided tanks.

With the new virtual position of KG'' , the value GM belonging to Eq. 92 and the roll spring coefficient

𝐾44 (see Eq. 91) have to be corrected as below in order to obtain the lever arm considering free surface

effects (see Fig 28 and Fig 29 below).

GM fluid = GM'' = GM − GG'' (94)

Figure 30. Schematic of GZ -Curve, corrected for free surface effects within tanks with liquid.



63


4.5.13. Linear Harmonic Ship Response to regular wave excitations

Each mode of motion defined in (Sec. 4.5.7) has its own linear harmonic response on and about the

centre of gravity location; as presented next:

Harmonic displacements as response to the external loads are written as:

rk(ωe,t) = rak∙ cos(𝜔𝑒) ; for k = 1,2,…,6 (95) Where rak is the maximum amplitude of a given mode of motion during the period T of a wave.

Harmonic ship velocity is given by:

��k(ωe,t) = − 𝜔𝑒∙ rak∙ sin(𝜔𝑒) ; for k = 1,2,…,6 (96)

And finally, harmonic ship accelerations are presented as well:

��k(ωe,t) = − 𝜔𝑒

2∙ rak∙ cos(𝜔𝑒) ; for k = 1,2,…,6 (97)

It is concluded that, in a harmonic response of the ship to a given regular excitation, knowing the

maximum amplitude of a given mode of motion, the maximum harmonic velocity and acceleration are

known as well by multiplying them by 𝜔𝑒 and 𝜔𝑒2 respectively.

4.5.14. Coupled Equations of Motion

For a given vessel, the equation of motions with respect to the steadily-translating coordinate system

are derived from Newton’s second law. The coupled equation of motions accounting for the

translations of and the rotations about the centre of gravity are built by expressing the linear

hydrodynamic forces in terms of the hydrodynamic mass and damping coefficients (See Eq. 83, 84),

the linear hydrostatic restoring forces in terms of the spring coefficients (See Eq. 91); and taking into

account the total wave loads coming from a determined sea-state. It gives place to the linear equations

of motion for each of the 6 modes of motion and is expressed in the frequency domain; see Eq. 98:

∑{(mj,k+aj,k(ωe)) ��k (ωe,t)+ (bj,k(ωe)) 𝑟��(ωe,t)+ (Kj,k(ωe)) r(ωe,t)}= Xwk ;

6

j=1

for k = 1,2,…,6

(98)



In which:

k = counter which stands for the 6 directions of the mode of motion;

mj,k = 6 x 6 matrix of inertia, in [kg];

bj,k = 6 x 6 damping matrix, in [kg/s];

Kj,k = 6 x 6 stiffness matrix, in [kg/s2];

𝑟k = ship harmonic displacement, in [m/s2] for translating modes or in [rad/s2] for rotating

modes;

��k = ship harmonic acceleration, in [m/s2] for translating modes or in [rad/s2] for rotating modes;

��k = ship harmonic acceleration, in [m/s2] for translating modes or in [rad/s2] for rotating

modes;

𝑋𝑤𝐾 = wave forces or moments, in [N].

4.5.15. Relative Wave Elevation

It accounts for the level of the surface elevation seeing from the deck of the ship.

𝑠𝑝(ωe) = 𝜁𝑝(ωe) − 𝑧 + 𝑥𝑏 ∙ 𝜃 − 𝑦𝑏 ∙ φ (99)

Where:

𝑠𝑝(ωe) = relative wave elevation.

𝜁𝑝(ωe) = sea surface elevation, in [m].

4.5.16. Total Inertia matrix of the ship

The ship mass as well as its distribution over the ship is considered to be constant during the time.

Small effects, such as the decreasing mass due to fuel consumption, are ignored. The total inertia

matrix of a ship at a given loading condition (see BV rules for defined loading conditions, Sec. 2.3) is

given below in term of the mode of motion coefficients:

m =

(

ρ∇ 0 0 0 0 00 ρ∇ 0 0 0 00 0 ρ∇ 0 0 00 0 0 Ixx 0 00 0 0 0 Iyy 0

0 0 0 0 0 Izz

)

(100)



65


In order to have a diagonal matrix, it is assumed that center of buoyancy and centre of gravity of the

ship are coincident, and zero-trim angles is encountered.

4.5.17. Response Amplitude Operator

For a given mode of motion represented by k = 1,2,…6; and defining a counter as n = 1,2,…,N; a linear

ship response 𝑟𝑘𝑛(ωe,t) is related to wave amplitude 𝐴𝑛(ωe,t) for a given frequency of encounter (ωe)

by a linear transfer function, Eq. 101 below. It strongly depends on the characteristics of the system,

namely ship hull dimensions and shape, loading conditions, forward velocity of the ship, V, and angle

of encounter, μ. Commonly, it is also referred as Response Amplitude Operator (RAO) of first order

motions.

𝑅𝐴𝑂(ωe,t) = ((𝑟𝑘𝑛𝐴𝑛) (ωe,t)) ; 𝑓𝑜𝑟 𝑘 = 1,2, … 6 (101)

4.6. Response in Irregular Waves

First a wave history in the time domain was built. Then, knowing the transfer function of a given k-

motion-mode for each n-regular-wave, the first order response history is built as well (see Eq. 102).

rk(t) = limN→∞

∑(𝑟𝑘𝑛𝐴𝑛)𝐴𝑛cos(ωent)

N

n=1

; 𝑓𝑜𝑟 𝑘 = 1,2,… 6 (102)

In which:

rk(t) = first order response history of a given mode of motion k;

(𝑟𝑘𝑛𝐴𝑛) = first order transfer function (RAO) of a given mode of motion k;

ωen = frequency of encounter, in [rad/s];

𝐴𝑛 = wave amplitude of a n-regular-wave with a frequency corrected for ωen, in [m].

In Eq. 102, after obtaining a series of n-regular-waves from the wave spectrum and forming series of

n-first-order response using a transfer function (Eq.101), a response spectrum analogous to wave

spectrum (eq. 52) can be defined in the frequency of encounter domain as:



𝑆𝑟𝑘(ωe) = |𝑟𝑘𝑛𝐴𝑛(ωe)|

2

∙Sζ(ωe) ; 𝑓𝑜𝑟 𝑘 = 1,2, … 6 (103)

Where:

𝑆𝑟𝑘(ωe) = first order response spectrum of a given mode of motion k;

Sζ(ωe) = 3-hours wave spectrum, corrected for the frequency of encounter ωe.

r in the sub-index denotes that discussion is about a given response and will be used from now ahead.

An harmonic regular response signal rk(ωe) of first order, for a given mode of motion k; is derived

from the first order response spectrum 𝑆𝑟𝑘(ωe). Basic characteristics are expressed in terms of spectral

moments are the following:

4.6.1. Response Amplitude and Period of a regular-response spectrum

The moments of a response spectrum are given similarly as done above for a wave spectrum:

𝑚𝑛𝑟𝑘 = 𝜎𝑟𝑘

2 =∫ 𝜔𝑒𝑛 ∙ 𝑆𝑟𝑘(ωe) ∙ 𝑑𝜔𝑒

∞

0

; 𝑓𝑜𝑟 𝑛 = 1,2, … ,𝑁 𝑎𝑛𝑑 𝑘 = 1,2,… ,6 (104)

Where:

𝑚0𝑟𝑘 = area under response spectrum, equal to its deviation, for a given mode of response k;

𝑚1𝑟𝑘 = 1st moment of area (centroid of spectrum) for a specified mode of response k;

𝑚2𝑟𝑘 = 2nd moment of inertia, for a particular mode of response k.

A response mean period, 𝑇𝑃𝑟𝑘 (peak period), can be found from the centroid of the response spectrum:

𝑇𝑃𝑟𝑘= 2π∙

𝑚0𝑟𝑘𝑚1𝑟𝑘

; 𝑓𝑜𝑟 𝑘 = 1,2,… 6 (105)

and the average zero-crossing response period, 𝑇𝑍𝑟𝑘, is found from the spectral radius of gyration:

𝑇𝑍𝑟𝑘 = 2π ∙ √

𝑚0𝑟𝑘𝑚2𝑟𝑘

(106)

Both 𝑇𝑃𝑟𝑘 and 𝑇𝑍𝑟𝑘

can be written as well in terms of 𝜔𝑃𝑟𝑘 (modal frequency) and 𝜔𝑍𝑟𝑘

.



67


4.6.2. Maximum short-term linear response

The short term definition corresponds to the duration of one sea state (typically 3 hours), which is

considered to be stationary. The probability density of a range of peak responses follows the Rayleigh’s

distribution. The maximal short term response, given in double amplitude and exceeded with a risk α

over a sea-state duration Dss; equals to:

𝑅𝑚𝑎𝑥𝑘 (𝐷𝑠𝑠, 𝑇𝑍𝑟𝑘, 𝛼) = 2 ∙ √2𝑚0𝑟𝑘

𝑙𝑛 (−1

((1 − 𝛼)1 𝑁⁄ ) − 1) (107)

Where:

𝑅𝑚𝑎𝑥𝑘 (𝐷𝑠𝑠, 𝑇𝑍𝑟𝑘, 𝛼) = Maximum short-term response of a given mode of motion k, in double

amplitude and unities given according the type of response;

𝐷𝑠𝑠 = total duration of the sea-state, in [s]. Typically it is equal to 3 hours = 10800 s;

𝑁 = 𝐷𝑠𝑠 𝑇𝑍𝑟𝑘⁄ Number of cycles during the time 𝐷𝑠𝑠, accounting for the number of times per

hour. that a response risk of exceedance 𝛼, repeats.

𝛼 = desired risk of exceedance, imposed as the risk that a given value of the response happens

only once during 𝐷𝑠𝑠.

4.6.3 Long Term Statistics

Considering that a short term analysis, as above described, is performed for a list of sea states observed

during a reference period named return period, 𝑇𝑅𝑘. The long term distribution can then be obtained

by cumulating the results from the short term analysis. This method consists in counting, over all sea-

states, all their maximum responses. It is given as the value that could happen only once during the

imposed period.

𝑃𝑟𝑜𝑏(𝑅 > 𝑋𝑘) = ∑ (1 − 𝑒𝑥𝑝 {−𝑅2

8𝑚0𝑟𝑘})

𝑁𝑠𝑠

𝑆𝑆=1

([(𝑇𝑅𝑘)𝑦𝑒𝑎𝑟]∙[365.24

𝑑𝑎𝑦𝑦𝑒𝑎𝑟]∙[24

ℎ𝑜𝑢𝑟𝑑𝑎𝑦 ]∙[3600

𝑠ℎ𝑜𝑢𝑟]

𝑇𝑍𝑟𝑘)

(108)



Where:

P 𝑟𝑜𝑏(𝑅 > 𝑋𝑘) desired risk of exceedance, imposed as the risk that a given value of the response

happens only once during 𝑇𝑅𝑘.

𝑆𝑆 = 1,2,3,…,Nss represents the number of sea-states to be studied.

𝑇𝑅𝑘 = Return Period, in years. For the present study, it will be considered equals to 17 years as

defined above.

4.7. Higher order analysis

Below is presented a scheme resuming the available approaches to study a ship response under the

influence of a given sea-state. Analyisis of higher order response for a given ship due to a higher order

excitation, follows a series of steps as listed bewlow and shown in Fig. 31.

- Non-linear wave excitation: From wave spectrum available, wave record 𝜁(𝑡) in time domain

will have to be re-generated by choosing different series of random phases. It triggers endless series

of time records. The largest wave, very important for an extreme response, may occur early in the

record, or later or even not at all (during a finite record). This can have a significant influence on

the response extreme values.

- Nonlinear wave excitation happens at different frequency than the one of belonging to the waves.

For instance, at lower frequencies, second order low frequency drift loads occurs. As well, second

order bound wave effects arise, which has twice the frequency with respect to first order regular

wave frequency, as defined in Sec 4.3.6.

- Nonlinear hydrostatic restoring forces occur due to the nonlinear surface elevation steep-shaped.

- The nonlinear motion of the ship generates nonlinear diffraction and radiation forces.

- A nonlinear response of seakeeping motions, loads and bending moment, are generated in

time domain. Each one of them has to be converted to some form of spectrum in order to obtain

convenient statistical data for distributions interpretation. These new spectrums depend a lot on the

random kind of waves generated. As the system is nonlinear, a spectrum of the output contains

energy at entirely different frequencies of the input wave spectrum, (which also needs to be

represented mathematicaly).

A naval architect needs a design response (an extreme dynamic internal load or displacement) with an

associated small probability which will be exceeded only once during ship lifetime. It strongly depend

on the highest wave occurrence in the time record. This implies that longer simulation time of dynamic



69


response is needed. Since such simulations often run at less than real time on even fast computers, the

computational effort becomes expensive.

Figure 31. Scheme of available approaches to study a ship response under sea influence.

In conclusion, it is more feasible to study linear response approach, then obtain an extreme value for a

desired response and finally apply corrections due to nonlinear effects.

4.7.1. Source of higher order effects in the Belgian Coastal Scatter Diagram

For values ε > 0.01, nonlinear effects should be taken into account to correct direct calculation results

coming from the long term value. For the range of navigation IN[0.6 < x < 2.0] and the range of wave

periods belonging to the Belgian Coastal sea state shown in Fig. 32, nonlinear effects are present.

Sinusoidal linear wave elevation does not represent the physical phenomena accordingly. In

conclusion, a correction must be provided to take into account these effects.

On the same hand, according to Ferrant (2014), the forward speed seakeeping problem, in a

‘Neumann-Kelvin’ linearized approach, is formulated in a similar way as the zero speed problem, with:

a) Modifications of the free surface conditions (thus specific Green functions in case of solution

based on Boundary Element Method).



b) Accounting for the wave encounter frequency, 𝜔𝑒, that is the wave frequency modified by the

Doppler effect due to the ship forward speed (as shown in Sec. 4.5.5).

The total potential in the moving frame of reference is expressed by separating the contribution due to

forward speed problem where the ship is modelled as stationary in a mobile frame of reference (wave

resistance potential), plus a ‘seakeeping’ contribution due to the incident wave and its interaction with

the ship. As in the present investigation only the encounter frequency 𝜔𝑒 correction is taken into

account, additional effects due to perturbations of the wave surface elevation need to be considered.

Figure 32. Belgium seaway spectrum up to HS = 2.0 m. Wave steepness at each HS value.

4.7.2. Additional Effects to be taken into account

Viscous Roll Damping

In the publication “A Practical Procedure for the Evaluation of the Roll Motions of FPSO’s

Including the Non potential Damping” made by J. M. Orozco (Bureau Veritas – Paris); C. V.

Raposo (Bureau Veritas – Rio de Janeiro) and Š. Malenica (Bureau Veritas – Paris), 2002; it was

mentioned that there are other sources of damping which are not taken into account by the potential



71


flow numerical model when assessing roll motions: dissipation of energy due to the drag, friction, flow

separation and some other effects. All of them cannot be modelled by the potential flow theory.

Numerical Results from J. M. Orozco et al. (2002):

In the study of J. M. Orozco et al. (2002), results are obtained for a typical FPSO configuration

(similar hull configuration than a tanker ship) with a Lpp = 320m. It was made a comparison of the

results for the damping coefficient obtained by the Ikeda–Himeno method with the results of the model

testing available. On Fig. 33 the results for two different loading conditions are presented:

Full loading conditions denoted by a letter “F”.

Ballast conditions denoted by a letter “B”.

Figure 33. Roll damping coefficient (in percentage of the critical damping) obtained by the model tests and

Ikeda-Himeno method (represented by the CALCUL curves).

Sub. Conclusions derivated from J. M. Orozco et al. (2002) research:

Results obtained by Ikeda-Himeno method overestimate roll damping. According to Fig. 33, roll

damping coefficient equals to 5% of the critical roll damping. Rudakovic (Bureau Veritas, 2015)

took this into account for the fully loaded condition analysis. On the same hand, 13% of critical roll

damping should be taken for ballast conditions.



Higher order effects in a seakeeping ship response analysis

State of the art in terms of seakeeping modes of motion simulation to accurately predict their values is

proposed in “Statistics analysis of ship response in extreme seas”, made by Bingjie Guo, Elzbieta

M. Bitner-Gregersen, Hui Sun and Jens Bloch Helmers. 2016.

Guo et al. (2016) research consisted in extending the work of Guo et al. (2013) to study statistically

the effect of ship forward speed on ship responses in extreme seas using 3D Rankine panel method.

Guo et al. (2016) Methodology of study:

In order to see the effect of nonlinear terms, the numerical results by this partially nonlinear method

(‘nonlinear’) were compared with those predicted by the linear version of the same 3D Panel method

(‘linear’) code and model test results.

The nonlinear simulations were performed in the time domain while the linear simulations were

performed in the frequency domain. Comparisons were made to verify the accuracy of the numerical

method and to check the effect of nonlinear terms. Additionally, extreme ship responses at several

forward speeds were calculated to investigate the effect they have on the results.

In order to study the effect of forward speed on ship responses in extreme seas, Guo et al. (2016) took

one irregular wave with large wave steepness kw∙(Hs/2) = 0.16 to simulate a sea state (see table 5).

Table 5. Irregular sea state characteristics. Guo et al. (2016).

A JONSWAP spectrum with a gamma parameter equal to 6 was mainly used for analysis. This sea

state is able to induce high nonlinear ship responses.

A LNG tanker was used for this study and its characteristics are presented in Table 6 for a CB equals

to 0.7. The numerical ship geometry and mesh distribution was kept the same for the linear and

nonlinear simulations.

ϒ Tp [s] Hs [m]

Full Scale 6 12.00 11.50

Model Scale 6 1.43 0.16



73


Table 6. LNG tanker characteristics at maximum draft. Bingjie Guo et al. (2016).

Guo et al. (2016) Results

a) Probability distribution:

The probability distribution of the irregular wave crests and troughs, measured for 30 minutes during

model tests, is shown in Fig. 34. Measured incident waves, fitted with the Rayleigh distribution (Eq.

58) to account for the statistics of linear potential theory, were also plotted. Results show that the wave

crest and trough distribution deviates significantly from the Rayleigh distribution when the steep waves

are present. Rayleigh distribution accounts only for linear effects and consider wave crests and troughs

having the same value; but the steeper the sea is, the higher is the difference between amplitude of

wave crest and wave trough, as shown in figure Fig. 34. Nonlinear effects are shown to become more

significant as the probability is lowered.

Figure 34. Probability Distribution. Irregular wave in model test (Tp = 12 s, Hs = 11.5 m).

Units Full Scale Model Scale

Scale [-] 1.00 1/70

Length over all (Loa) [m] 197.13 2.82

Length between perpendiculars (Lpp) [m] 186.90 2.67

Breadth (B) [m] 30.38 0.43

Depth (D) [m] 18.20 0.27

Draft (T) [m] 8.40 0.12

Displacement (Δ) [tons] 35675.00 103.83

COG_x [m] 94.87 1.36

COG_y [m] 0.00 0.00

COG_z [m] 8.26 0.12



b) Statistical analysis on ship responses in steep sea states:

The exceedance probabilities of ship responses in the irregular extreme sea state were studied. In the

numerical simulation, the irregular wave is modelled with the sum of 100 Airy wave. The numerical

simulations and the model test had the same time span in order to reduce the discrepancy due to

different sampling duration. The maximum time span of the model test is 7400 seconds (in full scale),

and all the numerical simulations have the same time span.

Ship heave and pitch motions of the LNG tanker with and without forward speed are illustrated in Fig.

35 and 36. In Fig. 35, the comparison of ship heave motions showed that crest and trough extreme

values were quite similar (see Exp curve). Both linear and nonlinear simulations underestimated the

ship heave motions compared to experimental results.

Figure 35. Heave motions of ship with/without forward speed.

In Fig. 35, nonlinear simulations gave better prediction of pitch crest for both forward velocities used

while underestimating pitch trough extreme values by almost 20 % for both cases.



75


In Fig. 36, when comparing the bending moment at mid-ship and waterline of the LNG tanker at V =

0 and V = 6 knots, it can be seen that ship motions increase significantly with increase of ship forward

speed. The comparisons showed that nonlinear simulations can give a reasonable prediction on ship

bending moments in general.

In conclusion, nonlinear method adequately predicts Bending Moment for hogging (at mid-ship and

waterline), Pitch and Heave crest; but underestimate Heave trough by 30 % (for all forward velocities

studied). These represents the maximum extreme values encountered taking into account nonlinear

wave elevation and nonlinear hydrostatic restoring.

Figure 36. Pitch motions of ship with/without forward speed.



Figure 37. Bending moment from waves. Ship with/without forward speed.

In the following table is presented the deviation between linear and nonlinear extreme values at V = 0

knots, according to results presented above, so they can be taken into account to correct linear results.

Table 7. Deviation of linear-method extreme-values with respect to the nonlinear ones at V = 0 kn.

Ship responses with different forward speeds:

The effects of forward speed on extreme ship responses were studied with different speeds: V = 3

knots, 6 knots, 9 knots and 12 knots, see Fig. 38. The ship responses at different forward speeds were

analysed with the time span equal to 90,000 seconds in order to obtain the reliable results. In

conclusion, the results showed that both 3-hour extreme ship motions and extreme hogging moment

increase with the increase of forward speeds. This is consistent to what was found in the model test.

Variable at 0 knots

Heave crest 1%

Pitch crest 14%

Bending moment sagging 15%



77


Figure 38. 3-hour extreme value with different forward speeds. Above-left: Heave amplitude. Above-right:

Pitch amplitude. Bottom: Wave load - Bending moment.

Effect of forward speed in roll motions

According to J. M. Orozco et al. (2002); when comparing wave linear theory damping coefficient

with model test results in irregular waves, it is expected that damping in waves at 0 knots speed is

higher than the damping obtained from calm water tests (as Ikeda-Himeno method). So, the values of

quadratic damping coefficient obtained in Fig. 33 are conservative as they are lower, leading to higher

values of roll amplitude.

This information regarding roll amplitude correction due to effects of forward speed is shown in J.M.J.

Journée et al, (2001): when comparing the speed dependent transfer functions of the roll motions in

beam waves and the pitch motions in head waves of a container ship with Lpp = 175 m; it is noticed

the opposite effect of forward speed on these two angular motions (see Fig. 39). Roll amplitudes

decrease when the forward speed increases. The authors affirm that this effect is caused by a with

forward speed strongly increasing lift-damping of the roll motions.



Figure 39. RAO’s of Roll and Pitch of a Containership

4.7.3 General conclusions. Additional effects to consider in linear analysis

In Rudakovic study (Bureau Veritas, 2015), it was only taken into account the correction due to wave

encounter frequency and non-potential roll damping correction. So, an additional need will arise from

the fact that empirical formulas to be proposed needs to be corrected to take into account nonlinear

effects for the right forward speed which they are meant to be suitable for (see shematic in Fig. 38).

This correction should be made according to the following specifications:

Nonlinear method underestimate Heave trough by 30 % compared to the model test results (for

all forward velocities studied: 0 knots and 6 knots). These represents the maximum extreme

values encountered taking into account nonlinear wave elevation and nonlinear hydrostatic

restoring. This corrections must be made to the linear potential flow direct calculation results.

Nonlinear extreme values of Heave and Pitch motions and Vertical Bending Moment, for V =

0 knots, should be taken into account to correct linear potential flow direct calculation results,

as shown in Table 6. Afterwards, a another correction for V = 10 knots as shown in Fig. 38

should be considered, to take into account the effects of wave resistance potential and

disturbance on sea surface when the vessel is advancing.

No correction to roll linear extreme values should be considered due to forward speed velocity

or nonlinear potential theory. Values of the quadratic damping coefficient obtained in Fig. 33

are already conservative as a lower damping is induced, leading to higher values of roll

amplitude.



79


On Fig. 40 it is shown a schematic on the kind of effects to be considered when correcting empirical

formulas due to the nonlinear surface elevation effects, namely the nonlinear restoring forces and

nonlinear spring matrix. As surge, sway and yaw represent in-plane modes of motion, second order

effects are present but not in a determinant manner when the ship is sailing, especially for the case of

Inland vessel with a high block coefficient (CB) which avoid sudden transversal section changes as it

is the case of seagoing container ship, so second order bound-wave effects (as parametric rolling) can

be neglected (even for Inland containerships). Second order drift forces generate the ship engine to

bring more power to sustain a given forward velocity, or in the case of sway and yaw motions,

manoeuvrability capacity of the ship together with ship rudder, maintain the course of the ship and

diminish second order effects. Low-frequency second order effects are determinant for fixed structures

or moored vessel, but that is not the case of the present study. So, accounting for the proposed scheme

will increase the accuracy of the real life effects that a ship could experiment in the Belgian Coastal

Sea State.

Figure 40. Schematic of non-linear correction to be considered on the proposed empirical formulas.



5. DIRECT CALCULATIONS USING HYDROSTAR CODE

5.1. Methodology

In order to obtain long term values for a given vessel response, each ship defined in the Table 4 is

modelled in Hydrostar for fully loaded conditions based on linear potential theory; with the scope of

building a long-term-response database in order to have enough data to proceed to a tendency study

afterwards. The scheme presented below will be followed to model each vessel:

1) For a maximum draft (of fresh water), mesh is built according panel method theory

From bodylines

From drawings: general arrangements and transversal sections.

Considerations:

Longitudinal measure of panels: [1-1.2] m.

Transversal measure of panels: [0.9-1] m.

2) Mesh quality and hydrostatic properties are checked.

3) Weight distribution is input. Displacement correction for salty water density.

Lightship weight assumed as weoght of the structure and empty tank weight. 0%

supplies. 0% cargo. 0% ballast.

Deadweight considered as 10% of supplies weight, cargo for maximum draft (< 100%

of total cargo weight).

4) Radiation/diffraction calculations are performed for a range of 𝜔𝑒 = [0.2-2.2] rad/s at 10 knots and

360º incident regular waves of unit amplitude.

5) Correction due to viscous effects on roll motion are imposed.

6) Calculations of motions, velocities and accelerations for radiation-diffraction conditions imposed.

7) Definition of locations on the ship where loads and relative wave elevations are desired to be

calculated.

8) Ships RAO are built for every mode of motion and load response at given locations imposed on the

hull

9) Long-term extreme values are obtained (in double amplitude) for a given response when subjected

to the Belgian Scatter Diagram conditions.



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5.2. ARGOS

ARGOS software is a naval architecture system which defines ship hydrostatics, stability and

longitudinal strength characteristics, belonging to Bureau Veritas. It is composed of a standard package

and some additional modules related to particular applications. Modules accounted in the software are

displayed in Fig. 41. The module of interest for the present research is Hydrostatic Particulars

5.2.2. ARGOS Hydrostatic Particulars

In this module, the results of the hydrostatic calculations of the vessel are displayed in terms of center

of buoyancy and centre of gravity coordinates. As the present study accounts for the fully loaded

conditions at maximum draught, trim is expected to be 0 [deg] and both centre of gravity and buoyancy

to have coincident longitudinal values, which is a key value to be taken into account.

Figure 41. ARGOS modules.



5.3. Hydrostar reference system

The following coordinate system is used by HydroStar:

Axis O-Xhs is positive in the forward direction;

Axis O-Yhs is positive to port side

Axis O-Zhs is positive upwards.

With the origin at the 0 m value of Length Overall, at the centerplane and at the free surface level for

a given draught and loading conditions; according to Fig. 42.

Figure 42. Hydrostar reference system.

5.4. Mesh Generation

Command execution: Hstar>proj>hsmsh - ship ship_name

The numerical code .hsmsh, which is the automatic mesh generator for Hydrostar, is devoted to the

automatic mesh generation of the ship type bodies, in the context of the seakeeping diffraction-

radiation codes based on the Boundary Integral Equation technique and 3D linear panel method code

as shown in Sec. 3.1.1.

5.4.1. Input to .hsmsh

.hul file: bodylines generated in ARGOS belonging to each ship are input. To represent them, it

is defined a (Obl , Xbl , Ybl , Zbl) coordinate system, see Fig. 43 (Obl is located in the 0 value of LOA

longitudinal position, just above the keel). For each section located at certain Xbl-longitudinal

position, a compound of (Ybl, Zbl) coordinates define a bodyline transversal profile. Only starboard



83


profiles are needed as vessels have symmetry respecting the center-plane (as mentioned above). It

is shown in Fig. 44 that coordinates for a given sectional position varies from the ship stern position

up to midship section (Fig. 44 left). Then midship section is kept constant up to a certain point,

where ship’s fore part is defined (Fig. 44 right). Additional values to be input in the .hul file are

the LOA, moulded breadth (B) and number of bodylines.

.mri file: takes into account the how to mesh the wetted hull surface, according subsequent defined

parameters:

Fore part shape, with the absence of bulbous bow.

Aft part shape, inclusion or absence of stern transom and additional shapes.

Draught of the ship for the present loading conditions, in [m], measured from above-keel

position according to BVNR217 rules. Once defined, it cannot be changed during the setting-

up process. Geometry obtained will be cut in the waterline level.

Trim of the ship, in [degrees].

Heel angle, in [degrees].

Meshing parameters to achieve a 3D panelled geometry. It is defined in number of elements

distributed in longitudinal direction Xlb, from [0 to LOA]; and number of element in transversal

direction Ylb along each section from [-B/2, B/2]. By default, the mesh generator takes into

account Lpp (or very similar value) value as it meshes the hull up to waterline level.

Figure 43. Coordinate system (Obl , Xbl , Ybl , Zbl) used to represent body-lines in .hul file.



Figure 44. Body-lines of vessel 43 at different sections. (Ybl, Zbl)-profile for aft (left) and fore (right).

5.4.2. Output of .hsmsh file

Input file for hslec (mesh), as in Fig. 45. Performing the command Hstar>proj>hslec

“ship_name”.hst permits reading the mesh and display information regarding the number of

panels it contains; and obtaining hydrostatic characteristic of hull geometry:

Center of buoyancy at the given draught (for fully loaded conditions);

Inertia matrix.



85


Figure 45. Vessel 20. LOA = 135 m; B = 22.5 m. Nlong = 110 divisions; Ntrans = 15 divisions.

To compute a preliminary check of generated mesh, Hschk command is used: Hstar> proj>

“ship name”.chk. Following verifications are performed:

Consistency of the normal vector orientation;

Panels with null area;

Panels over the free surface;

Panels at free surface;

Overlapped panels;

Holes (neighbour-absences);

A report is printed on the screen giving the number of panels presenting inconsistencies, see Fig. 46.

Figure 46. Hschk module report.



5.5. Additional Hydrostatic computation: Loading conditions

Command execution: Hstar>proj>hstat – “ship_name”.wld

Hydrostatic characteristic and inertia matrix is obtained from hslec module. In order to fulfil the matrix

[m] (Eq. 100) and to obtain hydrostatic restoring coefficients in stiffness matrix [K] (Eq. 91), the

longitudinal mass distribution of fully loading condition is input. Afterwards, it is checked the

hydrostatic correspondence of the model geometry with the real vessel characteristics input. In

addition, it is defined the sections where loads coming from hydrodynamic analysis are computed.

5.5.1. hstat input

A .wld file which contains the load distribution as well as the longitudinal division of the ship

in sections to compute the wave loads, is built as in Fig. 47. Following parameters are

considered:

From BV hydrostatic documents maximum draught loading conditions are imposed.

Case 1. Structure:

Lightship displacement (∆strucure) considering structural members only;

Longitudinal ship-structure’s centre of gravity, XGstructure; and vertical position of the centre of

gravity, ZGstructure (relative to the still water level, as defined in the hydrostar reference system);

Structure longitudinal gyration radius, assumed to be IXX = 0.35B. It is a valid assumption for

slender-body vessels with a high CB and long cargo (midship) section with vertical walls alongside.

Case 2. Cargo:

Displacement of total tank cargo + 10% supplies (fully loaded): ∆cargo. It is obtained subtracting

∆total − ∆strucure.

Cargo longitudinal gyration radius, taken as IXX = 0.35B as well.

Longitudinal coordinates of the cargo ends on the ship: X1 and X2; representing the aft-most cargo

bulkhead location and the fore-most cargo bulkhead location, respectively. These coordinates are

obtained from each vessel general arrangement plane, see Fig. 48.



87


Figure 47. Loading distribution and section definition for Vessel 20 at maximum draught (fully loaded).

Figure 48. Total length scheme of the cargo-midship section of Friendship.

ZFSURFACE 0 m

#Length Overall 135.00 m

#Draught 4.30 m

DISMASS TYPE 1

# case Description Δ [kg] X1 [m] X2 [m] XG [m] KXX [kg*m^2] ZG [m]

1 Structure 1606757 0.00 135.00 64.19 5.25 -1.24

2 Cargo 6550793 18.00 124.20 69.13 5.25 -0.50

# TOTAL 8157550 68.16 -0.65

ENDDISMASS

SECTION XPOSITION [m] YGTOTAL [m] ZGTOTAL [m]

1 0.00 0.00 -0.65

2 6.75 0.00 -0.65

3 13.50 0.00 -0.65

4 20.25 0.00 -0.65

5 27.00 0.00 -0.65

6 33.75 0.00 -0.65

7 40.50 0.00 -0.65

8 47.25 0.00 -0.65

9 54.00 0.00 -0.65

10 60.75 0.00 -0.65

11 67.50 0.00 -0.65

12 74.25 0.00 -0.65

13 81.00 0.00 -0.65

14 87.75 0.00 -0.65

15 94.50 0.00 -0.65

16 101.25 0.00 -0.65

17 108.00 0.00 -0.65

18 114.75 0.00 -0.65

19 121.50 0.00 -0.65

20 128.25 0.00 -0.65

21 135.00 0.00 -0.65

ENDSECTION

ENDFILE



Longitudinal and transversal centre of gravity of the Cargo. As this includes the weight of the tank

contents for maximum draught as well as 10% supplies, it is necessary to define each of them by

taking longitudinal and transversal moment of displacements:

𝑋𝐺𝑐𝑎𝑟𝑔𝑜 = [(𝑋𝐺𝑡𝑜𝑡𝑎𝑙 ∗ ∆𝑡𝑜𝑡𝑎𝑙) − (𝑋𝐺𝑙𝑖𝑔ℎ𝑡𝑠ℎ𝑖𝑝 ∗ ∆𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒)] (∆𝑐𝑎𝑟𝑔𝑜)⁄ (109)

𝑍𝐺𝑐𝑎𝑟𝑔𝑜 = [(𝑍𝐺𝑡𝑜𝑡𝑎𝑙 ∗ ∆𝑡𝑜𝑡𝑎𝑙) − (𝑍𝐺𝑙𝑖𝑔ℎ𝑡𝑠ℎ𝑖𝑝 ∗ ∆𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒)] (∆𝑐𝑎𝑟𝑔𝑜)⁄ (110)

Where:

𝑋𝐺𝑐𝑎𝑟𝑔𝑜 : longitudinal centre of gravity of the total tank content plus 10% of supplies.

𝑍𝐺𝑐𝑎𝑟𝑔𝑜 : transversal centre of gravity of the total tank content plus 10% of supplies.

∆𝑡𝑜𝑡𝑎𝑙 : total ship displacement. It is obtained from Argos, in the Capacity Plan module

(Fig. 41). This displacement is corrected for salty water (maintaining zero-trim), for an

invariant draught and geometry differences (as mentioned above in body lines hsmsh

definition).

𝑋𝐺𝑡𝑜𝑡𝑎𝑙 : longitudinal centre of gravity for ∆total

𝑍𝐺𝑡𝑜𝑡𝑎𝑙 : transversal centre of gravity for ∆total. Correction due to free-surface effects is

taken into account (see. Eq. 94 and Fig. 30). Value GG’ is provided in the hydrostatic

manual of each vessel.

Vessel’s length overall (LOA) has been divided into 21 equidistant sections whose coordinates

have been adapted to the mean still water level (Hydrostar frame of reference).

5.5.2. hstat output

After the commands hstat is run by the module hstat “ship_name”.wld, it is displayed again the total

hydrostatic properties coming from the loading distribution, where the difference between total centre

of gravity and total center of buoyancy is shown. Hydrostar offer the option to correct the calculated

centre of gravity and make it coincident to the center of buoyancy location, in order to make the matrix

[m] linear and keep zero-trim condition for fully loaded case.



89


Figure 49. Fully-loaded conditions for Vessel 20. Hydrostatic difference with hslec module.

Differences arise from the fact that values provided in the owner’s hydrostatic documents are for a

fresh water density. A correction for salty water is then carried out. Another important difference

comes from the dimension differences between computational geometry built using the hsmsh

modulus and the real hull. Nevertheless errors coming from mass distribution input and hydrostatic

calculations is often less than 2%, as show in Fig. 49. After the correction is performed, centre of

gravity and buoyancy are coincident and diagonal values Ixx and Iyy are shown as in Fig. 50.

Following characteristics are also calculated:

Wetted Hull Surface

Waterplane Area

Waterplane Inertia

Distances between the centre of buoyance and the metacentre (BM).



Figure 50. Correction of hydrostatic values from input loading conditions. Vessel 20.

5.6. Diffraction radiation computation

Executing the command: Hstar>proj>hsrdf “ship_name”.rdf

The module hsrdf of HydroStar is used to solve the problem of diffraction and radiation around fixed

and floating ship hull. It is taken into account the first order potential theory of wave surface elevation,

the integral equations of boundary element method and the evaluation of associated Green functions

on hull wetted surface. After the hydrostatic properties are known, the module hsrdf is run to obtain

radiation, diffraction and hydrostatic restoring loads for a series of sinusoidal waves of unit amplitude

and various frequencies, as defined in Sec. 4.5.6. (Loading Superposition) and Fig. 25.

5.6.1. Radiation solution

Representing the potential flow around a vessel hull when it oscillates in calm water. It is calculated

the added-mass, defined as the load on the vessel due to its unit acceleration and the linear radiation

damping, which represents the ratio between the load and vessel’s velocity. The matrices of added-

mass and radiation damping are of 6 x 6 dimensions for a determined ship hull (Eq. 83, 84, and 91).



91


5.6.2. The diffraction solution

Accounting for the potential flow around a vessel hull remaining immobile in incoming waves. The

wave excitation loads are obtained by integrating the dynamic pressure on the fixed vessel in incoming

waves, (see Sec. 4.5.10).

5.6.3. hsrdf input

Extension of the file that will store the results;

Wave frequencies: from 0.2 to 2.2 [rad/s];

Wave headings: from 0 to 350 degrees, for an increasing step of 10 degree;

Water depth (sea bed) assumed to be constant at 10 m;

Ship forward speed imposed at 10 knots. The encounter-frequency approximation is considered,

based on the use of the Green function associated to it (see Eq. 89, Sec. 4.5.10).

5.6.4. hsrdf output

Added-mass matrix.

Linear radiation damping.

Wave excitation loads.

5.7. Motion Computation

Executing the command: Hstar>proj>hsmcn “ship_name”.mcn

The modulus hsmcn allows the computation of each mode of motion on and about a ship centre of

gravity (according to the Eq. 98, presented in sec. 4.5.14), for the compound of wave frequencies with

unit wave amplitude defined in the hsrdf modulus. The response amplitude of each mode of motion is

obtained.



5.7.1. hsmcn input

To build the motion of each ship for a given excitation (unity wave amplitude), hydrostatic properties

coming from hstat modulus results together with the hydrodynamic properties calculated in hsrdf

modulus are used as input. These are listed below:

Centre of gravity position with respect to the origin of the reference system. All the results

obtained in hsrdf modulus in the center of buoyancy are transferred to the centre of gravity,

consistently, to enable the motion computations.

Gyration Radius described with respect to the centre of gravity. Values Kxx, Kyy and Kzz

are defined from hstat modulus as well.

Inertia Matrix: defined at the centre of gravity and having diagonal shape.

Stiffness Matrix: from hstat modulus.

Damping Matrix: The damping due to a linear radiation potential is computed by HydroStar in

the hsdmp module: in addition to the radiation damping, for roll modes of motion, a correction

is made according to J. M. Orozco et al. (2002) work shown on Fig. 33 (for the full loaded

case) and scheme presented in Fig. 40. The roll damping coefficient is defined as 5% of the

critical roll damping. Critical damping is calculated in an additional modulus named hsdmp,

were the inputs are the same as the present modulus. Results of the critical damping calculations

are shown below for Tristan. Linear and quadratic coefficients are not taken into account.

Figure 51. Vessel 43. Quadratic roll damping estimation using hsdmp.

5.7.2. hsmcn output

Output: motions, velocities and accelerations of a given vessel under study, defined in the steadily

translating frame of reference (G,x,y,z).



93


5.8. Pressure and wave elevation computation

Command computation: Hstar>proj>hsprs “ship_name”.prs

Using the module hsprs, relative wave elevation (RWE) and pressure acting on a vessel hull wetted

surface are defined in specific points in order to assess the loads and vertical bending moment coming

from the sea. These values are obtained previously with reference to the centre of gravity using the

module hsrdf and hsmcn. Then, applying the coupled motion equations (Eq. 98 in sec 4.5.14), they are

found in each desired point. Imputing coordinates on the ship according to the scheme presented

following.

5.8.1. hsprs input

Coordinates of points on the hull wetted are proposed taking into account the governmental standards

of the Federal Public Service of Transport and Mobility in Belgium and the Ministery of Ecology,

Sustainable development and Energy in France.

Longitudinally, the hull is divided in 5 sections according Fig. 52 A on Hydrostar reference system:

Stern, at 0 m.

CARGO_A: Aft section at 0.33∙L. Measured on port (_AP) and Starboard (_AS) sides.

CARGO_M: Midship section at 0.5∙L. Determined on port (_MP) and Starboard (_MS) sides.

CARGO_F: Fore section at 0.75∙L. Obtained for port (_FP) and Starboard (_FS) sides.

Bow: at Lvalue.

Transversally, the hull is divided in 5 sections according Fig. 52 B (Hydrostar reference system): 0.5B,

0.25B, 0, -0.25B and -0.5B. Vertically (Fig. 52 C), sections are represented at 2Tmax , 1.5Tmax , Tmax ,

0.5Tmax and 0Tmax ; being Tmax located at Zhs = 0m.



Figure 52. Proposed locations to assess relative wave elevation, wave loads and vertical bending moment. A,

Longitudinal divisions. B, Transversal division. C, vertical divisions.

5.8.2. hsprs output

Pressure and RWE at selected points.

5.9. Construction of the transfer functions

Command execution: Hstar>proj>hsrao “ship_name”.rao

After performing the calculations, the following transfer functions are constructed by using the

command “hsrao”, according definition presented in Eq. 101. RAO’s are given in term of 𝜔𝑒.

Motion, velocity and acceleration RAOs;

Wave kinematics at a point around ship hull;

Loads at a defined point around the ship hull.



95


RAOs are defined according following system of reference:

Figure 53. Ship system of reference relative to wave angle incidence.

The relation ship between the wave direction βgeo, the azimuth α relative to Hydrostar reference is :

𝛽𝑅𝐴𝑂 = −𝛽𝑔𝑒𝑜 + 𝛼 + 180º (111)

On Figure 52, for roll acceleration (αR), it can be identified 3 regions that typicaly can be encountered

in a linear transfer function: a low frequency area, with vertical motions dominated by the restoring

spring term, wave length is large when compared with the horizontal length; a natural frequency area,

with vertical motions dominated by the damping term, yielding a high resonance can be expected in

case of a small damping and a high frequency area, with vertical motions dominated by the mass term.

This yields that the waves are “losing” their influence on the behaviour of the body.



Figure 54. Roll-acceleration transfer function for Vessel 20.

Figure 55. Relative wave elevation transfer function for Vessel 20, at CARGO_A.

0.0

5.0

10.0

15.0

20.0

25.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

αR/A

[ra

d/s

2/m

]

ωe [rad/s]

0° 10°

20° 30°

40° 50°

60° 70°

80° 90°

100° 110°

120° 130°

140° 150°

160° 170°

180° 190°

200° 210°

220° 230°

240° 250°

260° 270°

280° 290°

300° 310°

320° 330°

340° 350°



97


5.10. Long term value for a given response

Command execution: Hstar>proj>hspec “ship_name”.spc

HYDROSTAR hspec module computes different kind of results. The following data are relevant to

the spectral calculation :

5.10.1. hspec input

RAO files coming from hsrao module.

Belgium Scatter Diagram with all sea states defined in terms of peak period, representative

wave height Hs and main angle of spreading 𝛽𝑔𝑒𝑜.

Azimuth, defined as specified in Section 3.3.

Desired locations of long term calculations, defined according hsprs module, Fig. 52.

Calculations are carried out using expressions JONSWAP spectrum defined for 𝛾 = 1 (see Eq. 60),

spreading cosine rule as Eq. 61, short-term value (see Eq. 107) for each sea-state belonging to the

Scatter Diagram and long term value expression (see. Eq. 108) for a return period of 17 years.

5.10.2. hspec output

Long term values (in term of double amplitude) are obtained for each mode of motion translating and

rotating about the steadily translating reference. Relative surface elevation (RWE) and loads at are

given for each desired location on the ship hull input in the module hsprs. Below is shown an example

of the long term value calculated for heave and pitch accelerations of Friendship vessel with an

associated probability of occurrence of only one time during the return period (see Fig.54).

It is built a data-base containing all the long-term value for each mode of motion, relative wave

elevation and vertical bending moment defined as Figure 50A, and accelerations defined in all

specified regions.



Figure 56. Heave acceleration long-term-value of Vessel 20.

Figure 57. Pitch acceleration long-term-value of Vessel 20.



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6. EMPIRICAL FORMULAS

6.1. General Scheme

Present analysis will be carried out only for long-term motion and acceleration entities, including long

term value of relative wave elevation. As a data-base is already created, this study can easily be

continued for the extreme-value loads and bending moment encountered at the return period of 17

years. A common pattern in all of the motion and acceleration BV NR217 formulas can be observed:

EntityX= 𝑎𝐵(𝑋,𝐻𝑠)∙𝑌𝑋 (112)

in which:

EntityX = any motion amplitude, linear acceleration or angular acceleration for a given vessel X.

Y = rest of the formula for a vessel X.

𝑎𝐵(𝑋,𝐻𝑠)= motion and acceleration parameter for a determined vessel X and a limit of navigation Hs.

6.2. Developing the problem

Looking at the BV NR17 rules and accounting for Sway and Heave acceleration (Eq. 6 and 7), both

equations have a sloped-shape defined by ‘Y’:

𝑌𝑋 = Entity

X

𝑎𝐵(𝑋,𝐻𝑠) (113)

Where:

𝑌𝑠𝑤𝑎𝑦𝑋 = 7.6

𝑌ℎ𝑒𝑎𝑣𝑒𝑋 = 9.81

And for Yaw acceleration, 𝑌𝑦𝑎𝑤𝑋 = 15.5 L⁄ .

It is desired to build a representation of all long-term values in terms of a slope to check if the obtained

slope can be addressed in terms of main ship characteristis. So it is important to understand which

variables are acting on the parameter aB.



6.3. Characteristis of the long-term-response

Following are presented the long term results for 2 vessels belonging to the built database:

Table 8. Characteristics of Vessel 17 and 31. Inland Navigation Vessels defined in Table 4.

Vessel’s Reference 17 31

L [m] 107.95 84.47

B [m] 11.41 9.64

T [m] 3.80 3.16

hw [m] 9.29 8.03

Δ [ton] 4297.35 2313.25

CB 0.88 0.86

Figure 58. Vessel 17 IN vessel. Long-term values. Motion; accelerations; Relative Wave Elevation vs Hs.



101


Figure 59. Vessel 22 IN vessel. Long-term values. Motion; accelerations; Relative Wave Elevation vs Hs.

For all entities shown in Fig. 58 and 59 for 2 vessels having different characteristics, it is evident that

larger long-term values are obtained when increasing the limit of navigation (defined in terms of Hs),

having as well linear behaviour with a positive slope, for the range of navigation studied (defined in

Table 3). This phenomena is encountered in all of the 46 inland vessel studied.

The motion-acceleration parameter aB is function of the forward velocity (V = 10 knots for all studied

cases), length between perpendiculars and representative wave height Hs. Thus, for a given ship, this

parameter will vary only in terms of Hs. Long-term values are supposed to be encountered at the

maximum limit of navigation imposed, or very close to it.

It is shown in Fig. 60 the deviation of the long-term representative wave height with respect to the

limit of navigation value, imposed at IN(2.0) and plotted for each ship length. The limit of navigation

represents the maximum value the scatter diagram can have (value input in aB(Hs)). In general, each

deviation value (belonging to each mode of motion) represents how close the RAO peak is with respect

to the scatter diagram peak, in 17 years of operation. As it is evidenced, maximum deviation for all

ship lengths at a given mode of motion, does not overtake 5 % (it only happens for yaw acceleration



for the shortest vessel encountered in the set, which will be discarded for yaw analysis). So, as

deviations are small enough, parameter aB defined in terms of maximum Hs (limit of navigation)

represents a good assumption. Therefore, linear representation according Eq. 112 is adequate.

It is important to remark the fact that only the limit of navigation IN(2.0) has been taken into account

in Fig. 60, which represents the shortest band of peak periods available for the extreme response to

occur: 𝜔𝑝 = [0.79 – 1.05] rad/s, see Sec. 3.2; which is corrected for the forward Velocity of V = 10

knots afterwards. Moreover, a good assessment of the Belgian Scatter Diagram is obtained. Note that

if the errors in Fig. 60 were all 0%, it would signify that the maximum response that the set of ship can

encounter (in 17 years) is achieved. This supposed response would not be able to be overtaken in any

scatter diagram of the rest of the world, due to the limited range of navigation an Inland vessel has.

So, it can be concluded that the Belgian Coastal Scatter Diagram, under the conditions of V = 10 knots,

represents an adequate excitation to obtain the maximum long-term value that any mode of motion can

face. Only for Vessel 38 (the smallest vessel), yaw acceleration this threshold is superated. It is set

aside the analysis of load and bending-moment deviations from each limit of navigation studied. It is

concluded the created database represents a very good estimate of the maximum extreme responses.

Figure 60. Deviation from Hs maximum assumed in Table 3.



103


6.4. Methodology

A linear behaviour is expected for every vessel when plotting each entity’s long-term value vs

parameter aB. This fact match with the shape of the Eq. 112. The objective becomes obtaining the

slope for all the cases at each entity and understand which variable would influence them (whether

ship length, breadth, draught,…, or any combination of them). A methodology is detailed below:

I) It is obtained the slope EntityX ⁄ (aB)X, Hs

. Once obtained the corresponding long-term value of a

given Entity for each of the 15 limits navigation defined in Sec. 3.5, a slope can be plotted.

Figure 61. Heave acceleration vs aB. Range of slope values is assesed.

Above is shown the relation of direct calculation results for extreme-values of heave acceleration vs

parameter aB, both taken for representative wave height Hs from 0.6 m to 2.0 m (from IN(0.6) to

IN(2.0)). In order to measure the accuracy of the current BV NR217 formula: aH = 9.81aB , it is

added a linear trend-line for the bottommost and uppermost tendencies and obtained the value of the

slope aHeave aB⁄ for each of them. They represent the limits of slope range. As shown on Fig. 61,

y = 10.369xR² = 0.9956

y = 45.971xR² = 0.995

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 0.025 0.05 0.075 0.1 0.125 0.15

ahe

ave

[m

/s2]

aB



minimum slope-value found is 10.37 belonging to vessel 45 and maximum one is 45.97 corresponding

to vessel 22. By simple observation it can be concluded that BV-NR217 underestimate the values of

heave acceleration as aH aB⁄ = 9.81 is out of this range.

Table 9. Characteristics of Vessel 45 and 22. Inland Navigation Vessels defined in Table 4.

Vessel’s reference 45 22

Slope aheave/aB 10.37 45.97

L [m] 131.86 33.70

B [m] 22.80 6.40

T [m] 5.20 2.20

Displacement [ton] 14728.90 405.00

GM [m] 7.02 0.69

KM [m] 11.20 2.77

L/B [-] 5.78 5.27

Table 9 contains a list of vessel main characteristics of the slope range limits imposed by vessel 45

and vessel 22 regarding heave acceleration. By simple observation of Fig. 61 and without entering in

further analysis, it can be observed that a smaller vessel in terms of length and breadth (and

consequently lesser displacement) will undergo greater accelerations due to a bigger motion amplitude

for every wave height limit assessed.

II) After defining the slope (Entity)X ⁄ (aB)X , Hs

for each entity under investigation in order to verify

Eq. 6, 7 and 15 and understanding the path to be adopted in order to improve current BV NR217

motion and acceleration formulas; it is proceeded to find the slope for further combinations as

disclosed in Fig.62. The best way to quick assess the influence of each variable on a vast group of

results obtained for each ship’s entity, is to reagroup them. Calculating several slopes and plotting

them against main ship characteristics (and combinations of them) will make it easier to find a good

tendency for a given entity: it will allow first of all to independize all possible development from

representative wave heigh (Hs). It is verified for each current BVNR217 formula if underestimations

(or overestimations) are encountered for the slope values (as explained with heave acceleration). The

most important advantage of following this scheme is that possible linear equations can be developed.

It is desired that the proposed empirical formulas are built in terms of the ship’s main characteristics

(shown in Table 4), including length between perpendiculars (L), moulded beam (B), maximum



105


draught (T) and displacement for maximum load condition (Δ). On the same hand, shape

characteristics as ship slenderness (L/B) and block coefficient (CB) are taken into account. And finally,

some combinations of main characteristic variables are as well considered as already mentioned.

Figure 62. 11 x 18 matrix built after combining the proposed slope vector with main ship’s features.



Figure 63. Surge acceleration vs combination: aB·B. Linear behaviour is verified.

It was developed a Visual_Basic program which provide the value of R2 values (meassure of the

deviation of a set of data with respect to a tendency curve) for all the proposed combination. R2 is

imposed to be ≥ 0.99 for all the cases (all the vessels studied for the present loading conditions) as a

requirement to be considered linear in order to obtain its slope afterwards. Above was showed the

behaviour of surge acceleration (aH) vs the combination (aB·B).

III) For the motions and accelerations having no linearity when carrying out previous steps (I and II),

it is proceded to suggest any variation of the current formula by first validating it and then proposing

step-by-step variations of its parameters. This situation can be encountered in Roll Amplitude Extreme

Values, see Fig. 64. It is presented on Fig. 64 that for some vessels, linearity is achieved ( R2 > 0.99 ),

while for other ones it is not ( R2 < 0.99 ). Further tendencies and proposals based on this methology

(assuming linear behaviour according Eq. 112) will lead to considerable errors due to the propagation

of error induced by cases with R2 < 0.99, ruining general tendencies and proposal accuracy.

y = 0.0935xR² = 0.9979

y = 1.5069xR² = 0.9992

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.4 0.8 1.2 1.6 2.0 2.4

asu

rge

[m

/s2]

aB B [m]



107


Figure 64. Roll Amplitude vs parameter aB.

Knowing that Roll Amplitude is defined in BV-NR217 as:

AR = aB√E, with E = 1.39GM∙B

0.35∙B

It is proposed to verity the behaviour of the value 1.392 for each vessel, if this value is achieved or a

different one:

AR∙0.35∙(B0.5)

aB∙(GM0.5)

= (1.392) (114)

If it is differemt, it is proceeded to evaluate the tendendy of AR∙0.35∙(B0.5)

aB∙(GM0.5) for each representative wave

height (Hs) against a parameter that may affect this entity, being physically stable and adequate from

the designing point of view; meaning that variables as Draft (T) can be directly discarded as it is not

allowed to be increased as well as Rule Length (L) which will not affect at all rolling motions and

accelerations. GM and GM corrected are proposed as a starting point.

y = 246.27xR² = 0.9979

y = 135.62xR² = 0.958

y = 89.882xR² = 0.9843

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0.00 0.03 0.05 0.08 0.10 0.13 0.15

gro

ll [r

ad]

aB [-]



6.5. Proposed Equations

Following models are developed considering the scheme disclosed in the Fig. 62. These are built in

terms of ship’s main characteristics, trying to provide an easy way for the assessment of the physics

involved behind. For instance, by decreassing in length L, breadth B or displacement Δ, it will lead to

higher amplitudes and accelerations. This principle should be kept in mind when examining each

proposal. Tendencies and validations related to each proposal are given in Section 1 and 2 of the

Appendices.Validations are undertaken in two ways:

Validation against direct calculation results of the ships presented in the Table 2, for Rudakovic

(2015) direct calculations.

Validation against the total database developed in the present study, including Rudakovic (2015)

results.

A good compromise between the empirical formulas proposed and the long-term-response direct

calculations results has been achieved. Assessment of all NR217 rules was carried out, evidencing an

underestimation in all the entities studied, except for the case of surge acceleration where the rule

proposes a constant value equals to 0.5 m/s2, representing it a great overestimation. Important to

highlight for the following proposed equations: Block Coefficient CB is found not to be determinant in

the general behaviour of the ship-modes-of-motion; its value is very hight (above 0.82 for fully loaded

case), due to slender vessels with a very marked parallelepiped section. It will not influence in

significant sense as done for sea-going vessels.

6.5.1. Sway Acceleration, in [m/s2]

If 𝛥 < 1700 ton :

𝑎𝑆𝑊 = 𝑎𝐵 ∙ (17.388 − 0.6165𝐵) (115)

If (1700 < 𝛥 < 5000) ton :

𝑎𝑆𝑊 = 𝑎𝐵 ∙ (15.074 − 0.4297𝐵) (116)



109


If (𝛥 > 5000) ton :

𝑎𝑆𝑊 = 𝑎𝐵 ∙ (13.644 − 0.3347𝐵) (117)

6.5.2. Surge Acceleration, in [m/s2]

𝑎𝑆𝑈 =300.47 ∙ 𝑎𝐵

𝐿 (118)

6.5.3. Heave Acceleration, in [m/s2]

If 𝐿 𝐵 ≤ 6⁄ and 𝐿𝐵 < 900 m2

𝑎𝐻 = 𝑎𝐵 ∙ (1566

𝐿− 0.019) (119)

If 𝐿 𝐵⁄ ≤ 6 and 𝐿𝐵 > 900 m2

𝑎𝐻 = 1374.2 ∙ 𝑎𝐵 ∙ (1

𝐿) ∙ 𝑓𝐻 (120)

Where:

𝑓𝐻 = correction factor due to Δ: If ∆ < 1900 ton; 𝑓𝐻 = 0.8; otherwise 𝑓𝐻 = 1.0 .

If 6 ≤ 𝐿 𝐵⁄ < 9

𝑎𝐻 = 𝑎𝐵 ∙ (1575.7

𝐿+ 0.0006∆) (121)

If 𝐿 𝐵⁄ > 9

𝑎𝐻 = 𝑎𝐵 ∙ (185.64

𝐵+ 0.0005∆) (122)

Factors (0.0006∆) and (0.0005∆) in the last two above expressions represent a correction for their

expressions. Even though heave acceleration have to decrease while displacement increases, the rate

at which 1575.7

𝐿 and

185.64

𝐵 decreases is higher.



6.5.4. Pitch Amplitude, in [rad/s2]

If 𝐵 < 13.5 m :

𝐴𝑃 =47.543 ∙ 𝑎𝐵

𝐿 (123)

If 𝐵 > 13.5 m :

𝐴𝑃 =86.091 ∙ 𝑎𝐵

𝐿1.037 ∙ (∆ ∙ 𝐵)0.037 (124)

6.5.5. Pitch Acceleration, in [rad/s2]

𝛼𝑃 =1266.6 ∙ 𝑎𝐵𝐿1.653

(125)

6.5.6. Pitch Period, in [s]

If (𝛥 < 8000)ton and (𝐵 < 13.5)m

𝑇𝑃 = √(47.543

1266.6∙ (2𝜋)2) ∙ 𝐿0.653 (126)

If (𝐿

𝐵< 6)ton and (𝐵 > 13.5)m

𝑇𝑃 = √(86.091 ∙ 𝐿0.616

1266.6 ∙ (∆𝐵)0.037∙ (2𝜋)2) (127)

6.5.7 Vessel relative motion

Ship stern at x = 0 m

ℎ1_𝑆𝑡𝑒𝑟𝑛 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (293.01

𝐿− 0.4996) (128)



111


Aft cargo position “CARGO_A” at x = 0.33L, see Fig. 52.A

ℎ1_𝐶𝐴𝑅𝐺𝑂_𝐴 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (179.45

𝐿+ 0.9823) (129)

Midship cargo position “CARGO_M” at x = 0.5L, see Fig. 52.A

ℎ1_𝐶𝐴𝑅𝐺𝑂_𝑀 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (154.93

𝐿+ 1.179) (130)

Fore Cargo position “F_CARGO” at x = 0.75L, to Fig. 52.A

ℎ1_𝐶𝐴𝑅𝐺𝑂_𝐹 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (207.02

𝐿+ 0.767) (131)

Vessel relative motion on Bow position at x = L, to Fig. 52.A

ℎ1_𝐶𝐴𝑅𝐺𝑂_𝐹 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (394.39

𝐿− 0.336) (132)

6.6 Nonlinear Correction

The source of nonlinear corrections is the nonlinear surface elevation and hydrostatic restoring forces.

In consequence, nonlinear heave and pitch motions have higher amplitudes compared to linear ones.

It is supposed that the same proportion between nonlinear and linear amplitudes will be encountered

in the case of heave and pitch accelerations. Below it is specified how corrections are accounted for:

According to Table 7, an extra percentage should be considered as a result of merely nonlinear

effects taken at V = 0 knots. Nonlinear Correction Coefficients are proposed.

Corrections due to the increment in forward velocity to V = 10 knots (shown in Fig. 38) are defined

in terms of Forward Velocity Coefficient.

Relative motion (RWE) Nonlinear Coefficient and Forward Speed coefficient are defined as the

average between heave and pitch coefficients for both cases.

Correction due to 30% of difference between nonlinear and experimental results for heave

acceleration, according Fig. 35; for any forward speed.



To bear in mind: nonlinear corrections come from long-term calculations obtained for a LNG Tanker

with CB equals to 0.7. In consequence, proposed correction should be accounted in the future from

inland-vessel simulations considering nonlinear potential theory, for a set of few representative

geometries that allows to make correction to all developed linear long-term data-base. This nonlinear

study intends to have a first insight on the topic.

6.6.1 Heave Nonlinear Coefficient

𝑓𝑛𝑙𝐻 = 1.01 (133)

6.6.2 Pitch Nonlinear Coefficient

𝑓𝑛𝑙𝑃 = 1.14 (134)

6.6.3 Relative Motion Nonlinear Coefficient

𝑓𝑛𝑙ℎ1 = 1.065 (135)

6.6.4 Heave Forward Speed Coefficient

𝑓𝑉𝐻 = 1.58 (136)

6.6.5 Pitch Forward Speed Coefficient

𝑓𝑉𝑃 = 1.50 (137)

6.6.6 Relative Motion Forward Speed Coefficient

𝑓𝑉ℎ1 = 1.54 (138)

6.6.7 Heave Experimental Coefficient

𝑓𝑒𝑥𝑝𝐻 = 1.30 (139)



113


6.6.8 Total Corrections

Corrected Heave Acceleration, in [m/s2]

𝑎𝐻∗ = 𝑎𝐻 ∙ 𝑓𝑛𝑙𝐻 ∙ 𝑓𝑉𝐻 ∙ 𝑓𝑒𝑥𝑝𝐻 (140)

Corrected Pitch Amplitude, in [rad]

𝐴𝑃∗ = 𝐴𝑃 ∙ 𝑓𝑛𝑙𝐻 ∙ 𝑓𝑉𝐻 (141)

Corrected Pitch Acceleration, in [rad/s2]

𝛼𝑃∗ = 𝛼𝑃 ∙ 𝑓𝑛𝑙𝐻 ∙ 𝑓𝑉𝐻 (142)

Corrected Relative Motion

ℎ1𝐿𝑂𝑁𝐺𝐼𝑇𝑈𝐷𝐼𝑁𝐴𝐿 𝑃𝑂𝑆𝐼𝑇𝐼𝑂𝑁∗ = (ℎ1𝐿𝑂𝑁𝐺𝐼𝑇𝑈𝐷𝐼𝑁𝐴𝐿 𝑃𝑂𝑆𝐼𝑇𝐼𝑂𝑁 ∙ 𝑓𝑛𝑙𝐻 ∙ 𝑓𝑉𝐻) (143)



7. CONCLUSIONS

It was built a data-base containing all the long-term values for each mode of motion, relative wave

elevation and vertical bending moment, defined as in Figure 52A for fully-loaded conditions.

Lightship conditions should be simulated according Section 2.3.1 parameters.

The Belgian Coastal Scatter Diagram, under conditions of wave encounter for V = 10 knots,

represents an adequate excitation to obtain the maximum long term value that any mode of motion

can face. Only for yaw-acceleration of Vessel 38, threshold of 5 % of error is overtaken when

comparing long-term representative-wave-height against the limit of navigation at Hs = 2.0 m

(belonging to the defined Range of Navigation). It was set aside the analysis of the load and

bending moment long-term-representative-wave-height deviations from the imposed limits of

navigation.

According to Fig. 33, roll damping coefficient equals to 5% of the critical roll damping represents

a good estimate for fully loaded case. In Rudakovic (Bureau Veritas, 2015) and the present

study, this was taken into account. On the same hand, 13% of critical roll damping has to be taken

for successive simulations of Lightship conditions.

Calculating several slopes Entity vs aB, and plotting them against main ship characteristics (and

combinations of them) made possible to find good tendencies for the studied entities. Proposed

empirical formulas were developed accordingly.

A good compromise between the empirical formulas proposed and long-term direct-calculations

has been achieved. Assessment of all NR217 rules was carried out, evidencing an underestimation

of the values, except for the case of surge acceleration, where the rule proposes a constant value

equals to 0.5 m/s2, representing it a great overestimation.

Good agreement is achieved between first-order direct calculations and proposed formulas.

For the range of navigation IN[0.6 < x < 2.0] and the range of wave periods belonging to the

Belgian Coastal sea state shown in Fig. 32, nonlinear effects are present. Sinusoidal linear wave

elevation theory does not represent the physical phenomena accordingly. A correction must be

provided to take into account these effects.

A second order correction is proposed: according to table 7, an additional percentage should be

taken into account due to nonlinear effects, taken at V = 0 knots. Nonlinear Correction Coefficients

were proposed.



115


Corrections due to V = 10 knots forward speed shown in Figure 38 are defined in terms of Forward

Velocity Coefficient.

Nonlinear corrections come from long-term calculations obtained for a LNG Tanker with CB

equals to 0.7. In consequence, proposed correction should be accounted in the future from inland-

vessel simulations considering nonlinear potential theory, for a set of few representative

geometries that allows to make correction to all developed linear long-term data-base. The

nonlinear corrections proposed, intended only to shed some light on this topic.



REFERENCES

J. Journée and W. Massie. Delft University of Technology. Offshore Hydromechanics. 2001.

O. Hughes and J. Paik. Ship Structural Analysis and Design. 2010.

Hydrodynamic Analysis of Inland Navigation Vessels. Stefan Rudakovic. Bureau Veritas, Inland

Navigation Management. Antwerpen. 2015.

Bureau Veritas Rules for the Classification of Inland Navigation Vessels (NR 217 B1 R04). Part B.

DNV GL Rules Classification for Inland Navigation Vessels Part 3 Chapter 2 and 3.

J. M. Orozco (Bureau Veritas – Paris); C. V. Raposo (Bureau Veritas – Rio de Janeiro) and Š.

Malenica (Bureau Veritas – Paris): “A Practical Procedure for the Evaluation of the Roll Motions

of FPSO’s Including the Non potential Damping”. 2002.

Ikeda Y., Himeno Y. & Tanaka N.: "A Prediction Method for Ship Roll Damping", Report of

University of Osaka, 1978.

Prevosto M.: "Distribution of maxima of non-linear barge rolling with medium damping", ISOPE

2001.

Bingjie Guo, Elzbieta M. Bitner-Gregersen, Hui Sun and Jens Bloch Helmers. “Statistics analysis

of ship response in extreme seas”. 2016.

Pierre Ferrant. Seakeeping. Ecole Centrale de Nantes Option Ocean/ Master EMSHIP 2013 – 2014.

Kim, K., Kim, Y., 2011. Numerical study on added resistance of ships by using a timedomain

Rankine panel method. Ocean Eng. 38 (2011), 1357–1367.

Guo, B.J., Steen, S., Deng, G.B., 2012. Seakeeping prediction of KVLCC2 in head waves with

RANS. Appl. Ocean Res. 35, 56–67.

Guo, B.J., Bitner-Gregersen, E.M., Sun, H., Helmers, J.B., 2013, Prediction of Ship Response

Statistics in Extreme Seas Using Model test data and Numerical Simulation based on the Rankine

Panel Method. In: Proceedings of the 29th International Conference on Ocean, Offshore and Arctic

Engineering, OMAE 2013, June 9-14, Nantes, France, OMAE 2013-10351.



117


APPENDICES



APPENDIX A. TENDENCIES.



119


Heave Acceleration

If 𝐿 𝐵 ≤ 6⁄ and 𝐿𝐵 < 900 m2 :

𝑎𝐻 = 𝑎𝐵 ∙ (1566

𝐿− 0.019)

If 𝐿 𝐵⁄ ≤ 6 and 𝐿𝐵 > 900 m2 :

𝑎𝐻 = 1374.2 ∙ 𝑎𝐵 ∙ (1

𝐿) ∙ 𝑓𝐻

Where:

𝑓𝐻 = correction factor due to Δ: If ∆ < 1900 ton; 𝑓𝐻 = 0.8; otherwise 𝑓𝐻 = 1.0 .



If 6 ≤ 𝐿 𝐵⁄ < 9

𝑎𝐻 = 𝑎𝐵 ∙ (1575.7

𝐿+ 0.0006∆)

If 𝐿 𝐵⁄ > 9

𝑎𝐻 = 𝑎𝐵 ∙ (185.64

𝐵+ 0.0005∆)



121


Sway Acceleration:

If 𝛥 < 1700 ton :

𝑎𝑆𝑊 = 𝑎𝐵 ∙ (17.388 − 0.6165𝐵)

If (1700 < 𝛥 < 5000) ton :

𝑎𝑆𝑊 = 𝑎𝐵 ∙ (15.074 − 0.4297𝐵)



If (𝛥 > 5000) ton :

𝑎𝑆𝑊 = 𝑎𝐵 ∙ (13.644 − 0.3347𝐵)

Surge Acceleration:

𝑎𝑆𝑈 =300.47 ∙ 𝑎𝐵

𝐿

y = 300.47xR² = 0.9928

0.0E+0

5.0E-5

1.0E-4

1.5E-4

2.0E-4

2.5E-4

0.0E+0 2.0E-7 4.0E-7 6.0E-7 8.0E-7

a SU /

aB.B

∙Δ

1 / (B∙Δ∙L) [(ton∙m2)-1]



123


Pitch amplitude

If 𝐵 < 13.5 𝑚 :

𝐴𝑃 =47.543 ∙ 𝑎𝐵

𝐿

If 𝐵 > 13.5 𝑚 :

𝐴𝑃 =86.091 ∙ 𝑎𝐵

𝐿1.037 ∙ (∆ ∙ 𝐵)0.037



Pitch acceleration:

𝛼𝑃 =1266.6 ∙ 𝑎𝐵𝐿1.653

Vessel relative motion on ship stern at x = 0 m:

ℎ1_𝑆𝑡𝑒𝑟𝑛 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (293.01

𝐿− 0.4996)

y = 293.01x - 0.4996R² = 0.8516

0

2

4

6

8

10

12

0.00 0.01 0.02 0.03 0.04

h1

Ste

rn /

0.0

8.C

.n

1/L [m-1]



125


Vessel relative motion on Aft cargo position “CARGO_A” at x = 0.33L, according to Fig. 52.A

ℎ1_𝐴 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (179.45

𝐿+ 0.9823)

Vessel relative motion on Midship cargo position “CARGO_M”, x = 0.5L, according to Fig. 52.A

ℎ1_𝑀 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (154.93

𝐿+ 1.179)

y = 179.45x + 0.9823R² = 0.9144

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.00 0.01 0.02 0.03 0.04

h1

carg

o_

AS

/ 0

.08

.C.n

1/L [m-1]

y = 154.93x + 1.179R² = 0.9138

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.00 0.01 0.02 0.03 0.04

h1

carg

o_

MS

/ 0

.08

.C.n

1/L [m-1]



Vessel relative motion on Fore Cargo position “CARGO_F” at x = 0.75L, according to Fig. 52 A

ℎ1_𝐹 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (207.02

𝐿+ 0.767)

Vessel relative motion on Ship Bow at x = L, defined according to Fig. 52 A

ℎ1_𝐹 = 0.08 ∙ 𝐶 ∙ 𝑛 ∙ (394.39

𝐿− 0.336)

y = 207.02x + 0.767R² = 0.9283

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.00 0.01 0.02 0.03 0.04

h1

carg

o_F

S /

0.0

8.C

.n

1/L [m-1]

y = 394.39x - 0.336R² = 0.9783

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.00 0.01 0.02 0.03 0.04

h1

Bo

w /

0.0

8.C

.n

1/L [m-1]



127


APPENDIX B. VALIDATIONS.



Sway Acceleration

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

Sway

Acc

ele

rati

on

Pre

dic

tio

n [

m/s

2]

Sway Acceleration Direct Calculation [m/s2]

Victrol ProposalTanya ProposalSpecht ProposalOdina ProposalMozart ProposalSmack ProposalArc_en_ciel ProposalMontana ProposalNew_York ProposalEuroports ProposalGSK57 ProposalOural ProposalBosphore ProposalVictrol NR217Tanya NR217Specht NR217Odina NR217Mozart NR217Smack NR217Arc_en_ciel NR217Montana NR217New_York NR217Euroports NR217GSK57 NR217Oural NR217Bosphore NR217

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.50 1.00 1.50 2.00 2.50

Sway

Acc

ele

rati

on

Pre

dic

tio

n [

m/s

2]

Sway Acceleration Direct Calculation [m/s2]



129


Surge Acceleration



Heave Acceleration



131


Pitch Amplitude:



Pitch Acceleration:



133


Relative Motion at CARGO_A (Starboard):

0.0

1.0

2.0

3.0

4.0

0.00 1.00 2.00 3.00 4.00

RW

E_C

AR

GO

_A P

red

icti

on

[-]

RWE_CARGO_A Direct Calculation [-]


0.0

1.0

2.0

3.0

4.0

0.00 1.00 2.00 3.00 4.00

RW

E_C

AR

GO

_A P

red

icti

on

[-]

RWE_CARGO_A Direct Calculation [-]



Relative Motion at CARGO_M (Midship):

0.0

1.0

2.0

3.0

4.0

0.00 1.00 2.00 3.00 4.00

h1

_CA

RG

O_M

Pre

dic

tio

n [

-]

h1_CARGO_M Direct Calculation [-]


0.0

1.0

2.0

3.0

4.0

0.00 1.00 2.00 3.00 4.00

h1

_CA

RG

O_M

Pre

dic

tio

n [

-]

h1_CARGO_M Direct Calculation [-]



135


Relative Motion at CARGO_F:

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00

h1

_CA

RG

O_F

Pre

dic

tio

n [

-]

h1_CARGO_F Direct Calculation [-]


0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00

h1

_CA

RG

O_F

Pre

dic

tio

n [

-]

h1_CARGO_F Direct Calculation [-]



Relative Motion at Stern:

0.0

1.0

2.0

3.0

4.0

0.00 1.00 2.00 3.00 4.00

h1

_Ste

rn P

red

icti

on

[-]

h1_Stern Direct Calculation [-]


0.0

1.0

2.0

3.0

4.0

0.00 1.00 2.00 3.00 4.00

h1

_Ste

rn P

red

icti

on

[-]

h1_Stern Direct Calculation [-]



137


Relative Motion at Bow:

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00

h1

_Bo

w P

red

icti

on

[-]

h1_Bow Direct Calculation [-]


0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00

h1

_Bo

w P

red

icti

on

[-]

h1_Bow Direct Calculation [-]

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