Aalborg Universitet

Wave Forces and Overtopping on Crown Walls of Rubble Mound Breakwaters

Pedersen, Jan

Publication date:1996

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Citation for published version (APA):Pedersen, J. (1996). Wave Forces and Overtopping on Crown Walls of Rubble Mound Breakwaters: anExperimental Study. Aalborg: Aalborg Universitetsforlag. (Series Paper; No. 12).

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SERIES PAPER 12 ISSN 0909-4296

Jan Pedersen

Wave Forces and Overtopping on Crown Walls of Rubble Mound

Breakwaters

- An Experimental Study -

Hydraulics &t Coastal Engineering Laboratory Department of Civil Engineering

Aalborg University

August 1996

Hydraulics & Coastal Engineering Laboratory Department of Civil Engineering

Aalborg University Sohngaardsholmsvej 57

DK-9000 Aalborg, Denmark

ISSN 0909-4296 SERIES PAPER No. 12

Wave Forces and Overtopping on Crown Walls of Rubble Mound Breakwaters

- An Experimental Study -

by

J an Pedersen

August 1996

Published 1996 by Hydraulics & Coastal Engineering Laboratory Aalborg University

Printed in Denmark by Centertrykkeriet, Aalborg University

ISSN 0909-4296 SERIES PAPER No. 12

Preface

The present report Wave Forces and Overtopping on Crown Walls of Rubble Mound Breakwaters - An experimental Study is submitted as one of the require­ments for the degree of Ph.D. according to notice no. 9~9 of 11th December 1992 from the Danish Ministry of Education and Research. The thesis has been publicly defended at Aalborg University on 31st May, 1996.

The study was financed by the Danish Technical Research Council in connection with the frame work programme Marin Teknik, 1989-1992 and carried out from 1st March 1990 to 31st January 1993 at the Department of Civil Engineering, Aalborg University, Denmark under the supervision of Professor H. F. Burcharth.

The author wishes to thank everybody in the department for their support and assistance during the study. Also thanks to the staff in the Hydraulics & Coastal Engineering Laboratory at Aalborg University for their work and help during the experimental period.

Special thanks to Harry Luke and Tue Hald for reading and correcting the manuscript and to my girlfriend and children for their pat ience and support throughout the study.

Aalborg, 30th August

Jan Pe.dersen

i

.:i! : ., .

Contents

Preface

Contents

List of Symbols

Abstract

Sammenfatning

1 Introduction 1.1 Rubble mound breakwater- Concept . . . . . . .... .

1.1.1 Failure modes of rubble mound breakwaters . . . 1.1.2 Detailed description of crown wall failure modes

1.2 Experienced crown wall failures .. . . ......... .

2 State of the art 2.1 Wave action on rubble slopes . . .. .

2.1.1 Wave run-up assessment ... . 2.2 Wave force estimation on crown walls

2.2.1 Parametric investigations ... 2.2.2 Spatial distribution of wave pressure 2.2.3 Wave force assessment . ...... . 2.2.4 Conclusion on general level of knowledge.

2.3 Wave overtopping . . . . . . . . . . . . . . . . . . 2.3.1 Admissible overtopping rates .. . ... . 2.3.2 Overtopping on breakwaters with superstructures .

3 Model tests 3.1 Purpose of model study

3.1.1 Investigated parameters 3.2 Sea states . . . . . . . .

3.2.1 Wave generation . 3.2.2 Wave analysis . . .

3.3 Investigated cross sections

lll

1

iii

vu

XI

xiii

1 2 3 4 7

11 11 12 14 15 18 21 25 26 26 28

31 33 33 34 34 34 38

3.4 Measurement of pressures . . . . . . . . 3.5 Measurement of overtopping discharges

41 43

4 Wave pressures and forces on crown walls 45 4.1 Distribution of wave pressures . . . . . . . . . . . . . . . 45 4.2 Wave force components . . . . . . . . . . . . . . . . . . 51

4.2.1 Force distributions and statistical force estimates 55 4.2.2 Correlation between force component estimates 58

4.3 Parametric investigation . . . . . . . . . 62 4.3.1 Influence of wave height . . . . . . . 62 4.3.2 Influence of wave period/length . . . 63 4.3.3 Influence of armour crest freeboard . 64 4.3.4 Influence of slope angle . . . . 65 4.3.5 Influence of armour layer type . 65 4.3.6 Influence of crown wall height . . 66 4.3. 7 Influence of armour berm width . 68 4.3.8 Resume of parametric investigation . 69

4.4 Conclusions drawn from model study . . . . 70 4.5 Design equations for wave load components 70

4.5.1 Assumed pressure distribution . . . 71 4.5.2 Design equation for horizontal wave force 73 4.5.3 Design equation for overturning moment . 80 4.5.4 Design equation for base pressure . 81

4.6 Summary of wave force design equations . 82 4.6.1 Conclusions 84

4. 7 Scale effects . . . . . 85

5 Geotechnical response 89 5.1 Soil failure . . . . . . 90 5.2 Soil deformations . . 92

5.2.1 Mathematical model 93 5.2.2 Evaluation of damping and stiffness terms. 96 5.2.3 Evaluation of soil parameters. . . . . . . . 97 5.2.4 Simplified model for crown wall structure 98 5.2.5 Evaluation of dynamic amplification 100

6 Wave overtopping 105 6.1 Parametric investigation 105

6.1.1 Influence of wave period 106 6.1.2 Influence of wave height 106 6.1.3 Influence of wall crest freeboard . 107 6.1.4 Influence of slope angle . . . . . 108 6.1.5 Influence of armour crest berm width 108 6.1.6 Influence of armour layer type 108

lV

6.2 Design equation for overtopping discharge 6.2.1 conclusion .... . . .. . .. .. .

7 Summary and conclusions

Bibliography

A Table of measurements

B Photos and drawings

V

110 112

113

117

121

131

a , b, c,d

Ac

B

B*

Cm

dn ,50

dx

d<p

Dx

D'P

e

I l e

In ld ,cov.pled

lP Fh

··~

list of symbols

empirical coefficients

armour crest freeboard defined as vertical distance from SWL to armour crest

width of armour crest berm or width of foundation

horizontal distance from back of crown wall to interface between breakwater slope and SWL

wave celerity corresponding toT m

nominal stone diameter

damping coefficient for horizontal motion

damping coefficient for rotational motion

damping ratio for horizontal motion

damping ratio for rotational motion

void ratio

frequency

vertical distance from armour crest to top of crown wall face (un­protected part of crown wall face)

undamped natural frequency

coupled damped natural frequency

spectral peak frequency

horizontal wave force per metre run acting on crown wall

maximum horizontal wave force per metre run acting on crown wall

horizontal wave force per metre run acting on lower part of crown wall

horizontal wave force per metre run acting on upper part of crown wall

Vll

h

h,

hprot

ho

Hs

Ice

kx

kx ,static

k tp ,static

M

M c c

Mcc,max

M peak

Pm

Pu

kinematic friction force

static friction force

vertical wave force per metre run acting on crown wall

gravity force

modified dimensionless freeboard

gravity

dynamic shear modulus

maximum static shear modulus

water depth

height of crown wall face

height of stone protected part of crown wall face

vertical distance from crown wall base to center of gravity

significant wave height defined as Hs = 4Jffi0

mass moment of inertia around center of gravity

stiffness coefficient for horizontal motion

static stiffness coefficient for horizontal motion

stiffness coefficient for rotational motion

static stiffness coefficient for rotational motion

local wave length corresponding to wave period Tm

local wave length corresponding to wave period Tp

deep water wave length corresponding to wave period T m

deep water wave length corresponding to wave period Tp

mass

overturning moment per metre run acting on crown wall

moment around center of gravity

maximum moment around center of gravity

maximum overturning moment per metre run acting on crown wall

amplification factor for horizontal motion

overturning moment exceeded by x percent of the waves

amplification factor for rotational motion

hydrostatic wave pressure at crown wall face

stagnation pressure at crown wall face due to wave impact

uplift wave pressure at crown wall base

Vlll I • I

Q

Qm

Q:n Q* r

rox

Sp

So m

sop

t

t failure

trise

wave pressure at base of crown wall

wall base pressure exceeded by x percent of the waves

maximum wave pressure at base of crown wall

reaction force from soil due to horizontal motion

overtopping rate per metre run of crown wall

mean overtopping rate per metre run of crown wall

dimensionless mean overtopping rate per metre run of crown wall

dimensionless overtopping rate per metre run of crown wall

roughness factor

radius for equivalent circular footing having same soil contact area

radius for equivalent circular footing having same mass and mass moment of inertia

vertical distance from SWL to top of crown wall face

vertical run-up level above SWL

vertical run-up level above SWL exceeded by x percent of the waves

reaction force due to rotational motion

Owen's dimensionless freeboard

local wave steepness based on T m

local wave steepness based on Tp

deep water wave steepness based on T m

deep water wave steepness based on Tp

time, thickness of structure

duration of failure state

wave impact rise time

wave impact decay time

damped natural period

spectral mean period

spectral peak period

mean zero crossing period

up-rush velocity

volumes

reduction factor for wave impact

lX

Xb horizontal motion at crown wall base

xca horizontal motion at center of gravity

Xv excentricity of Fv

Xb horizontal velocity at crown wall base

xca horizontal acceleration at center of gravity

y vertical run-up wedge thickness

Yetr effective wave pressure impact zone height

zh excentricity of Fh

a breakwater front slope angle

{3 angle of incidence

1 peak enhancement factor in JONSWAP spectrum

'Yw specific weight of water

flt time lag

f strain

() apex angle of run-up wedge

e angle of friction

cp angle of rotation

cp angular velocity

cp angular acceleration

/-Lk kinematic coefficient of friction

J-Ls static coefficient of friction

v Poisson's ratio

e surf similarity parameter

em local surf similarity parameter based on T m

eP local surf similarity parameter based on Tp

deep water surf similarity parameter based on T m

deep water surf similarity parameter based on Tp

density

Pw density of water

O"o mean principal stress

a x standard deviation of x

T shear stress

X

J

Abstract

Wave Forces and Overtopping on Crown Walls of Rubble Mound Breakwaters- An Experimental Study

The scientific progress of our understanding of the interaction between coastal structures and the sea has greatly improved in the recent years. The present state of knowledge includes structural and financial optimization of the struc­tures based on reliability evaluations. The first requirement for such an evalu­ation is a mathematical formulation of the related structural failure modes. For rubble mound breakwaters several new reliable design formulae have been de­veloped over the last decade but at least one major task still remains unsolved - the assessment of the wave loading and hence the stabilty of rubble mound breakwater crown walls.

This background motivated the initialization of the present study on wave im­posed forces and wave overtopping on crown wall structures. The two subjects were investigated through an excessive parametric model study involving more than 370 long duration test series in the coastal laboratory at Aalborg University.

Based oo analyses of the experimental data a design method for assessing the maximum wave forces on the vertical face of crown wall structures has been developed as well as a new and more versatile design equation for the related overtopping discharges.

The stability of crown wall structures has also been investigated and a new methodology for the evaluation of the geodynamic response of the foundation is presented.

xi

I

I

Sammenfatning

Bfl)lgekrrefter pa og overskyl af bfl)lgeskrerme pa stenkastningsmoler - et eksperimentielt studium

Indenfor det vandbygningstekniske omrade er de videnskabelige fremskridt ved­r{Z!rende vor forstaelse af interaktionen mellem kystkonstruktioner og det om­givende hav blevet vresentligt forbedret indenfor de seneste ar. Det nuvrerende videnskabelige niveau indbefatter sikkerhedsteoretisk baserede strukturelle og 0konomiske optimeringer af kystkonstruktioner. Det basale krav til gennemf0-relse af en sadan optimering er tilstedevrerelsen af en fysisk formulering af de mulige brudmader, der ingar i problemstillingen. For stenkastningsmoler er der indenfor det sidste arti udviklet nye forbedrede design metoder for adskillige af brudmaderne, men isrer pa et punkt mangler der stadig information - fastsret­telsen af de dimensionsgivende laster pa b0lgeskrerme og dermed disses stabilitet.

Pa denne baggrund blev det besluttet at igangsrette det nrervrerende studium omhandlende laster og overskyl pa b0lgeskrerme. De to frenomener blev studeret i et omfattende eksperimentielt model-parametrisk studie inbefattende mere end 370 langtids fors0g med uregelmressige b0lger i Aalborg Universitets hydrauliske laboratorium.

Udfra analyser af de eksperimentielle data er der udviklet en banebrydende design metode til fastsrettelse af den maksimale b0lgekraft pa den vertikale b0lgeskrermsflade samt en ny og, i sammenligning med de eksisterende, mere alsidig design metode til beregning af det tilh0rende b0lgeoverskyl.

Endvidere er der fortaget en stabilitetsanalyse af b0lgeskrerme og i den sammen­hreng udviklet en ny metode til evaluering af den underliggende jords dynamiske respons.

Xlll

Introduction

Rubble mound breakwaters have for more than a century been used worldwide to protect mankind against the violent forces of the surrounding sea. Their applications are versatile, being used for the enclosure of harbour basins, in providing berthing facilities in deep water areas and for the protection of land (dikes) and beaches (offshore breakwaters).

Until the late 1960's rubble mound breakwaters were exclusively used in relati­vely shallow waters and a fair amount of experience and expertise was available for these situations. At that time the construction of the large oil tankers was in progress and a strong need for safe berthing facilities for these vessels arose. To fulfil these needs the traditionally shallow water structures were extended to deeper waters and thereby waves and wave forces for which no previous experi­ence was available.

This development resulted in the failures of several large rubble mound break­waters in the 1970's and early 1980's (see the examples in section 1.2) , which clearly demonstrated that the available design methods were inadequate.

The recognition of this inadequacy is clearly reflected by the vast amount of published research results concerning rubble mound breakwaters during the last two decades. Many new design concepts and design equations have been pro­posed of which several are generally accepted as being more reliable design tools when compared to the previous recommendations of the Shore Protection Manual (SPM 1984).

1

2 CHAPTER 1. INTRODUCTION

1.1 Rubble mound breakwater- Concept

One of the reasons that rubble mound breakwaters still constitute a major prob­lem in maritime civil engineering is that hardly two breakwaters in the world have been constructed in the same way. This is due to the difference in envi­ronmental loading_ and the natural availability of construction materials at the different sites. Hence the designation rubble mound breakwater covers a wide variety of structures though all having some common features.

When talking about rubble mound breakwaters we normally distinguish between conventionally designed structures with a stable armour layer and structures where the armour layer is alloved to reshape.

Reshaping breakwaters, often termed berm breakwaters, are constructed in such a way that the stones on the breakwater front face can be moved and rearranged by the waves whereby the breakwater face will adapt itself to the wave climate. Reshaping breakwaters are never constructed with a superstructure on the top and will therefore not be considered further.

Conventional breakwaters are constructed in order to withstand the wave loading without any significant movement of material. In contrast to reshaping break­waters these breakwaters are often constructed with a superstructure (crown wall). The term rubble mound breakwater will be used in the following for a conventional structure.

In figure 1.1 two typical rubble mound breakwater cross sections are illustrated. Although appearing different the structure of the cross sections have several points of resemblance. The innermost part, the core, is typically made of quarry­run or, if available, of gravel taken from the sea bed.

The outer seaward layer, the armour layer, consists of units sufficiently large and heavy to remain in posistion under wave attack. The units can be rocks (up to app. 20 tons) or made of concrete if heavier units are needed or if natural sources of rocks are not available. The inclination of the front slope is one of the parameters that determines the required mass of the armour blocks. Typical slope inclinations are in the range 1:1.5 to 1:3.5.

Between the core and the armour layer one or more filter layers are placed, the aim being to prevent the finer inner materials from being washed out through the gaps between the armour blocks and to improve the foundation for the armour blocks.

Figure l.l(a) shows a breakwater cross section primarily used in relatively shal­low waters or in cases where access on top of the breakwater is not required. This

1.1. RUBBLE MOUND BREAKWATER- CONCEPT

(a) Without crown wall structure (b) With crown wall structure

Figure 1.1: Examples of typical rubble mound breakwater cross sec­tions

3

type of structure is mainly used for harbour basin enclosures where ship berthing along the breakwater is not requested and for shore protection purposes. In both cases relatively large overtopping quantities can be allowed. The upper part of the structure is simply accomplished by extending the armour and filter layers over the top of the core and partly or fully down the back slope. The strong protection of the upper part of the rear slope is necessary due to the action of overtopping waves.

Figure 1.1 (b) shows an iri many ways similar structure when compared to figure 1.1(a). In this case the uppermost part of the breakwater is constituted by a concrete superstructure, termed a crown or capping wall. The wall has several purposes :

• The overall crest height of the structure is increased and the permeab­ility of the upper part of the breakwater is reduced. This results in less wave overtopping and less wave transmission through the breakwater which means that less wave energy is transmitted to the lee-side of the breakwa­ter. For similar overtopping conditions a breakwater with a crown wall has a significantly smaller volume than a breakwater without a crown wall.

• The crown wall structure constitutes an access road which can be used for repairment works, for traffic to and from the breakwater, and for carrying service installations such as pipelines, sanitary installations, electricity etc.

1.1.1 Failure modes of rubble mound breakwaters

A rubble mound breakwater is a very complex structure involving several struc­tural components, which obviously means that failure can occur in many ways. Failure will here be defined as any exceedence of prespecified structural or func­tional properties. This definition implies that any partial or total collapse of one

4 CHAPTER 1. INTRODUCTION

or more structural elements which can lead to a global failure of the whole struc­ture is considered to be a failure, but also for instance large wave overtopping discharges which might damage persons, ships or equipment on and behind the breakwater is regarded as a failure condition (functional) although the breakwa­ter itself might not be subject to any damage.

Figure 1.2 outlines the possible failure modes for a typical rubble mound break­water configuration including different soil and block layers, a berm and a crown wall.

Ovr·r lo pJH llg

l:: rosioll, b r t•ilktogc• ~ 13t~•akH(!f' . ' h di ug . IJ i liuj!

C:o1·c ::;t-ll.lerneul ................... ____ ...,""'

inst.abllit.y

~tth:ioil sf' t t.lf·nH~nt. ............ ..._ ____ ..,.,...

Figure 1.2: Possible failure modes for a rubble mound breakwater. Redrawn from Burcharth {1993).

Due to the many geometrical, structural and hydrodynamic parameters involved in the description of the failure modes outlined above it has not yet been possible to establish reliable design procedures for all of them.

During the PIANC working group 12 project on safety of rubble mound break­waters (see Burcharth (1991a) and (1991b)), in which the author participated, it was recognized that especially the failure patterns involving the crown wall was very poorly understood. Only some very general guidelines could be found in the literature. Therefore the design still has to be based on site specific model tests.

With these problems in mind it was decided to investigate the problems associ­ated with the crown wall structure more systematically.

1.1.2 Detailed description of crown wall failure modes

Like the rubble mound breakwater the crown wall itself can also be constuct ed in several ways. On smaller breakwaters wave screens of wood (figure 1.3(a))

1.1. RUBBLE MOUND BREAKWATER- CONCEPT 5

are often seen. On larger breakwaters heavy concrete structures, basing their stability on frictional resistance, are the only possible solutions (see figures 1.3(b) - 1.3(d)).

Wooden wave screen

(a) (b) (c)

Figure 1. 3: Examples of crown wall configurations.

(d)

As illustrated in fig. 1.3 concrete crown walls can take multiple forms. Com­monly the structures are comprised of a vertical or recurved wall connected to a horizontal base which transmits the wave loading to the underlaying soil. The vertical face can be fully or partly protected from wave attack by armour units placed in front of the structure. Crown walls are often constructed with a skirt penetrating into the soil to improve the apparent frictional resistance of the structure (figure 1.3( d)) by forcing the slip failure surface to go through the rubble mound.

On very large breakwaters the crown wall can even be constructed as a hollow caisson in which pipelines and other installations can be placed and thereby protected against the rough environment. An example of such a structure is given in figure 1.4, showing the inside of the crown wall structure at the Punta Lucero breakwater in Bilbao, Spain.

From a safety viewpoint the stability of the crown wall is essential since a failure of this structure might lead to a total breakdown of the whole rubble mound breakwater. The forces exerted on a crown wall from the waves occur in two ways. The primary action takes place on the vertical front face giving rise to large horizontal forces and large overturning moments. Secondly the wave penetrates into the soil leading to an increase in pore pressures which, if the underside of the wall base is placed close to the mean water level, may reach the structure and hence act as a vertical loading on the structure. These loading mechanisms result in the possible failure modes depicted in figure 1.5.

Sliding (figure 1.5(a)) is probably the most common reason for failure of crown walls. It occurs when the horizontal loading exceeds the frictional resistance, which may be altered by rising pore pressures, between the structure and the underlaying soil. On many breakwaters this failure pattern can be obeserved as small dislocations of some of the wall sections.

6 CHAPTER 1. INTRODUCTION

Figure 1.4: Inside of the Punta Lucero breakwater crown wall.

Another possible failure mode is overturning or tilting of the entire wall sec­tion (figure 1.5(b) ). This phenomenon may be difficult to identify since it will

__.. Sliding

(a)

t Overturning/ tilling

(b)

Cracking

(c)

Figure 1. 5: Crown wall failure modes.

Gcolecnical failure

slip fa~;-~-;~··········· ··············

(d)

1.2. EXPERIENCED CROWN WALL FAILURES 7

start with small rocking movements of the wall. These movements will decrease the frictional soil/structure resistance at the front and at the back of the struc­ture, whereafter sliding of the strucure is likely to occur. Another possibility is the generation of strong water flows in the gaps formed at the structure ends (venting) which may lead to undermining of the structure and the armour layer.

Another aspect in the design of crown walls is the structural strength of the wall itself (figure 1.5(d)). In this context also the deteoration of the material during the life time of the structure must be taken into account.

Finally, also geotechnical failures must be considered. Where the first three described failure modes can be solved by increasing the mass of the structure this is not necessarily the case for geotechnical failure . Therefore an optimization of the structure is required.

All of the described failure mechanisms are quite simple to account for from an engineering viewpoint. The major problem arises however in the assessment of the wave loading on the strucure.

1.2 Experienced crown wall failures

As mentioned in the previous chapter the cause of failure may be very difficult to determine due to the impossibility of inspecting the structure under extreme weather conditions and the often nearly totally damaged breakwater profile after failure.

In the period of 1971 to 1981 several failures of large rubble mound breakwaters, most of them with crown walls, were experienced.

All these breakwater failures were thoroughly analysed. In some instances the wave climate and hence the design wave conditions had been underestimated and in other instances structural parts of the breakwater had not been considered during the design phase, e.g. the Sines breakwater where the structural integrity of the applied Dolos units had fully been ignored.

In at least two cases the main cause of failure was directly related to the crown wall. Giinbak and Ergin (1983) studied the reasons for the total breakdown of the Antalya harbour breakwater in Antalya, Thrkey in december 1971. A typical design cross section and the section profile after destruction is shown in figure 1.6.

During the storm that damaged the breakwater, observations revealed that the crown wall sections started to slide backwards resulting in gaps between the

8 CHAPTER 1. INTRODUCTION

Figure 1. 6: Typical cross section of the Antalya harbour breakwater, Antalya, Turkey {from Giinbak and Ergin {1983)).

sections. The water penetrated through these gaps and rapidly eroded the back of the breakwater thereby reducing the stability of the crown walls. After the storm nearly all crown wall sections were found on the rear side of the breakwater. When the crown wall failed the unprotected top of the breakwater was directly exposed to the waves and a rapid destruction of the structure from the top to the zone slightly below Mean Water Level followed.

The failure of the Antalya Harbour breakwater is a typical example of the im­portance of evaluating the influence from one structural part of the breakwater on other parts. The very permeable layer, on which the crown wall is founded is , is during wave uprush filled with water which, if it cannot escape, as is the case with the breakwater section along the reclaimed area, gives rise to very high pore pressures in the layer. If the water cannot drain out of the layer before the next wave approaches a permanent setup of the internal water table will build-up. With the voids filled with water the wave dissipation decreases which inevitably leads to higher up-rush velocities and thereby larger forces on the crown wall. This is probably the explanation for the failure of the crown walls on the Antalya breakwater.

Giinbak (1985) also examined the damages on the Tripoli Harbour North West Breakwater. The inner app. 2000 m of the breakwater section is illustrated in figure 1.7(a) and the outer last app. 2500 m is shown in figure 1.7(b).

Behind the first 2000 m length of the breakwater there is a sand reclamation with harbour facilities. Between the rear slope and the sand fill a geotextile membrane was placed (see figure 1.7(a)). The outer 2500 m of the breakwater had no reclaimed area behind it and the backslope was protected by armour stones.

In two severe storms in january 1981 the breakwater was, after several minor damages during its construction, severely damaged. Over the whole length of the breakwater some armour diplacement and Tetrapod breakage took place. Along

1.2. EXPERIENCED CROWN WALL FAILURES

(a.) (b)

Figure 1. 7: Tripoli Harbour North West Breakwater cross sections (from Gunbak {1985}}.

9

the first 2000 m several crown wall elements failed accompanied by movements and cracks in the base slabs. Where the walls had failed excessive erosion and damage to the reclamation and service installations were observed. Venting, a phenomenon associated with the shooting out of water and air in the form of jets due to excessive pressure build-up under the wall base, was seen immediately behind the base slab along some parts of where reclamation was performed.

The reason for the damage of the 'fripoli Harbour NW Breakwater was primar­ily due to a severe underestimation of the wave heights and secondly due to insufficient model tests where the steep sea bed in front of the structure was not modelled.

10 CHAPTER 1. INTRODUCTION

Figure 1.8: Crown wall sections on The Tripoli Harbour NW Break­water after storms in january 1981.

I l

State of the art

The following chapter is devoted to a summary of the present status of knowledge concerning wave loading and wave overtopping on rubble mound breakwater crown walls. Both research topics are, as will be shown, governed by empirical relationships obtained mainly from small scale model tests in laboratories. The developed design equations all contain two or more constants which, among other parameters, are functions of the breakwater geometry and the armour layer roughness and permeability.

2.1 Wave action on rubble slopes

Wave impacts on the breakwater crest are highly influenced by the wave trans­formation/breaking processes on the rough porous front slope. As the waves ap­proach the slope the rapid decrease in water depth and the bottom friction cause the waves to steepen. Eventually the waves reach a steepness where they become unstable and finally break. The surf similarity parameter ~o = tanafJHs/Lo has been found to be a good descriptor of the type of wave breaking (~ is also often referred to as the Irribarren number or Battjes parameter). The wave breaking criteria given in figure 2.1 have been suggested by Battjes (1974) based on measurements of Galvin (1968).

In case of plunging or collapsing breakers, which will be the situation for storm waves in relatively deep waters and relatively steep rubble slopes (1 < cot a < 3) ,

11

12

0.5 < ~0 < 3 plunging breaker

CHAPTER 2. STATE OF THE ART

2.5 < ~0 < 3.5 collapsing breaker

~0 > 3 surging breaker

-~-'!Y_b _________ _

Figure 2.1: M a in wave breaking types

an extensive amount of energy dissipation takes place on the rubble slope and in the porous filter layers. This will result in a decrease of transmitted wave energy and in a reduction of the impact on the breakwater crest. Further, when a wave breaks on the porous slope a large volume of air will be entrained into the water, hence the uprushing wave impacting on the structure, will consist of a mixture of water and air bubbles. These complicated processes in combination with the many geometrical and physical parameters which would be required for a description of the breakwater crest region, are the main reasons why an analytical or semi-empirical model for prediction of the wave loading on crown walls have not yet been established.

One of the physical processes which can add to a better understanding of both wave loading and wave overtopping on crown walls, is the wave run-up on sloping structures.

2.1.1 Wave run-up assessment

Several investigations of run-up levels on different types of sloping structures have been performed. Wave run-up is defined as the vertical distance from SWL to the crest of the uprushing wave. For irregular waves a significance level is normally used, e.g. Ru2% which is the run-up level only exceeded by 2% of the waves.

Van der Meer (1988) measured run-up levels on several configurations of ar­moured rubble slopes. Like in earlier studies on smooth slopes it was found that the run-up level could be described by the surf similarity parameter ( CIRIA/ CUR 1991) :

Rux =a. ~m Hs

for ~m< 1.5 (2.1)

Rux = b · ~m c for ~m > 1.5 (2.2) Hs

2.1. WAVE ACTION ON RUBBLE SLOPES 13

For very permeable structures the run-up is limited to a maximum of :

(2.3)

where ~m = tana/JHs/Lom and Lom is the deep water wave length corres­ponding to the mean wave period.

Values for the coefficients a, b, c and d are given for different exceedence levels (x) in table 2.1 and level of 2% figure 2.2 shows for a relative run-up level of 2% the fit of equations (2.1) - (2.3) to the measurements of Van der Meer.

Table 2.1: Coefficients for calculation of run-up levels

Run-up level coefficients x% a b c d 0.1 1.12 1.34 0.55 2.58 1 1.01 1.24 0.48 2.15 2 0.96 1.17 0.46 1.97 5 0.86 1.05 0.44 1.68 10 0.77 0.94 0.42 1.45

0

2.5 0

X

2.0 X X

R.2\l. X

H, 1.5 equation (2.3)

1.0

0.5 o Impermeable core x Permeable core

0 0 2 3 4 5 6 7 8

~m

Figure 2.2: 2% run-up level on armoured rubble slopes.

14 CHAPTER 2. STATE OF THE ART

The calculation of run-up is important since wave impacts on the crown wall do not commence until the run-up level exceeds the level of the crown wall base slab and the run-up height must be expected to have a large influence on the wave force imposed on the wall.

2.2 Wave force estimation on crown walls

Wave forces on a crown wall structure exposed to irregular waves are of a stochastic and hence very complicated nature.

The imposed loads on a wall depends both on the characteristics of the waves and the geometry, including permeability and roughness, of the seaward face of the breakwater.

The distributions of wave induced pressure and the related resultant wave forces at a given instant on the wall are outlined in figure 2.3. The figure also defines relevant geometrical parameters influencing the wave load.

fe Re

Ac

B

ph

Fit

~

roughness, permeability

Figure 2.3: Definition of parameters and pressures. Redrawn from Burcharth (1993)

F.

The present study will solely address the pressures exerted on the vertical part of the wall (Ph) · The uplift pressure (Pv), acting below the base slab, cannot be determined in small scale flume tests because of strong scale effects related to the flow inside the porous mound. Commonly a triangular pressure distribution

2.2. WAVE FORCE ESTIMATION ON CROWN WALLS 15

is therefore assumed based on the pressure measured at the toe of the vertical face (Pb) and the hydrostatic pressure at the rear of the structure. The latter is obviously zero if the level of the base slab is above the internal water leveL In case of homogeneous and rather permeable soils the above assumption is believed to give a conservative estimate of the vertical load.

As previously mentioned, rather few investigations concerning wave loading on crown walls have been reported. In the following a summary will be given of the present knowledge.

2.2.1 Parametric investigations

The complexity of the wave breaking process and the following up-rush of the water/ air mixture on the front slope makes the establishment of an analytical expression for the wave impact on the breakwater crest impossible. Thus, the only accessible way of gaining information on the problem is by performing parametric studies in the laboratory. Such studies concentrate on obtaining an empirical relation between the response, e.g. the crown wall wave loading, and all parameters influencing the response.

Jensen (1984) reports the results of measurements of horizontal wave force from several site specific model investigations. In all the studies, variations in wave height (Hs), wave period (Tp) and and water level were investigated and related to the maximum wave force per meter wall for 1000 waves (F0 _1%)- Since relat­ively few waves were used in each test case (a little more than 1000) the Fo.I% force estimate is subjected to some uncertainty.

From analysis of the tests Jensen found that the influence from water level vari­ations could be expressed by the distance from SWL to the armour crest (Ac) and that the measured horizontal force was directly proportional to the ratio Hs/Ac- Concerning the wave period a clear tendency to an increase in wave loads for increasing Tp was found .

In one of the test cases Jensen (1984) also studied the influence of wave obliquity for long crested waves on the wall loading. The results for two different wave heights are depicted in figure 2.4.

The wave force clearly decreases with increasing angle of wave incidence. The almost linear decrease from 0° and upwards is in sharp contradiction to re­sults from overtopping experiments by Bradbury et al. (1988), who found that discharges were nearly unaffected by small variations ( < 25°) of (3. The same tendencies should be expected for the wave forces. The findings of Jensen might be explained by the model setup, where the force measurements were averaged

16

Hnriznntal fnn:e (t/m)

500

40<1

300

200

H =8m '

10 20 30 40 5!1

p (deg)

CHAPTER 2. STATE OF THE ART

Horizontal fnrce (t/m)

500

400

300

200

1!10

o 111 20 3o •o so

P<dcgJ

Figure 2.4: Influence of wave obliquity on crown wall loading. Redrawn from Jensen {1984).

over a 48 m (prototype scale) test section. With such a long section the maximum wave pressure along the wall does not occur simultaneously when attacked by oblique waves and hence the measured response will be less than for a smaller test section.

Bradbury et al. (1988) performed experiments with 5 different rock slope con­figurations in order to investigate the influence of breakwater geometry on the wall loading. To some degree their results support the conclusions by Jensen, e.g. proportionality between wave force and wave height and increase in wave force for increasing wave periods. The influence on wall loading from variations in structural geometry does not appear very clearly in their report and no defin­ite conclusions are made concerning the importance on the different geometrical parameters. The latter might have to do with measuring problems relater to the used force table (Bradbury et al. 1988). Several of the measurements had to be both highpass and lowpass filtered in order to get what visually seemed to be satisfying force recordings. Such filtering processes always have the risk that information in the original signals is lost.

Hamilton and Hall (1992) carried out a series of model tests to investigate the stability of precast concrete crown walls in small scale models. In their study the effect on the minimum mass of the structure to remain stable was investigated for different design parameters: wave height, wave period, crown wall height, water level and front slope inclination. Also the effect of positioning the crown wall either directly on the core or on top of the armour layer as well as the effect of stabilising skirts (cf. figure 1.3( d)) was investigated. The minimum

2.2. WAVE FORCE ESTIMATION ON CROWN WALLS 17

stable mass (MSM) is directly comparable to the wave loading exerted on the wall face. Most of the tests in their study were performed with regular waves. Figure 2.5 depicts some of the results from the investigation.

70

60

E so

--Cl 40 ~ ~ 30 (/)

~ 20

10

00

70

60

~ 50

Cl e.. 40

:::E 30 (/)

::2 20

w OQ

70

Hew= 0.100m (1)

60 Hcw•0.100m (2)

F = 0.025m .. . +

T = 2.25a

~ 50 Hcw•0.070m

Slope 1:1.5 40 On Cote ~

Hcw•0.040m

~ 30 (/)

:::E 20 T • 1.75• F • 0.025 m

10 Slope 1:1.5 I Rogul11tWaves OnCotw

0 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2 0.~ 0.3

Wave Height (m) Wave Height (m)

7 (3) . (4)

OnNmoUI 60Nolago

On Core

i 50 l.cw•0.010m

40 l.cw • 0 .030 m f 0

:::E 30 low • 0.050 m (/) .

Hew • 0.070m ::2 20 .. F & O.o50m

~ T -= 1.75s 10 Slope 1:1.5 • Regular Wavet

0.05 0 .1 0.15 0.2 0.25 0.3 0

0.05 0.1 0.15 0.2

Wave Height (m) Wave Height (m)

Figure 2. 5: Influence of (J}:wave height, (2}:crown wall height (3}:crown wall position and (4}:stabilising skirts. From Hamilton and Hall {1992}

Regular Waves

0.25 0.3

Regular WrNes

:

Hew • 0.100m F • 0,025m T • 2.25• Slope 1:1.5 On eo<.

0.25 0.3

A linear relationship between wave height and wall loading was observed as long as overtopping rates were moderate (figure 2.5.(1)). This linearity continues until the waves are large enough to induce a significant amount of green water overtopping. From that point the rate of increase in wave force decreases and a horizontal asymptote is approached.

Hamilton and Hall do not provide any definite conclusions about the wave period but simply state that in general the wave loading on the wall increases for larger wave periods.

A very interesting parameter, not investigated in any of the previous studies, is the height of the crown wall. Jensen (1984) assumed that the wave loading would be proportional to the wall height - see section 2.2.3. Hamilton and Hall examined the stability of three structures with different heights, figure 2.5.(2). The measurements show that for small wave heights where none of the 3 walls

18 CHAPTER 2. STATE OF THE ART

are overtopped the wave loading is identical. When the wave height is increased and overtopping commences on the lower walls a smaller wave force is recorded when compared to the forces on the still non-overtopped highest wall. Finally, a threshold wave height directly related to the initiation of green water overtopping is reached, and from that point wave forces remain nearly constant for still increasing wave heights.

Two values of front slope inclinations, cot a = 1.5 and 3.0 were tested. In general the measurements showed that wave forces were smaller for the gentler slope.

Figure 2.5.(3) shows the difference between placing the crown wall on the core material and placing it on top of the armour. The stability of the walls is clearly reduced when placed on the armour layer. It must be remembered that the used walls all have a smooth base slab, hence the situation might have been different if the walls were cast in-situ, where the concrete could penetrate into the gaps between the armour stones.

Finally, the effect of constructing the crown wall with a stabilising skirt penet­rating into the underlying soil layers was investigated, figure 2.5.(4). In all tests with walls without a stabilising skirt the failure was found to be a sliding failure, whereas crown walls with a skirt failed due to overturning. Also, the three tested skirt sizes had nearly the same effectiveness in increasing the stability of the wall. Pedersen and Burcharth (1992) also presented results from a parametric study on crown wall loading. The published findings are comprised in the analysis of the model tests for the present study and will not be commented further .

2.2.2 Spatial distribution of wave pressure

The distribution of wave exerted pressure on a crown wall is very complex, since pressure maxima at different locations do not occur simultaneously. Since the up-rushing water travels faster on the outside of the armour than inside the porous layer, the pressure loading on the wall is expected first to take place in the region just above the crest of the armour. A little later, when the pores in the armour- and sublayers have been filled with water, pressures also act on the lower protected part and beneath the base slab. The maximum load situation, or the pressure situation that is most critical for the stability of the wall, will, depending on the type of wave breaking and configuration of the armour, occur either at the moment where the water tongue hits the upper part of the wall or a little later when the wall is fully saturated and pore pressures have risen.

Jensen (1984) presented examples of maximum wave pressure distributions on a crown wall for different angles of wave incidence - figure 2.6. Pressures were measured by means of 5 pressure transducers mounted into the wall face. The

2.2. WAVE FORCE ESTIMATION ON CROWN WALLS 19

signals from the transducers were lowpass filtered in order to remove high tran­sient load components.

llm' Jn 20 tu >O 20 10

Figure 2. 6: Examples of measured pressure distributions by Jensen {1984}

Jensen found that for maximum horizontal loading, wave pressures acted over the full height of the wall and the pressure distributions tended to be even. The highest pressures were registered immediately above the armour crest level.

Burcharth et al. (1995) presented examples of time evolution of wave pressure on the wall face for two wave conditions, see figure 2. 7.

For a breaking wave impact a rapid rise of wave pressures on the upper part of the wall is registered, whereas the lower part, which is protected by the armour units situated in front of the structure, has not yet experienced any increase in pressure from the up-rushing wave. Approximately 1 second later the full height of the wall is exposed to wave pressures and the maximum horizontal force is reached.

The situation is a little different when considering a non-breaking (surging) wave impact. The pressure rise is more gentle and a clear difference between wave loading on the upper unprotected and the lower armour protected part of the wall is not present.

20 CHAPTER 2. STATE OF THE ART

Breaking Wave (Pressure in kPa)

o.l -0.7

2.5

IP.3

10.] 34.0 , 84 5 ~ 40 2' 1.3 28.7 5 8 s « 3

3 0 3.2 8 .2 20.1

19. I 20.3 24.5 3110

jo i.2 ~3 ~B ~-4 36 I 47.1 34.7 20.2 22.2

« .4 67.0 74 3 74 0 75.3

61 .0 89.P 62 8 63.6 84.8

20 13. 1 9.4 6.11 3.3 ~·8 j0.3 j0.4 ~03

78 4 74.3 70.P 88.3 85.11

86.4 84.2 62 8 81 .8 80 2

Non breaking Wave (Pressure in kPa)

48~8 33.0

81.2

71.7

+1 8.0

1.2 4~.o io.4 ~2.8

41.2 43.7 34.7 35.3

~2 ~~ ~~ n8

83. I 80.9 78.9 80.9

IPZ ~3.? ~7 3 38.1 32 2

94.8 P4.8 Q6.2

81.7 81 .7 82.3

Figure 2. 7 : Examples of measured pressure distributions by Burcharth et al. {1995 ). Prototype scale. All levels in m. Time lag between recordings : flt = 0.37s. Wave incidence angle : f3 = 20°. Armour crest is located in level + 14.0)

2.2. WAVE FORCE ESTIMATION ON CROWN WALLS 21

2.2.3 Wave force assessment

Giinbak and Ergin (1983) and Giinbak (1985) propose a method for assessing the wave pressure on a crown wall based on run-up calculations. In figure 2.8 the assumed pressure distribution is outlined.

Figure 2.8: Definition of pressures and pressure height

The pressure Ph represents the hydrostatic pressure intensity at the wall toe and Pm is the wave pressure component caused by stagnation pressures. From the crest of the armour down to the wall bottom the stagnation pressure is assumed to decrease to a value of 0.5Pm·

In order to calculate the pressures Pm and Ph a triangular wave run-up wedge is assumed on the breakwater slope. The run-up height Ru is calculated as the run-up height that would occur on an infinitely long rubble slope according to equations (2 .1) - (2.3).

The apex angle (8) of the run-up wedge is assumed to be 15°. The vertical distance (y) over which to calculate the hydrostatic pressure component can be calculated as

(Ru- Ac) y=

sma

sin8 (2.4)

cos( a- 8)

For calculation of the stagnation pressure Giinbak assumed that the velocity ( v0 )

of the up-rushing wave front can be determined as

vo = vg:y (2.5)

22 CHAPTER 2. STATE OF THE ART

and hence the stagnation pressure Pm= 9Pw · v5/(2g) can be written

9Pw ·y Pm= 2

(2.6)

Although the above procedure is very simple it satisfies many of the observations outlined in section 2.2.1, for example the reduction in wave load rate when excessive overtopping starts.

Giibak used the described procedure to analyse the Tripoli and Antalya break­water failures (see section 1.2), and found in both cases that the crown wall would fail under the given circumstances.

From analyses of the previous mentioned cite specific model studies, Jensen (1984) proposed an expression on the following form for estimating the horizontal wave force per meter width on a crown wall

- ---=a+b -Fh (Hs) 9PwhjLop Ac

(2.7)

where a and b are dimensionless coefficients taking into account the effect of slope inclination, wave direction, armour permeability /roughness and the geometry of the crest.

The above relationship is derived from a limited range of parametric variations and should only be used in accordance with this. The influence of wall height clearly limits the use of equation (2. 7) to situations with only moderate overtop­pmg.

Bradbury et al. (1988) fitted their measurements from 5 different breakwater configurations to Jensen's expression and found a reasonably good agreement, but the coefficients a and b had to be fitted for each geometry. Also they obtained different coefficients than J ensen for a similar structure. The measurements by Bradbury et al. (1988) were performed with a very poor instrumentation which might explain some of the differences between their and Jensen's findings.

Burcharth (1993) summarised the findings of Jensen and Bradbury et.al. His results are reproduced in figure 2.9.

2.2. WAVE FORCE ESTIMATION ON CROWN WALLS

Cross section

A (1) B (1) c (2) D (2) E (2)

Cross section A Cross secUon B

Ac

~ 4.95

f------::---:...--1111. i h .. =3.0

Cross section C

I 0.15 •I

Cross section D Cross section E

All measures in meters.

Parameter ranges in tests 0.1% exceedence values of coefficients in eq. (2. 7)

Ac (m) .&.. &. a b SOp= Lo Ac

5.6- 10.6 0.016 - 0.036 0.76 - 2.5 -0.026 0.051 1.5- 3.0 0.005 - 0.011 0.82 - 2.4 -0.016 0.025

0.10 0.023 - 0.07 0.9 - 2.1 -0.038 0.043 0.14 0.04- 0.05 1.43 app. 65% of values for C 0.18 0.04- 0.05 1.11 app. 25% of values for C

Figure 2.9: Empirical coefficients in equation {2. 1}. {1) : Jensen {1984} , {2} : Bradbury et. al. {1988) From Burcharth {1993}.

23

24 CHAPTER 2. STATE OF THE ART

Burcharth (1993) also proposed a method for calculating the wave induced pres­sures on the wall face. This procedure is like the method proposed by Giinbak and Ergin (1983) based on the assessment of wave run-up on an imaginary elon­gation of the front slope. Unlike Giinbak and Ergin, wave pressures are not separated into an impact part and a hydrostatic part. For simplicity Burcharth solely considered the imposed wave load to be a function of a hypothetical hy­drostatic pressure. Figure 2.10 outlines the procedure.

~--·

~-

.. ·

B •I

Figure 2.10: Definition sketch for calculation of wave pressure

The total hypothetical horizontal force is found to be :

(2.8)

From comparisons of Fh,hydro with actual measured wave forces as calculated from the data given in figure 2.9 Burcharth proposes the following function to obtain a central estimate of the measurements :

F ( Re - Ac)

h = 0.22 + 0.12 B Fh,hydro (2.9)

The 10% confidence limits are given by ±0.4Fh (Burcharth 1993).

Burcharth concludes that the assumed hydrostatic pressure distribution obvious­ly is not correct as large impulsive pressures from the impinging wave front on the unprotected part of the wall can not be expected to be modelled well by this

2.2. WAVE FORCE ESTIMATION ON CROWN WALLS 25

simple type of approach. FUrthermore, the method will yield a very conservative estimate of the pressures acting at the toe of the wall face and hence lead to very large uplift forces assuming a triangular pressure distribution below the base slab.

2.2.4 Conclusion on general level of knowledge

The preceeding section presented the methods presently available for assessing the wave loading on crown wall structures. The three methods can be used as to serve for a first estimate of the wave loading but cannot be used generally as they all exclude one or more geometrical or physical components influencing the loading. The methods proposed by Jensen (1984) and Burcharth (1993) are both based on analysis of model tests of quite a small number of structural layouts which means that only a limited number of parameters can be included in the design. The method by Giinbak and Ergin (1983) is based on a physical inter­pretation of the interaction between a rubble mound structure with a crown wall and the wave run-up on the breakwater front slope. A number of assumptions concerning the run-up wedge and the velocities of the water jet are made and a few structural parameters, e.g. the berm width, are not incorporated in the pro­cedure. The method has not been validated beside a few calculation examples of actually failed crown wall structures.

Generally it must be concluded that the present state of knowledge is very lim­ited and does not offer a reliable versatile solution for the design of crown wall structures. The poor situation is, as previously mentioned, primarily caused by the difficulty in obtaining an analythical/theoretical expression describing the physics of waves breaking/progressing on a breakwater slope and the impact between a solid structure and the water jet. As also pointed out earlier the only way of gaining more information about the subject is by means of carefully performed and selected model test studies where the different parameters influ­encing the wave load are investigated. Hence the main aim of the present study is to investigate the parametric influence of a variety of physical and geometrical parameters entering the problem. This will be accomplished by a parametric model investigation involving 12 different breakwater cross sections where each parameter is examined keeping all other influencing parameters constant. In this way a profound understanding of the importance of each parameter on the wave load exerted on the wall face can be established eventually leading to a new design concept.

26 CHAPTER 2. STATE OF THE ART

2.3 Wave overtopping

Wave overtopping is the main hydraulic action causing damage to the crest and rear of a rubble mound breakwater and must therefore be considered carefully. Information on overtopping quantities is especially important if reclaimed areas or structures are situated closely behind the breakwater where massive water volumes, often having high velocities, may cause extensive damage.

2.3.1 Admissible overtopping rates

The impact of overtopping water volumes on different obstacles situated on top of an overtopped structure has been investigated by several researchers in order to assess admissible discharge rates for different investigated objects.

Under random wave attack overtopping discharges vary with up to several orders of magnitude from one wave to another meaning that wave overtopping is a very non-linear function of wave height and wave period. This time variation is difficult to measure and quantify in the laboratory and hence overtopping discharges are frequently given by the mean discharge Qm expressed as discharge per meter run (m3 jmjs).

Based on prototype investigations consisting of wave climate measurements and expert impressions of the impact of overtopping volumes on different objects situ­ated on the top of breakwaters (Goda 1971 , Fukuda et al. 1974 and Goda 1985) the guidelines given in figure 2.11 were developed and adopted in the Japanese code of practice. The version given in figure 2.11 is taken from the Dutch/English "Manual on the use of Rock in Hydraulic Engineering" (CIRIA/CUR 1991).

2.3. WAVE OVERTOPPING

100

,......_ Ci) a

..;)--

8 10

"' 0 ....... "--'

Cl}

~ ...s:= u Ci) ·-"0 bO !::::: 0.1 ·-0.. 0.. 0 t: Cl}

> . Uncomfort· 0 No !::::: 0.01 able but not Minor damage ro

Unsafe at dangerous damage to Cl}

::E high speed fittings etc.

0.001 Wet, but not

Safe at all uncomfort·

No speeds

able damage

0.0001 Vehicles Pedestrians Buildings

Figure 2.11: Suggested critical overtopping discharges. {Redrawn from CIRIA/CUR {1991}}

27

100

10

No damage

Revelment 0.0001

seawalls

28 CHAPTER 2. STATE OF THE-ART

2.3.2 Overtopping on breakwaters with superstructures

The calculation of overtopping discharges is based on empirical expressions fitted to hydraulic model test results. Since the discharges depend not only on the environmental conditions (wave height , wave period and water level) but also on the material properties and geometrical layout of the breakwater it is evident that only a few specific cases have been investigated thoroughly.

Owen (1980), (1982a) and (1982b) presents an empirical method for the calcu­lation of the overtopping discharges for sea walls with smooth faces (roughness value r = 1) and without the presence of a crown wall. The developed design for­mula, based on a series of model tests with random waves, relates a dimensionless discharge Q* to a dimensionless freeboard parameter R* :

Q* = Aexp ( -B~*) (2.10)

where

Q* (2.11)

R* (2.12)

A and B are empirical coefficients taking into account the slope inclination and the crest configuration of the structure. This methodology implies that the coefficients must be determined for each specific structural layout.

Brad bury et al. · (1988) performed random wave model tests with rubble slopes and different configurations of crown walls and concluded that there was a stronger dependence of the dimensionless ratio jt than expressed in equation (2.10). To obtain a best fit to their data Owen's expression was modified to :

Q* Aexp ( -B~*) (2.13)

F* R* ( Re ) = ( Re ) 2

(S:: Hs Hs V?;; (2.14)

Bradbury et al. (1988) also suggest that an expression on the form :

Q* = AF*B (2.15)

could give a slightly better description of the Q* - F* relationship than does the

2.3. WAVE OVERTOPPING 29

exponential form in equations (2.10) and (2.13) . Also, Bradbury states that the above formulae in many ways are inadequate since the geometry of the structure is not taken into account. It is obvious that parameters like the berm width, the armour crest position, the slope of the face and the vertical wall freeboard will have an effect on the overtopping discharge and therefore should be included in a design equation. Anyway, it seems likely that these geometrical variations will only have a pronounced effect for relative small discharges, whereas, if the crown wall is inundated, small geometrical variations at the crest are expected to have only a minor influence.

Jensen (1984) reported overtopping measurements from 7 different breakwater/­crown wall configurations. From analysis of these measurements he proposed a relation of the following form :

(2.16)

B* being the horizontal distance from the back of the vertical face of the crown wall to the interface between the mound and the still water level.

As for the equations of Owen and Bradbury et al. equation (2.16) lacks informa­tion about the armour crest position and crown wall freeboard whereas the slope angle and berm width are included indirectly through the parameter B*. For waves in relatively shallow water Jensen's results show that the coefficient b is nearly equal to 1 giving a linear relationship between 9;;Jz and ~. For larger depths of water b decreases to a value lower than 1.

Although the main scope of the present study was to investigate the wave forces on a rubble mound breakwater crown wall, the experimental setup was easily modified to incorporate measurements of wave overtopping. Illustrations of the model setup can be found in section 3 and a discussion of the results and con­clusions from the measurements is given in section 6.

30 CHAPTER 2. STATE OF THE ART

Model tests

In order to improve the knowledge of how breakwater crown walls perform under random wave attack an extensive test programme was set up. In the tests, measurements of wave loading exerted on the vertical front face of the crown walls as well as the amount of water overtopping the structures were measured. A total of 373 tests were performed. The different tests and the obtained results are given in table A.1 in Appendix A. Each test had a duration corresponding to 5000 waves or more in order to quantify the low probabilistic wave force estimates (e.g. the 0.1% fractile of the horizontal wave force) with reasonable accuracy.

All tests were performed in a 1.6 m wide and 26 m long wave flume located at Aalborg Hydraulics and Coastal Engineering Laboratory, Aalborg University.

A breakwater model test section of 1.0 m width, delimited by wave guidance walls, was installed in the middle of the flume as far away from the wave generator as possible. Behind the breakwater a gravel beach with a 1:5 slope was laid out. This setup, shown in figure 3.1, was chosen in order to limit wave reflections in the flume.

The crown wall structure was constructed from 6 mm thick high-strength alu­minium plates which were mounted onto the wave guidance walls. The alu­minium structure was further strengthened by means of four triangular braces, see figure 3.2. Water levels varied between 0.51 m and 0.59 m. The base of the crown wall was in all tests located at level +0.55 m and the crest of the armour berm at level +0. 70 m - see section 3.3 for further details . In front of the

31

32 CHAPTER 3. MODEL TESTS

breakwater an array of 3 wave gauges was placed in order to analyse the wave field.

......

Wave ma ke r

,.

I l

L .L.-1.6 m •••

Wave gouges

w

i \

ave gu1 once

0. 3 m

1.0 m

0.3 m

19 m 26 rn

WO 11

c~~· )C'£ ·i} \()Pc'i(Pf r. 'F)( 'P' ':)t '(,'Pf.'!:.)c · .> Tt' .. lc • l ,(_ 'i .5il00.::'f-l< KS0-tf'>f

I ' I l r .:(,1 1 :;-', ).f:'t;,c:>·:·v 'CXX~ff'<...

l . ... t l /(. ~ /( J. : . _l/

I Grave l beach

\

Figure 3.1 : Outline of breakwater setup in wave flume

Figure 3.2 : Crown wall seen from the lee-side.

3.1. PURPOSE OF MODEL STUDY 33

3.1 Purpose of model study

The main aim of the model study was to get a better insight and understanding of the physics governing wave loading and wave overtopping on crown walls. Since a detailed description of the wave breaking and wave transformation phenomena that takes place on the rubble slope can not be obtained, a better understanding of the problem can only be achieved by studying how wave forces and wave over­topping are influenced by variations in the governing physical and geometrical parameters.

3.1.1 Investigated parameters

In the following a brief listing of the investigated parameters is given. Figure 2.3 sketches all the relevant parameters with the exception of the sea state para­meters. Since waves were generated from a predefined spectrum the spectral estimates Hs and Tp were chosen as representatives of the wave field. In table 3.1 the investigated parameters and their respective variational ranges are given.

Table 3.1: Investigated parameters and their variations.

Parameter

armour

Range 0.10 m - 0.18 m 1.20 s - 2.20 s

0.51 m - 0.59 m random: rock, Dolos, cubes smooth: cubes

0.15 m - 0.33 m 0.11 m - 0.19 m 0.11 m- 0.37 m 0.18 m - 0.36 m

1.5 - 3.5

Ratio Range

~m 1.1 - 5.1 Hs/Ac 0.5 - 1.7 Rc/Ac 1 - 2.6 Ac/B 0.3- 1.1 cot a 1.5- 3.5

The above variations were chosen in order to cover the most common structural variations observed in prototype structures. Tests with the highest wall (hJ = 0.33 m) are in that sense unrealistic, but were performed in order to assess the total momentum in the waves.

34 CHAPTER 3. MODEL TESTS

3.2 Sea states

3.2.1 Wave generation

The model waves were generated according to a version of the JONSWAP spec­trum. The applied version, specified in The Danish Code of Practice (DS-449 1983), is a 3 parameter spectrum defined by Hs, /p (= ,J.. ) and the so-called

p

peak enhancement factor r which in all tests was kept constant at a value of 'Y = 3.3

S(f) = \4

156 H; fir'"!" exp [ -~ ( 1) ']

where :

exp (- (!- /p)2

) 2 a2 j 2 f p

{ 0.10 if f ~ /p 0.50 if f > /p

(3.1)

Wave board control signals were generated by the software package PROFWACO (Frigaard et al. 1993) developed at the hydraulic laboratory. The control signals are calculated by convolution of a white noise signal through a digital filter with the characteristics of equation (3.1). The white noise filtering method has the advantage that very long time series can be generated without any signal repe­tition. The latter requires that the random number generator used for creating the white noise signal does not repeat itself. It was verified that this was not the case. As a different seed number for the random number generator is used for each test series the wave fields of two tests with identical Hs and Tp are not identical.

The wave board is a hydraulic driven piston type paddle operating over the full water depth.

3.2.2 Wave analysis

The irregular wave field, being composed of both incident and reflected waves, was recorded simultaneously by three wave gauges with a logging frequency of 16

3.2. SEA STATES 35

Hz. The gauges were placed in a row pattern just in front of the structure. This setup enables the recorded signals to be separated into incident and reflected wave trains by means of the reflection analysis procedures given by either Goda and Suzuki (1976) or by Funke and Mansard (1980) . In the present study both methods were adopted and average values of the reflection estimates were used.

In figures 3.3 and 3.4 results of the wave analysis from two tests with identical spectral peak periods and water levels are presented for wave heights of Hs ~ 0.10 m and Hs ~ 0.18 m respectively.

From the incident wave spectrum the incident significant wave height Hs ~ Hmo = 4Jffi0 and the spectral peak period Tp and mean period T m = )m0 jm2

were derived. Average reflection amplitude coefficients, weighed with respect to the incident wave energy at each frequency, ranged in all tests between 22% and 25% depending on the actual sea state.

At the time when t he model tests were carried out an active absorption system was not available in the hydraulic laboratory, and hence incident wave trains are infiltrated by re-reflected waves from the wave board. This infiltration, which is most pronounced for long wave periods as they are the most difficult to absorb on the breakwater slope and the spending beach, can be seen as an increased amount of low frequent energy in the incident wave spectra.

From the incident wave spectra inverse fourier transforms were performed in order to obtain the incident wave elevation time series. These series were then analysed with respect to the distribution of wave heights and wave periods by zero down-crossing analysis. In tests with small significant wave heights (figure 3.3) the wave height distribution clearly followed the theoretical Rayleigh distri­bution, whereas for the largest significant wave heights (figure 3.4), where some wave breaking in the form of spilling waves occurred, the observed wave height distributions deviated quite significantly from the Rayleigh distribution.

36 CHAPTER 3. MODEL TESTS

Incident variance spectrum

1.4

~ 1.2

~ 1.0 0 g 01 0.8 ·c ~ 0.6

-;;; l3 0.4 ~

V,) 0.2

0.0 0 0.5 1.5

Frequency (Hz)

Reflection amplitude spectrum

50

~ ._., 40 ~ a :a 30 § t:: 20 .9 u 0

10 !;:: 0

IX

0 0 0.5 1.5

Frequency (Hz)

Wave height distribution 0.20 .--------- --- ------------------,

>. ·~ 0.15 t:: 0

'1:)

g 0.10

~ 0:: 0.05

Rayleigh . '\.

0

Measured

5 10 15 20 Wave height (cm)

Figure 3.3: Example of wave analysis results -test no . 132. Hs = 0.105 m , Tp = 1.6 s , W L = +0.55 m 5349 waves, mean reflection coef. = 24.9%

3.2. SEA STATES

N' ]_ 4.0

t)

~ 3.0 ·~ > 2 0 § . u & 1.0

tl)

50

~ ';;' 40 "0 2 :g_ 30

~ c 20

.S? B c 10 ~

0

0

f- . ~ f-

- !·•

' - I··

0

Incident variance spectrum

0.5 1.5 Frequency (Hz)

Reflection amplitude spectrum

~ i·

~~ IT

~ ~~~~~ rr ::: ~ ~ ~

11~ ~-,l""~ljll~~ r:'-L'•, ~ I!~ I· ~ ~ ~·

i~. ~ f.~ ~ ~; ~ l" ~ ~ ~ ~~ ~ ·~ ~ [·1 l<;

i'C !I ~ ~ ~ ~~ ~ ~· ~ ~ {\,

0 .5 1.5 Frequency (Hz)

Wave height distribution 0.10 ,------- - - - - --- --------------,

c 0.08 ·v; c .g 0 .06 .~ :g 004 -<'I • .D

£ 0.02

0

Measured

5 10 15 20 25 Wave height (cm)

Figure 3.4: Example of wave analysis results -test no. 136. Hs = 0.181 m , Tp = 1.6 s , W L = +0.55 m 5086 waves, mean reflection coef. = 23.1%

37

38 CHAPTER 3. MODEL TESTS

3.3 Investigated cross sections

For all tested breakwater configurations the core was constructed of relative coarse sand (dn,so ~-2mm). Between the core and the inner filter layer (10 mm thick and dn,so = 5 mm) a geotextile was placed to prevent out-washing of the core material. The second filter layer had a thickness of 40 mm and consisted of stones with a nominal diameter of dn,so = 12 mm. Finally a 100 mm thick armour layer with different units in different tests was placed. To ensure the armour remained stable under all wave conditions a thin chicken wire was placed on top of the layer- see figure 3.2.

Figure 3.5 illustrates the principal structure of the breakwater cross section. Also shown in the figure are the geometrical variations listed in table 3.1.

AH measures In m

landward Seaward

1.21

,~,;~~{~:~~~4?:i]t0!sf~i ~~1Gi:~'::- .

2.81-4.21

I I I I 0.55 I I I I I I

0.96- 2.08

Figure 3. 5: Sketch of breakwater cross section with overall measures of geometrical variations.

In order to examine the influence from the parameters given in table 3.1 the 12 different breakwater cross sections outlined in figure 3.6 were tested. Most of the tests were carried out on the cross sections 1-4 where the influence of the parameters H 8 , Tp, hf, Re and Ac was investigated. Cross sections 2, 5, 6 and 7 were used to assess the influence of the armour crest width B. The slope angle a was studied by comparing cross sections 2, 8 and 9 and finally the influence of the applied type of armour (roughness/permeability) was investigated with the sections 8, 10, 11 and 12.

In Appendix B, figure B.1 two photographs of one of the tested breakwater configurations are shown.

3.3. INVESTIGATED CROSS SECTIONS

Section 1

All measures in mm.

Section J

, .. 180 .. , 0

~ 0 I<') I<')

+550 0 Ill -

All measures in mm.

Section 5

~ +550 "'

, .. 240 .. ,

0

"' 0 Ill

All measures In mm.

Section 2

1.,. 180 ~I

0

+550 ~

0 m

0 11')

All measures in mm.

Section 4

, .. 180 .. ,

Ill .,; ...

Ill

+550 "' 0 Ill

All measures in mm.

Section 6

0 ... +550 "'

0

"'

300

All measures in mm.

Figure 3. 6: Cross sections 1 - 6

... ,

39

40

Section 7

, .. 360

0

"' 0 "<t 0 +550 N 11>

All measures in mm.

Section 9

0 -.t

+550 N

, .. 180 .,

0 a>

0 11>

All measures in mm.

Section 11

0 "<t

+550 N

,. 180 .. ,

0

"' 0 11>

All measures in mm.

..,

CHAPTER 3. MODEL TESTS

Section 8

0 -.t

+550 N

, .. 180 . ,

0

"' 0 11>

All measures in mm.

Section 10

0 "<t

+550 N

, .. 180 .. ,

0 0>

0 If)

All measures in mm.

Section 12

0 • +550 N

0

"' 0 If)

All measures in mm.

Fig. 3.6 continued: Cross sections 1- 12

3.4. MEASUREMENT OF PRESSURES 41

The four investigated wall heights - cross sections 1-4 - are shown in figure 3. 7 below. Detailed drawings of the walls are given in Appendix B, figures B.2 -B.5.

Figure 3. 7: Photography of the 4 used walls.

To protect the thin membrane on the pressure sensors against the filter and armour stones a thin steel net was placed approximately 5 mm in front of the sensors.

3.4 Measurement of pressures

Initial tests with a wall section suspended in a dynamometer showed that this type of setup introduced several errors in the measurements of forces exerted on the crown walls. The problems are caused by the wide frequency range that must be covered by the force table without introducing any dynamic amplification in order to measure wave impact forces of very short duration as well as hydrostatic forces having a duration in the order of the wave period. In practice it turned out to be impossible to construct a dynamometer sufficiently rigid (high natural frequency) to avoid dynamic errors and at the same time have a measurable output from the system. Hence it was decided to measure the wave forces by means of pressure transducers.

42 CHAPTER 3. MODEL TESTS

To avoid disturbances in the pressure field on the wall, the pressure cells were built into the walls and aligned with the vertical face. The sensors should prefer­ably have a smooth surface and a quite large diameter to smoothen out local pressure transients. With these constraints in mind 16 Phillips P13-0EM pres­sure transducers with a diaphragm diameter of 18 mm were chosen. The sensors have an operational pressure range between 0 and 4 bar (0- 40 kN/m2

) which is around 5- 10 times the maximum pressures expected in the model tests.

Figure 3.8 shows 2 pressure cells mounted with a water proof chasing and 8 other cells mounted into one of the walls.

0 1 2 3 4 5 6 7 8 9 1 0 cm

11 I I 11 11 11 11 I I 11 I I 11 I

Figure 3.8: Phillips P13-0EM pressure transducers.

All pressures were measured relative to the atmospheric pressure. The dynamic response of the sensors was not tested other than visually verifying that the transducers responded to short duration impact loadings. The transducers were powered from a signal amplifier and the sensor output was amplified by a factor of 1000. The amplifier was checked to be linear up to 0.5 kH z. Prior to the experiments the it was verifies that the pressure transducers did not suffer from

3.5. MEASUREMENT OF OVERTOPPING DISCHARGES 43

temperature drift and nonlinearity. The sensors were calibrated at least twice each week and showed very l~ttle variation. Figure B.7 in Appendix B shows the electronic devices for powering the wave gauges and the pressure sensors.

Initial tests showed that a sampling rate of 256 H z was sufficient in order to avoid loosing information about the signals. In some of the tests with the lowest wall, where rapid pressure rises do not occur since the wall is fully protected by the armour stones, a sampling rate of 128 H z was used.A voltage regulator was used to feed all electronic devices in order to minimise voltage gradient effects in the measurements. With the regulator in circuit it was not necessary to filter the pressure signals. Due to the large amount of data the pressure signals were instantly transformed to resultant wave forces and only the pressure distributions for the 3 largest wave impacts in each test serie were stored.

3.5 Measurement of overtopping discharges

Overtopping measurements were conducted by measuring the amount of water falling into a 0.6 m wide and 0.8 m long tank placed immediately behind the crown wall - see figure 3.9 below and figure B.8 in Appendix B. Only green water overtopping was measured as the wind field could not be simulated in the laboratory.

Flow meter

Crown Wall

Figure 3. 9: Sketch of setup for wave overtopping m easurements.

From the tank the water was pumped back into the wave flume through a wa­ter clock. Since many of the tests were run as batch jobs during nights the recordings of the water clock were done by means of computer controlled camera which was t riggered to take a picture after each run, see Appendix B, figure B.9. Unfortunately, this remote registration did not work properly in all the tests

44 CHAPTER 3. MODEL TESTS

resulting in lack of overtopping measurements in some of the series. Only the mean overtopping quantity from each test was measured.

Wave pressures and forces on crown walls

In the following the results obtained from the performed model tests will be presented. As the measurements were performed by measuring the pressures exerted on the front face of the wall informations on both spatial and temporal pressure distributions have been obtained as well as information of the resulting forces imposed on the wall.

4.1 Distribution of wave pressures

In each test series time evolutions of pressures exerted on the wall face corres­ponding to the 3 maximum wave force loadings were stored. Depending on the wave climate and the geometry of the actual cross section the spatial distribution of wave induced pressure on the wall face develops differently. In figures 4.1 -4.3 evolutions of wave pressure are shown for identical wave conditions for 3 of the 4 crown wall heights used. In the figures the maximum wave loading occurs at time t = 0.000 s . The figures are representative of typical developments of maximum pressure on the respective cross sections.

For a crown wall fully protected by the armour units in front of it (figure 4.1)

45

46 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

the pressure rise is relatively gentle and an almost even pressure distribution at the time of maximum loading is observed.

High crown walls with an upper unprotected wall part are subject to high impact pressures in the region just above the armour crest berm- figures 4.2 and 4.3. A typical evolution of wave pressure on these type of high structures is illustrated in figure 4.2. Initially (t < -0.027 s) no wave pressures act on the wall face. In less than 0.006 s the pressures rise from 0 to approximately 6 kN/m2 in a narrow region immediately above the armour crest. Within the next 0.03 s the high-pressure impact zone is widened and an increase in more slowly varying pressures on the lower protected part of the wall as the water starts to penetrate into the voids in the armour is observed. At time t ~ 0 s the maximum load on the wall .is reached. The distribution of pressure is characterised by a high pressure intensity on the upper part and an almost even distribution of half or less the intensity on the lower unprotected part.

Figure 4.3 shows a very similar situation except that the pressures on the pro­tected part are somewhat higher when compared to figure 4.2. These higher pressures are caused by water from the preceeding wave still being present in the voids in the armour layer when the next wave approaches. This situation is typical for tests with the highest crown wall where all uprushing water is stopped by the wall and hence more time is required to allow for backflow and drainage of the voids.

To some extent the above observations support the assumptions about the dis­tribution of wave pressure on the wall face made by Giinbak and Ergin (1983) - see section 2.2.3. Also the measured pressure distributions compare well with those measured by Jensen (1984), compare figures 4.3 and 2.6, for the pressures at time of maximum overall loading and for a wall configuration with a high unprotected upper part. The unprotected wall part in the wall configuration used by Burcharth et al. (1995) (figure 2.7) only constitutes a small fraction of the total wall height and hence the measured pressures should be compared with the pressure distributions shown in figure 4.1 for a wall fully protected by the armour. For the maximum overall loading the distributions of pressure are seen to compare quite well.

Three consecutive snapshots of an impinging wave on the highest of the four used walls (cross section 3 in figure 3.6) are shown in figure 4.4. The photographs can directly be related to the pressure evolution in figure 4.3.

4.1 . DISTRIBUTION OF WAVE PRESSURES

F=29N/m M=0.7Nm/m Pb= 0.70kN/m2

150 -.---..-------,

IOOf--+---i

50\

0 +-"--t--i 0 2.0 4.0

time =-0.0547s

F=159N/m M=8.5Nm/m Pb= 1.79kN/m 2

150 .---,r---,

100 ~ 50 1 0 -+-_.+--_,

0 2.0 4.0

time =-0.0156s

F=l71N/m M=I0.2Nm/m R = l.69kN/m2

150 R 100 '

50 \

0 -+-_.._+-_, 0 2.0 4.0

time= 0.0234s

Test no. 89 Hs = 0.182 m TP = 2.2 s Ac = 0.15 m

F=45N/m M= 1.7Nm/m Pb=0.8lkN/m2

150 -.---.,---,

I 00 f----:-+---1

50~-t---1

0 -t-L-+----i 0 2.0 4 .0

time =-0.0469s

F=209N/m M=12.6Nm/m Pb=l.86kN/m 2

150 j)

100 ' 50 -+--T-+-----i

0 -t----''1----i 0 2.0 4.0

time =-0.0078s

F=l60N/m M=9.3Nm/m Prl .63kNfm2

150 ....--.,---,

100\

50 \

0 +---"-+---1 0 2.0 4 .0

time = 0.0313s

F= 63N/m M=2.5Nm/m Pb= l.OOkN/m2

150 -.---.,---,

I 00 _.)'---+---!

l 50 \

0 +-......_+----1 0 2.0 4.0

time =-0.0391s

F"'229N/m M=I4.4Nm/m Ph=1 .91kN/m2

150~

I 00 -t----..if----1

·~ 50 -t--f---1

0 -t----"'1---i 0 2.0 4.0

time = O.OOOOs

F=154N/m M=8.8Nm/m 11.= 1.59kN/m2

150 ........--.,---,

100 \

5: l 0 2.0 4.0

time= 0.0391s

No. pressure cells : 16 Vert . cell spacing : 8.5 mm Hor. cell spacing : 40 mm Logging frequency : 128 H z

150

F= 88N/m M=3.9Nm/m P.= 1.21kN/m 2

I 00 ..,_)_+----l

50 \

0 \ 0 2.0 4.0

time =-0.0313s

F=215N/m M=I3.6Nm/m Ph=l.88kN/m 2

150 1\ 100 ""' 50 -t--f+---1

0 -t----''r----1 0 2.0 4 .0

time = 0.0078s

F=151N/m M= 8.6Nm/m Prl .59kN/m2

150 .... ,,----.---,

I 00 -tl}r.---+--1

50 \

0 \ 0 2.0 4.0

time= 0.0469s

Section I

F=I24N/m M=6.1Nm/m Pb= 1.46kN/m2

150

100 -t~r-t---1 t

50 +-._+----l

0 +--'\'-+---1 0 2.0 4.0

time =-0.0234s

F=I89N/m M=I2.4Nm/m Ph=l.81kN/m2

150 l 100 ' 50 -t---\t---1

0-t--llt---f 0 2.0 4.0

time = 0.0156s

F=I56N/m M= 8.9Nm/m ll.=1.6lkN/m2

150 -.r--..-----,

100 )

50 +-+-+----l

0 +--"-+----1 0 2.0 4.0

time= 0.0547s

, ... 180 .. ,

All measures In mm.

Figure 4.1: Pressure distribution on crown wall for max. wave force loading in test no. 89. See figure B.2 for details on pres­sure cell placement.

47

48 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

F=85N/m M=8.5Nm/m Jl,=0.77kN/m2

195 )

130

65

0 0 2.0 4.0 6.0

time =-0.0273s

F=389N/m M=48.8Nm/m Ph= 1.58kN/m2

195

p ...... 130

I 65

0 \ 0 2.0 4.0 6.0

time =·0.0078s

F=584N/m M=56.9Nm/m Ph =3 .34kN/m z

195 -~ ~

130

~ 65

' 0 0 2.0 4.0 6.0

time= 0.0117s

·Test no. 202 H8 = 0.173 m Tp = 2.2 s Ac = 0.11 m

F=I43N/m F=188N/m M=I7.1Nm/m M=23.1Nm/m P.,=0.89kN/m z R. = 1.03kN/m2

195 195 """"'

,-~ ,_

~ ~ 130 130

..-

65

\ 65

' 0 0 0 2.0 4.0 6.0 0 2.0 4.0 6.0

time =-0.0234s time =·0.0 195s

F=496N/m F=586N/m M=59.1Nm/m M=65.0Nm/m Ph=l .99kN/m2 Ph =2.51 kN/m2

195 195 ~

../ ~ • ~ 130 130

< ... ~, 65

' 65

~ 0 0 0 2.0 4.0 6.0 0 2.0 4.0 6.0

time =-0.0039s time = O.OOOOs

F=576N/m F=559N/m M=55.1Nm/m M=52.6Nm/m lt=3.32kN/m2 P"=3.32kN/m2

195 195 i' ~ V !.t""

130

~ 130

~ 65 65

1 J 0 0 0 2.0 4 .0 6.0 0 2.0 4.0 6.0

time= 0.0156s time=0.0195s

No. pressure cells : 16 Vert. cell spacing: 11 mm Hor. cell spacing : 40 mm Logging frequency : 256 H z

F=235N/m M=29.6Nm/m R.=l . l4kN/m2

195 ~

...... ~ ~ 130

• 65

\ 0 0 2.0 4.0 6.0

time =-0.0 156s

F=621N/m M=64.6Nm/m Ph=3.09kN/m2

195

~ ~ 130

l 65

~ ~ 0

0 2.0 4.0 6.0

time = 0.0039s

F=526N/m M=48.5Nm/m lt=3.23kN/m2

195

[/ 130 ,

65

1 0 0 2.0 4.0 6.0

time = 0.0234s

Section 4

Ill

+550 01 0 Ill

All measures In mm.

F=304N/m M=38.6Nm/m R. = 1.30kN/m2

195 -::> ~

130 ....

65

0 ' 0 2.0 4.0 6.0

time =-0.011 7s

F=608N/m M=60.6Nm/m ~ =3.38kN/m2

195

P' ~.-"

130

'"' 65

~ 0 0 2.0 4.0 6.0

time= 0.0078s

F=495Nim M=45.2Nm/rn lt=3.12kN/m2

195 -i' 130

65

1 0

0 2.0 4 .0 6.0

time= 0.0273s

Figure 4.2: Pressure distribution on crown wall for max. wave force loading in test no. 202. See figure B.3 for details on pressure cell placement.

4.1. DISTRIBUTION OF WAVE PRESSURES

F=416N/m M=48.2Nm/m Ph=l.85kN/m2

330

~ 220 I"

110

0 0 2.0 4.0 6.0

time =-0.0273s

F=862N/m M=I31.2Nm/m Ph=2.60kN/m2

330

~ ~ 220

...... ,-,

110

0 0 2.0 4.0 6.0

time =-0.0078s

F=901N/m M=140.7Nm/m Ph =3.01 kN/m2

330 ...........

220 ~ I 110

\ 0 0 2.0 4.0 6.0

time=0.0117s

Test no. 166 Hs = 0.183 m Tp = 2.2 s Ac = 0.11 m

F=521N/m F=612N/m M=64.9Nmlm M=79.9Nmlm Ph =2.11 kN/m2 Ph =2.33kN/m2

330 330

""' \,.,.,

220

' 220 r;::::.

110 110

0 0 0 2.0 4.0 6.0 0 2.0 4.0 6.0

time =-0.0234s time =-0.0 195s

F=907N/m F=931N/m M=I42.3Nmlm M=l48.5Nm/m Ph=2.68kN/m2 Ph=2.77kN/m2

330 330 "" f"'oo. ~ r-....

220 ~

220 ' ~ ~ ~~

110 110

0 0 ' 0 2.0 4.0 6.0 0 2.0 4.0 6.0

time =-0.0039s time= O.OOOOs

F=894N/m F=870N/m M=I37.4Nm/m M=J30.9Nm/m Ph=3.14kN/mz Ph=3.25kN/rn2

330 i""""oo 330 ......._

220 i\ 220 \

If 110

1\ 110

0 0 0 2.0 4.0 6.0 0 2.0 4.0 6.0

time= 0.0156s time= 0.0195s

No. pressure cells : 16 Vert. cell spacing : 20.2 mm Hor. cell spacing : 40 mm Logging frequency : 256 H z

F=707N/m M=96.1Nmlm ~ =2.57kN/m2

330

' 220 ....... r->

110

I 0

0 2.0 4.0 6.0

time =-0.0 156s

F=921N/m M=I46.8Nm/m Ph=2.86kN/m2

330 ....... 'I\. 220

r" 110

0 \ 0 2.0 4.0 6.0

time = 0.0039s

F=860N/m M=126.7Nm/m Ph=3.37kN/rn2

330 r""''

220 1\ ~

110 I) 0

0 2.0 4.0 6.0

time= 0.0234s

Section 3

+550

0

"' "'

0 Cl)

0 .,.,

F=806N/m M=116.8Nm/m Ph=2.63kN/m2

330

~ ...... 220 'I r--

110

0 0 2.0 4 .0 6.0

time- 0.0117s

F=909Nim M=I43.6Nm/m Ph=2.93k.N/m2

330

""' ~ 220

I 110

0 t 0 2.0 4.0 6.0

time= 0.0078s

F=857N/m M=I22.9Nm/m lt=3.50kN/m2

330 ......._

220 ~ I

110

) 0

0 2.0 4.0 6.0

time= 0.0273s

All measures In mm.

Figure 4 .3: Pressure distribution on crown wall for max. wave force loading in test no. 166. See figure B. 5 for details on pressure cell placement.

49

50 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

Flow and impact corres­ponding to maximum load­ing on crown wall. The situation corresponds to the pressure distribution at time t = 0.000 s in figure 4.3. Velocity paths are par­allel to the slope and rel­atively few air bubbles are visible in the fluid.

The wave approximately 0.2 seconds after the max­imum wave impact. Pres­sures are reduced signific­antly on the upper half of the unprotected part of the wall and the thick­ness of the uprushing water tongue has decreased. The flow field is very stochastic and a large amount of air bubbles are visible.

The wave approximately 0.4 seconds after the max­imum wave impact. Pres­sures have decreased on the whole of the wall and the flow has reversed down­wards the slope.

Figure 4.4: Consecutive snapshots of a typical large wave impact on cross section 3 - see figure 3. 6. The photographs can be compared to the pressure distributions in figure 4.3

4.2. WAVE FORCE COMPONENTS 51

4.2 Wave force components

Since the amount of data from the pressure sensors is enormous - approximately 50Mb disc storage per test - the pressure recordings are transformed to the following 3 resultant wave force components by numerical spatial integration over the wall height :

• Horizontal wave force Fh

• Overturning moment M

• Wall base pressure Pb.

The pressure Pb at the base of the wall is stored since this component is normally used to assess the uplift force acting beneath the base slab of the structure, see section 2.2.

Calculations of the overturning moments are solely based upon the wave pres­sures acting on the vertical wall face, i.e. contributions from uplift pressures are not included.

Time series of Pb and Fh are not filtered or smoothed in any way. The overturning moment M is smoothed by averaging the sum of the value at each time step and its neighbouring values. Examples of time series of the three force components are shown in figures 4.5 - 4. 7 for different load situations.

Figure 4.5 illustrates a typical lapse of time during a wave attack on the lowest wall type which is fully protected against impact forces by the armour stones in front of the wall. The example shows the maximum load condition recorded in test no. 55 - see table A.l p. 123. In all tests with this wall configuration relatively smooth force component time series without a distinct peak around the time of maximum load were observed. Also it was noticed that the peak values of the 3 components occurred simultaneously in the majority of the test series. Rise times, i.e. the time from zero loading to maximum loading (see figure 4.8) are typically in the range 0.1 s < trise < 0.2 s.

In figure 4.6 a time sequence from test no. 149, breakwater cross section 3, is shown. The force signals are seen to be double peaked with app. 0.03 seconds between the two peaks. Such situations were only experienced in very few of the tests and only in test series with the highest wall, i.e. cross section 3 in figure 3.6. The magnitudes of the horizontal force Fh are nearly identical for the two peaks whereas the overturning moment M has a clear maximum at the first peak and the base pressure Pb reaches its maximum around the second peak. This time history indicates that the wave attack progresses in the following way :

52 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

1200

960

E' ....... 720 z -u.. 480

240

0 125

100 -E 75 E ~ so ~

25

0 6.0

4.8 ........ C\1

E ....... z 3.6

~

~ 2.4 c::

1.2

. . . . . I I I t

· ··· · · ········i·············-~··············~·-·············~·-·· · · I I I t .

Fnax: 217 N/M Corresponding: 11 : 14 .3 HJ1/M

Lag (s]

0.00 ····· ····· ..... : .......... ..... : ........ ····· . .:,.. ............ --~ ·•··•· Pb = 1. 78 kM/n2 0 • 00

: l I L...--:::---~=----:-: ---l I I I I I I I ............... ~- · ..... ······· ~······ ··· ·····t············· ·t········ · ····-~··· ·· ········-~ ............. -~·- · .......... . I I I I I I I t I I I 1 t o I I I I I I I e I I I I I I

·········· .. ··r-····· .. r···---···r············r············r········-.... ~ .............. ~ ---· ·· -· · -.. ·-. . . ttl1ax: 14.3 Hnll'l

............... ; ............... ~ ...... .......... ~ ...... .......... ~---··· Corresponding: I t I I

: : : F : 217 HIM . . . -·--······· · · -:·-·-----······i··· --· ·····--·t··········--·~·-·-·· Pb : 1. 79 kN/1'12

t I f t I

' . . ' ' f I I I I I

Lag [s]

0.00 0.00

············································-··········································'··············'·············· I I I I I I I

f I I I f I I I I t I I I I I t I I I • I I I I I

··············~··············:·············· ~···· · ··· · ······; .............. } ................ ; .............. ; ....... ······. I o t I I . . ' . . . . .

Pnax: 1.78 kNIM2 Lag . . ········ ······i· ············-~ · ·············1···············~······ Corresponding: . . .

: ! F : 217 HIM . . ······ ········~··············~··· ··· · ·· ··· · ·+···· ··········~······ 1'1 = 14.3 ""'/"

lsl

0.00 0.00

: ~ ~ '"-•:-----,-: --__,...---1 ·········· · ···t······ ·· ··· ···~······· · ··••o ·~········ ····· ··~·· ·· ··· ·· ·····~············ ··!··············t··············

I t I I I 0 I I t I o

I I I 0 I I I I o

··············1··········· -~· · ············:· · ~-············~··············r········-·····~· · ·· ······· · ·· t I o t I I I I

~====+==--·~-~·--~·--~--~--~--~ 0-+ ' . • 0 0.20 0.40 0.60 0.80 1.0 1.2 1.4 1.6

time (s)

Figure 4. 5: Example of typical force time series on the lowest protected wall. From test no. 55.

• a solid water impact in the form of a water hammer impinges on the up­per unprotected part of the wall. This impact causes a large overturning moment and a large horizontal force impact whereas the base pressure is only slightly affected.

• the water hammer is reflected from the wall and all 3 force components decrease.

• the still progressing wave now fully covers the wall resulting in large mag­nitudes of all the force components. Fh and Pb have their maximum values at this stage whereas the overturning moment is somewhat smaller than before.

. - ' -

4.2. WAVE FORCE COMPONENTS

__..., E ...._ ~ l1..

__..., E ...._ E ~ ~

-"' .€ z 6 0...0

1200

960

720

480

240

0 125

100

75

50

25

0 6.0

4.8

3 .6

2.4

1.2

. . . ' ' ' ' Corresponding: ··············:············••!•••···········:···············:······ : : : • t1 = 56 , 5 Ht1/n . . .

········ · · ··· ·~ ············ ··i· · ···· ····· ···f······ ·······+······ Pb : 3.81 kN/"2 . . . I t I I I t

I t I t t I I I I I o

Lag lsl

0.03 o.oo

................. !.............. .. ············~···············~··· ····· · ···· ·~ ·· ··· ······· · ·:··········· ·· ··:······ · · ······ t I t I I I I t t . . . ' I I I t I I I t I .......................... -r········· ..... : ............. r ............ T ............ T ............ .

' : : Corresnond. 1' ......, ', ····· · ··· · ···· ~ ··· ··· ········t············ ··~·· · ··· · ······· ·~ ····· · . .... ··~

I I I t

, F : 557 N/" . .............. ; .............. ; .............. f ............. ~ ...... Pb : 1.87 kH/1'12 I I t I t I o I I t I t I t I I • t • • •

lsl -() .03 -() .03

................ ................ .. . ......................................................................................... . J I J t t o I t t t t

: : : : : ····t········ · ···~····· ·········r··· ···· ·· ···· ·~·-··· ··· ······r············· · . . . . . .

~ax: 3.91 kH/1'12 ............. + ............. , .............. ~ .............. f...... Corresponding: : : : F : 571 H/1'1

.............. ;.............. .. .......... + ............. ~...... H : 56.5 Nl'll" . . . t t I I 0 f

t o t I t t t t I t t t

Lag [s]

o.oo 0.03

··············:·············· ..... ····· · ···:· ···· ··· · ····· ~· ······ ····· · ·r · .. ······ ·· ···· ~ ······· ······ · : · ···· · ··· ···· · . . ..................... , ............... . . . . , ............................... _ ....... .. . . . . . . . . 0 .... . ......... .. , ................ , . ............. . . . .

o!===~~~--~:~--~--~--~--~~ 0 0.20 0.40 0.60 0.80 1.0 1.2 1.4 1.6

time (s)

Figure 4· 6: Example of double peaked force impact on high wall. From test no. 149.

53

The relatively few wave impacts giving rise to double peaked force recordings all have maximum values somewhat lower than the maximum recordings in the respective test series.

Figure 4. 7 shows the force time series for the maximum force record in test no. 149 from where the double peaked forces in figure 4.6 also originates. The loads are clearly defined by a single peak and all 3 force components reach their maximum values at the same time. This type of force development, which corresponds to the pressure evolution shown in figure 4.3, is registered for the majority of the impacting waves on high crown walls.

In an extensive study concerning wave loading on caisson breakwaters Bagnold (1939) found that the horizontal wave force imposed on such structures typically had a lapse of time as shown in figure 4.8. This observation was later verified

~~ ~--- - ~

54 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

E' -.. z ..._.

u.

E' E z ._..

~

"' E ....... z :::.. a:'

1200

960

720

480

240

0 125

100

75

50

25

0 6 . 0

4.8

3 .6

2.4

1.2

. . . ··············~··············!··············1···············r······

I I I I . . . . . . .................. , .................................. ....................... . I I I I . . . . ' . ' . '

Corresponding: " = 99.0 Hrl/1'1 Pb : 3.7! kN/1'12

lsl o.oo o.oo

··············;··········· ·;·· ··········r··············r··············r······-.. ······r············-·1··········-··· . . .

• ""•uc= 99 .o Nrl/1'1 Lag . .............. ; .............. : ............. ; .............. ;...... Corresponding: ls l

: : F : 781 H/1'1 0. 00 . . .............. : ............. : . ........... ~ ............. ~ ...... Pb = 3.71 kH/112 0.00

: I : :

I I I I I I f I I ............................................................ -............................................................ .. . I I I o I I I I t I I I I I

: : I ; : : :

......... ·······~ · -········-···~······ .......... ; ............. .

. . .

. .

.............. j .............. j···· ····· · ·~··i··· · ··· .. ·····t······ Corresponding: : : , : F : 781 N/1'1 .............. ( ........... i ............. t ............ -f' ..... H • : 99.0 NM/ 1'1

: : : I :

Lag [sl

0.00 0.00

··············!············ ~--··· ·······:···-·········-~··· ··· · · ······r·············-~ · -·· ········· ·; ········· ·····

: : : : : I

I I I I I I I ..................... , ............... ................................................... ~.. . ........ , ................ , ............... . ~---~:--,_J . ! ! 1

0 . . .

0 0.20 0.40 0.60 0.80 1.0 1.2 !.4 1.6 time (s)

Figure 4· 7: Example of typical force time series on highest wall. From test no. 149.

by Oumeraci and Kortenhaus (1992) and Marinski and Oumeraci (1992) in a very comprehensive study involving both wave force assessment and stability evaluations of caisson breakwaters. The same principal shape of the wave force evolution is found in the present study with the exception of the relative few tests where double peaked loads were observed. Hence the time evolution of the horizontal wave force component can be described by the peak force (Fh ,peak),

the more slowly varying (semi-static)part of the impact (Fh,static) and the two periods trise and tdecay.

The rise times trise were, for all the configurations with an upper unprotected wall part (cross sections 2- 12 in figure 3.6) and exposed to waves sufficiently large to impact on this part, found to lie in the range 0.01 s < t rise < 0.1 s. For

4.2. WAVE FORCE COMPONENTS 55

Figure 4.8: Schematised wave force evolution.

milder wave conditions, where only the lower protected wall part is loaded, rise times are similar to those found for cross section 1, i.e. 0.1 s < trise < 0.2 s. For the high walls the decay time is typically within the range 0.1 s < tdecay < 0.25 s whereas for the lowest wall type the decay of the force components is in the order of the wave period.

4.2.1 Force distributions and statistical force estimates

Examples of distribution curves for the three force components are given in figures 4.9 - 4.10 for the lowest and highest wall types respectively and for almost identical wave conditions. It is noticeable how the distribution curves turn out to be straight lines in the logarithmic (log10) probability plots for the highest crown wall where hardly any overtopping occurs. A study of figure 4.9 representing the lowest crown wall shows that this linearity does not exist due to the excessive wave overtopping taking place on this configuration.

For further analyses two statistical estimates of each force component have been extracted :

• the 1% force fractile, i.e. the force which in average is exceeded by 1% of the waves.

• the 0.1% force fractile, i.e. the force which in average is exceeded by only 0.1% of the waves.

In table A.l these probabilistic estimates are given for each force component (Fh, M and Pb) in all the tests. The notation in table A.1 is given as F1% = F10; FO.l% = F1 etc.

56 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

Prohiihility n( cxc.:~Lk.·nc.:c Pn>bohility of e>tccetlence

~~~c----------------------------, lc~KI ..,...,-------------------------------,

I c-Ol le-Ill

lc·02

~~~ ------------------ ---lc-114 +-.--.--.--,--.--,--.--,--.--.--.--.--.--.--.--1

() 50 100 150 200 250 31Xl 4 I ll 12

Huriwnt•lli~rcc (N/m) ()vertumint; mnmcnl (Nmhn)

Ptoh•hility of cxc..,.)encc J,-+(KJ -.-----::-----------------------,

lc~ll

lc-{12

11,11

Test no. 89 Cross section 1 Hs = 0.182m Tp = 2.2s Ac = 0.15m

115

Excessive overtopping

1.11 I .S 2.11 2.5

Section 1

1 .. 180 .. ,

All measures in mm.

Figure 4.9: Force distributions from test no. 89 (see table A .l p. 124).

14 16

4.2. WAVE FORCE COMPONENTS 57

Pruhahilily nf excccdencc Pn>t..hility of excee<Jence

~~~~--------------------------~ le~~~----------------------------~

Ie-O I

l e-02

le-03

le-01

le-02

lc-03

n 1 on 200 300 4CXI 500 600 700 scx1 900 1 ~

Hnriznnl:ll force (Nim)

0 W ~ 00 W lOO lW I~ I ~

Ovenuming mumcnl (Nmlm)

Pruhahllily of excecdeoce

le-+00 ,...-----:::-----------------------,

I c-Ol

lc-02

lc-03

I c-(14 -h-rrm-r-rm"rT"r"TTTT'I'TTT'T"T"'TT'M'TTT"r"TTTTT....-rl

0.0 0.5 1.0 I.S 2.0 2.5 3.0 3.5 4.0

Wall ha.<e pre-<Surc (kNim2)

Section 3

14 180 .. I

Test no. 166 0

Cross section 3 CO

0

Hs = 0.183m I') I')

+550 0

Tp = 2.2s Lf)

Ac = O.llm Qm =52 · 10- 6 m3 /m/s

All measures in mm.

Figure 4.10: Force distributions from test no. 166 (see table A.1 p. 125}.

l I i

' \ : .

I: . . \. ,. ,

:t, ., : ~ i' ',• ~ 11

I . ,: :

58 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

4.2.2 Correlation between force component estimates

Scatter plots outlining the relation between the three extracted force components from the tests with cross section 1, tests no. 1 - 89 in table A.1, are shown in figures 4.11 and 4.12. Each figure contains two sets of scatter points, one set with e.g. the maximum horizontal force (Fh,peak) from each wave attack plotted against the corresponding moment (M at time of Fh,peak) and another scatter point set the other way around, i.e. Mpeak versus the corresponding value of Fh. Where the two sets of scatter points do not coincide there is a time lag between the peaks of the respective force components, i.e. a situation similar to figure 4.6. If the scatter points create a narrowly shaped "body" the two force components have a strong mutual correlation.

Studying figure 4.11 it can be concluded that time lags between Fh,peak and Mpeak only occur for relative small wave loadings. For large wave loadings the two scatter sets coincide very well. Also, it can be concluded that Fh and M are almost linearly correlated.

Thrning to figure 4.12 where Pb is plotted versus Fh the situation is somewhat different. The correlation between Pb and Fh (and hence between Pb and M) is much lower and the deviation between the two scatter clouds is more pronounced, meaning that the peak occurrence of Fh (or M) and Pb is often not simultaneous.

Plots similar to figure 4.11 and figures 4.12 are shown in figures 4.13- 4.14 for the tests with cross section 3 (highest crown wall, tests no. 114- 166). Again, a very strong correlation between Fh and M is observed and also the peaks occur simultaneously. Like figure 4.12 the plot of Pb vs. Fh in figure 4.14 shows that the base pressure does not strictly follow the two other force components.

In figures 4.15 - 4.16 Mo.t% vs. Fh,O.t% and Pb,o.t% vs. Fh,o.t% are plotted respectively for all 373 test series. A very strong dependency between Fh,O.l%

and Mo.t% is observed.

As observed in the scatter plots for the individual waves (figure 4.12 and figure 4.14) the correlation between Pb and the other two force components is also weak when considering the 0.1% estimators. For small wave loadings the correlation between Pb,O.l% and Fh,O.t% is quite high but for higher load intensities large deviations are observed.

4.2. WAVE FORCE COMPONENTS

20,---------------------------------,

15

~ z 10 ..._.,

5 -

0 50 100 150

D M at time of Fh,peak

-~p Fh at time of Mpeak

200 250

Fh (N/m)

300

Figure 4.11: M plotted vs. Fh for all waves with cross section 1.

3.0 ..

+

2.5

2.0 ,-...

('l

.§ m a

~ 1.5 ..._., ..0

A..

1.0

0.5 0 Pb at time ofFh ak ,pe 0~ ·+ Fh at time of Pb,peak

Jl(o o 0.0

0 50 100 150 200 250 300

Fh (N/m)

Figure 4.12: Pb plotted vs. Fh for all waves with cross section 1.

59

:

' · !

::

! ''

I ~ !

I I.

60 CHAPTER 4. WAVE PRESSURES AND FORCES ON CROWN WALLS

200

I 0

0 • o

150 cni

m.b~c " &. (]

,-... El fir

~ z 100 .._.,

:=E

50 ...

0 M at time ofFh k ,pea Fh at time of M peak

0

0 250 500 750 1000 1250 1500

Ph (N/m)

Figure 4.13: M plotted vs. Fh for all waves with cross section 3.

,..-... N

_§ ~ .._.,

.D tl..

5.00 --,-----------------~

3.75 +

2.50

1.25

0 250 500

ID

0 Ph at time of F h,pcak

+· Fh at time of P h,pcak

750 1000 1250

Fh (N/m)

1500

Figure 4.14: H plotted vs. Fh for all waves with cross section 3.

4.2. WAVE FORCE COMPONENTS

0 250 500 750 1000

Fh.O.I% (N/m)

Figure 4, .15: Mo.l% plotted vs. Fh,o .I%·

,.-., N

5000

~ 3750 .......,

1250

0 250 500 750 1000

Fh.O.l% (N/m)

Figure 4,.16: Pb,0.1% plotted vs. Fh,O.l % ·

61

1250

1250