deisgn of caisson

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D E S I G N C A I S S O N B R E A K W A T E R

A n

evaluation of the formula of

G o d a

C a r l i t a L. Vis


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D E S I G N

C A I S S O N

B R E A K W A T E R

A n e v a l u a t i o n o f t h e f o r m u l a o f G o d a

C L .

V i s

D e l f t , A u g u s t 1 9 9 5

D e l f t U n i v e r s i t y o f

T e c h n o l o g y


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page ii


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summary

S U M M A R Y

The g row ing need fo r b reakwate rs in deep wa te r due to the inc reas ing d raugh t

o f l a rge vesse ls d raws the a t ten t ion to ca isson b reakwate rs . These mono l i th i c

s t ruc tu res a re more economica l compared to rubb le mound b reakwate rs .

Espec ia l l y i n deep wa te r l ower cons t ruc t ion and ma in tenance cos ts and

considerab le savings in construct ion t ime can be rea l ised. A ca isson is bu i l t on

shore and towed ou t to the ac tua l o f fshore s i te . Un fo r tuna te ly , damage a t a

ca isson i s o f ten p rogress ive . Th is causes an abrup t co l l apse o f the s t ruc tu re . By

unders tan d ing the dynam ic p rocesses invo lve d , the des ign o f the s t ruc tu re can

be sound ly based .

The fo rmu la o f Goda (19 85 ) is a wo r ldw ide used des ign meth od fo r ve r t i ca l

b reakwate rs based on the quas i -s ta t i c approach . H is des ign method i s ve ry

usefu l as a f i rs t ind icat ion for the d imensions of the ca isson. In order to be ab le

to ana lyse Goda 's method , the des ign o f a ca isson b reakwate r i s rough ly

d iv ided in three phases. Fi rs t the crest e levat ion of the ca isson, the design

wave and the des ign wa te r dep th , a re de te rmined w i th p robab i l i s t i c

cons ide ra t ions abou t the economy o f the ha rbour . Subsequen t l y the wave load

fo l l ows f rom the wave p ressure fo rmu lae . Th i rd l y , the w id th o f the s t ruc tu re

se ts the we igh t o f the s t ruc tu re wh ich de f ines the sa fe ty aga ins t fa i l u re .

Goda se ts the des ign pa ramete rs on de f in i te va lues regard less the cos t -bene f i t

ana lys is o f the harbour . H is design wave is the h ighest wave in the design sea

s t a t e ,

which is based on the pr inc ip le that a breakwater should be designed to

be sa fe aga ins t the s ing le wave w i th the la rges t p ressure among s to rm waves .

From the compar i son o f the measured wave fo rces o f the hydrau l i c mode l

s tudy and the va lues ca lcu la ted w i th the wave p ressure fo rmu lae o f Goda and

o f the l i near wave theory no conc lus ions can be d rawn. Th is i s pa r t l y due to the

c lose resemb lance o f the resu l ts o f the l i near wave theory and Goda 's fo rmu la

fo r the cond i t i ons a t Europoor t Ro t te rdam and par t l y caused by the sca t te r i n

th e m e a su re m e n ts .

An exper imen t abou t the fa i l u re mechan isms o f the ca isson con f i rms the

in t roduc ing o f uncer ta in t i es concern ing the p lac ing o f the ca isson on the rubb le

m o u n d fo u n d a t i o n .

Goda 's wave p ressure fo rmu lae tu rned ou t to be in fac t des ign fo rmu lae . No t

on ly h i s des ign pa ramete rs bu t the fo rmu lae themse lves inc lude sa fe ty

cons ide ra t ions . Eva lua t ion o f God a 's fo rmu la i s the re fo re on ly va l i d w he n the

who le des ign p rocess i s taken in to accoun t .

I t i s no ted tha t the accuracy o f the ca lcu la ted wave p ressure on the wa l l i s ve ry

good w i th respe c t to the uncer ta in t i es in t roduce d in the found a t ion fo rces and

the de te rmina t ion o f the des ign pa ramete rs .

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page iv


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table of contents

T A B L E O F

C O N T E N T S

S U M M A R Y P

a

g

e

'»

P R E F A C E page vi i

1 I N T R O D U C T I O N

Page 1

1.1 W hy bu i ld ing a ca isso n bre akw ater ? page 1

1.2 Design process page 2

1.3 A im s of th is s tu dy page

4

1.4 Out l ine o f co nte nts page 5

2 D E S I G N P R I N C I P L E S

Page 7

2.1 Fa i lure me cha nism s page 7

2 .1 . 1 Break wate r s l i d ing page 9

2 .1 .2 Break wate r ove r tu rn ing page 11

2.2 Probab i l is t ic design process page 12

2 .2 . 1 The des ign pa ram ete rs resu l t f rom an econom ic

decis ion prob lem page 12

2 .2 .2 D imens ions o f ca isson page 14

2.3 Design acc ord in g to Goda page 16

2 .3 .1 Des ign pa ram ete rs page 16

2. 3. 2 Resis tan ce again st fa i lure page 17

2 .3 .3 D imens ions o f ca isson page 8

2 .3 .4 Rubb le mo und foun da t io n page 20

3 H Y D R A U L I C D E S I G N C O N D I T I O N S page 23

3.1 W av e sta t is t ic s in open sea page 23

3 .1 .1 D is t r i bu t ion o f wa ve he igh ts page 25

3 .1 .2 D is t r i bu t ion o f wa ve per iods page 26

3 .2 Tra ns fo rma t ion o f deep wa te r da ta to da ta a t the s i te . page 29

3 .3 Chance tha t des ign wa ve he igh t

H

d

is excee ded page 32

4 L I N E A R W A V E T H E O R Y

Page 35

4.1 Formu lae of wa ve pressure page 35

4 .1 .1 The re f lec t ion o f i ncom ing wa ve s page 36

4 . 1 . 2 W ave p ressure on the f ron t o f the ve r t i ca l wa l l . page 37

4. 1. 3 Wav e pressure on the base of the ca iss on page 41

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table of contents

4 . 2 Spec t ra l ana lys i s page 42

4 .3 Ca lcu la t ion fo r Europoor t Ro t te rd am page 45

F O R M U L A O F G O D A page 49

5.1 Formulae of w av e pressure page 49

5.1 .1 W av e pressure on the f ront o f a ver t ica l wal l . . . page 49

5 .1 .2 W ave p ressure under a wa ve t roug h page 53

5 .2 Ca lcu la t ion fo r Europoor t Ro t te rd am page 53

E X P E R I M E N T S I N W A V E C H A N N E L page 57

6 .1 A im s o f exper im en ta l s tud y page 58

6 .2 Sca l ing cons ide ra t ions page 59

6.3 Exp er ime nta l set up page 61

6 .3 .1 Co ns t ruc t ion o f the ca isson mode l page 61

6 .3 .2 Meas ur ing sys tem page 62

6 .4 Exper imen t 1 : De te rm ine hor i zon ta l dynam ic wa ve fo rce page 64

6 .5 Exper imen t 2 : De te rm ine the ho r i zon ta l dynam ic wa ve fo rce a t

m om en t o f fa i lure o f the ca isso n page 66

R E S U L T S A N D D I S C U S S I O N page 69

7.1 An alys is o f Go da 's design pr inc ip les page 69

7 .2 L inear wa ve theo ry com pared w i th God a 's fo rm u la . . . page 70

7 .2 .1 W ave p ressure fo r wa ve c res t page 70

7 .2 .2 W ave p ressure fo r wa ve t roug h page 72

7 .3 The exper im en ta l da ta com pared w i t h the theo ry page 72

7.3 .1 Exp er ime nt 1 page 76

7 .3 .2 Exper imen t 2 page 77

7 .4 R e co m m e n d a t i o n s p a ge 7 7

C O N C L U S I O N S Page 79

L I S T O F

S Y M B O L S

page 81

R E F E R E N C E S page 87

page vi


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preface

P R E F A C E

This repor t is wr i t ten as a par t o f my study for the MSc. degree at the Facul ty

of Civ i l Engineer ing at the Del f t Univers i ty o f Technology, Hydraul ic engineer ing

group .

Part o f th is s tudy about the design of a breakwater is per formed at the Imper ia l

Co l lege fo r Sc ienc e , Tec hno logy and Med ic ine in London . In the hydrau l i cs

labora to r ies o f the de par tm en t o f C ivi l Eng ineer ing , a hydrau l i c m ode l s tud y has

been carr ied out in order to compare exper imenta l resu l ts w i th the resu l ts o f

the ore t ica l ca lc u la t ions . I rea l ise tha t learn ing about the Engl ish w a y of

hydrau l i c eng ineer ing a t Imper ia l Co llege was a g rea t opp or tu n i ty , fo r wh ich I

am very g ra te fu l .

The advice and pract ica l ass is tance of Professor P. Holmes and Dr. D. Hardwick

o f Imper ia l Co llege o f Sc ien ce , Tec hno logy and Med ic ine a re g ra te fu l l y

a c k n o w l e d g e d .

Fina l ly , I w ou ld l ike to expre ss my gra t i tude to Professo r J .K. Vr i j l ing and Mr.

K .G. Bezuyen o f the De l f t Un ive rs i ty o f Techno logy fo r the i r superv i s ion .

D e l f t , Au g u s t 1 9 9 5

Carl i ta L. Vis

page vi i


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page vi i i


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I N T R O D U C T I O N

A b re akw ate r ca n be des igned fo r severa l d i f fe re n t pu rpose s . In the f i r s t sec t ion

o f th i s chap te r backgrou nd in fo rma t ion i s g i ven abou t ca isson b reakw ate r s in

gene r a l . The des ign p rocess o f a ca isson b reakwate r i s d i scussed in the second

s e c t i o n .

Subsequen t l y , the ob jec t i ve o f th i s s tudy i s de f ined in the th i rd sec t ion

fo l l owed by the exp lana t ion o f the ou t l i ne o f th i s repor t i n the las t sec t ion .

W h y b u i l d i n g a

c a i s s o n

b r e a k w a t e r ?

T h e

m a i n f u n c t i o n s o f a b r e a k w a t e r

The bas ic fun c t ion o f a b rea kwa te r i s to p rov ide p ro te c t ion aga ins t w av es . Th is

protect ion may be necessary for an approach channel or for a harbour i tse l f , in

order to prov ide a suf f ic ient t ranqui l harbour basin for sh ips to navigate and

moor . Other pu rposes o f a b reakwate r can be :

• Reduce the am ou nt o f dredg ing requi red in a harbour entra nce by cu t t in g

o f f the l i t to ra l t ranspor t supp ly .

• Guide the cur ren t in the app roac h chan nel or a long the co as t .

Reduce the grad ient o f the cross current in an approach channel in order

to make the sh ips enter ing the harbour bet ter s teerab le .

In th i s s tudy the p ro te c t ion aga ins t wa ve s i s cons ide red to be the on ly fun c t io n

o f a b reakwate r .

A c a i s s o n

b r e a k w a t e r

The cho ice o f a b reakwate r type

for a g iven s i tuat ion depends on

m a n y fa c to r s . Tw o t yp e s o f

b reakwate rs can be

d is t i ngu ished :

• Ca isson b reak wate r

• Rubb le mo und b rea kwa te r

iiirfflTin

1

i h

iiirfflTin

a . JBfflmm

mÊMËsêmm^

caisson

rubble mound

Figure 1.1

Caisson breakwater and rubble mound

breakwater

Mono l i th i c s t ruc tu res , the so

ca l led ca isson b reakwate rs , have

ma jo r advan tages compared to

rubb le mound b reakwate rs in

deep wa te r . Fo r i ns tance the

volume of a ca isson in deep water is less than that needed for a rubble mound

breakwate r because the la t te r i nc reases w i th the square o f the wa te r dep th

see f i gu re 1 1 Mono l i th i c b reakwate rs a re a l so more economica l because o f

the i r l ower cons t ruc t ion and ma in tenance cos ts and the i r cons ide rab le sav ings

in cons t ruc t ion t ime , fo r a ca isson i s bu i l t onshore and towed ou t to the ac tua l

o f fshore s i te . A rubb le mound b reakwate r can on ly be bu i l t o f fshore wh ich i s

page 1


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introduction

cons ide rab ly more expens ive . Rubb le mound b reakwate rs are never th e less more

popular in w e s te rn co u n t r i e s b e ca u se th e y can fu l f i l the i r func t ion e ven w he n

t h e y

are

se ve re ly d a m a g e d . D a m a g e

at a

ca i sso n b re a kw a te r

is on the

o ther

h a n d m o s t

of the

t ime s p rogress ive , wh ich resu l ts

in an

ab rup t co l l apse

of the

m o n o l i t h .

1 . 2 D e s i g n p r o c e s s

The des ign of a ca isso n b re a kw a te r is an i te ra t ive

p ro ce ss .

It can be

d iv ided rough ly in to thre e

p h a se s ,

see

f i gu re

1.2.

F i rs t ly

the

des ign

paramete rs have to be d e te rm i n e d in a cco rd a n ce

w i t h the des ign p r inc ip les . The des ign pr inc ip les

co n s i s t of econ om ica l cons ide ra t ions because an

very h igh

and

h e a vy s t r u c tu re

is not

favourab le .

S e c o n d l y ,

the

des ign pa ramete rs

are

used

as

i npu t for the w a ve p ressu re fo rm u l a e , w h i c h

resu l ts in a d e s i g n w a v e l oad . Su b se q u e n t l y the

d i m e n s i o n s of the ca i sson can be c a l cu l a te d .

These d imens ions have

to be

c h e c k e d w i t h

the

design pr inc ip les again because

the

o p t i m u m

s t re n g th of the s t r u c tu re is re la ted to the s tab i l i t y

of the s t ru c t u r e , w h i c h is prov ided on ly by the

w e i g h t of the s t r u c tu re . In o th e r w o rd s , the

o p t i m u m ra t i o b e tw e e n

the

w i d t h

and the

he ight

of

the

s t r u c tu re

has to be

d e te rm i n e d .

design principles:

design parameters

dimensions :

width

height

Figure

1.2

General design

process

A s tandard des ign method for ve r t i ca l wa l l

b r e a k w a t e r s

wa s

deve loped

by the

Japanese Goda

[ref 4] and is

used

w o r l d w i d e .

He

m a d e

his

fo rm u la a f te r numerous hy drau l i c mode l s tud ies .

The

wave p ressure fo rmu lae

of

Goda were emp i r i ca l l y de r i ved

and

va l i d a te d

by the

p e r fo rm a n ce of p ro to t yp e b re a kw a te r s .

D e s i g n

p r i n c i p l es

A b re a kw a te r is a ssu m e d to have fa i l ed when the m a i n fu n c t i o n is no longer

f u l f i l l ed .

T h a t

is, the

p ro te c t i o n a g a i n s t w a ve s

is

less than requi red.

Ove r to p p i n g of w a v e s is the re fo re con s ide red as fa i l u re . A d i s t i n c t i o n has to be

made be tween fa i l u re in the sense of to ta l co l l apse of the s t r u c tu re and

m a l f u n c t i o n of the b r e a k w a t e r .

The s t reng th aaa ins t co l l apse of a s t ruc tu re

shou ld be des igned in su ch a way t h a t the

s t r u c tu re can res is t the ex t re me hydrau l i c des ign

load on the s t ru c tu re , o the rw ise u l t ima te fa i lu re

occurs . These chosen ex t reme des ign cond i t i ons

d e te rm i n e th e re fo re the needed s t reng th or

s tab i l i t y of the ca i sson .

h a r b o u r

Figure 1.3

Sliding

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introduction

The fu l f i lmen t o f func t ion ing o f the s t ruc tu re under no rma l l oad ing cond i t i ons i s

depend ing on the c res t e leva t ion o f the b reakwate r . The amoun t o f a l l owed

over top p ing shou ld resu l t f rom a cos t -b ene f i t ana lys i s in re la t i on w i th ' the

economy ' o f the ha rbour and the hydrau l i c des ign cond i t i ons . In o the r words ,

the ques t ion needs to be answered i s : 'Wha t i s the accep ted downt ime o f the

harbour resu l t i ng f rom over topp ing o f waves fo r the l i fe t ime o f the ca isson ' .

Th is des ign cond i t i on se ts the min imum cres t e leva t ion o f the b reakwate r .

The s t reng th aga ins t ex t reme load ing resu l ts f rom

the u l t ima te fa i l u re mechan isms w i th i nc lud ing

sa fe ty measures aga ins t fa i l u re . The judg ing o f

the accep ted chance o f u l t ima te fa i l u re o f the

caisson dur ing i ts l i fe t ime is a lso an economic

dec is ion p rob lem, because the p robab i l i t y o f

exceedance o f the des ign pa ramete rs se ts the

probabi l i ty o f u l t imate fa i lure o f the ca isson.

The most impor tan t u l t ima te fa i l u re mechan isms

fo r a ca isson b reakwate r a re :

• Sl id ing (see f igu re 1.3)

• Ov er tur n ing (see f igure 1 .4)

• Fa i lure o f the fou nd at io n ( f igure 1 .5 g ives

two fa i lure possib i l i t ies)

h a r b o u r

Figure 1.4

Overturning

h a r b o u r

Figure 1.5

Failure of the

foundation

D e s i g n

p a r a m e t e r s

Once the amoun t o f a l l owed over topp ing i s de te rmined , the c res t e leva t ion o f

the ca isson i s es tab l i shed w i th tak ing in to accoun t the requ i rements f rom

mar ine rs . Accord ing ly the des ign wa te r dep th and the des ign wave w i th an

accep ted p robab i l i t y shou ld be chosen .

W a v e l o a d i n g

The wave load on the s t ruc tu re can be ca lcu la ted w i th the wave p ressure

fo rmu lae i f the fo l l ow ing paramete rs a re known:

• cres t he igh t

• chara c te r i s t i cs o f the rubb le mou nd foun da t io n

• wa te r dep th

• wa ve he igh t and wa ve per iod

D i m e n s i o n s

c a i s s o n

The proba bi l is t ic load ing on the ca isson dete rmin es the prob abi l i ty o f u l t im ate

fa i l u re th rough the fa i l u re mechan isms. S l id ing and over tu rn ing a re caused by a

hor i zon ta l wa ve fo rce ac t ing on the exposed f ro n t and by a ve r t i ca l up l i f t fo rce

ac t ing on the base o f the ca isson . Bo th fo rces a re resu l t i ng f rom the dynamic

wa ve p ressure . Immed ia te to ta l fa i lu re of a ca isson b rea kwa te r occ urs by

de f in i t i on when the des ign cond i t i ons a re exceeded . The w id th o f the ca isson

se ts the to ta l we igh t o f the s t ruc tu re wh ich p rov ides the des igned s tab i l i t y fo r a

cons tan t he igh t o f the ca isson .

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introduction

1.3 A i m s of t h i s s t u d y

In th i s s tudy

the

des ign p rocess

of a

ca i sson b reak wate r w i l l

be

ana lysed .

Goda 's des ign method w i l l the re fo re be co m p a r e d w i t h a d e s i gn m e th o d th a t

m a k e s use of the l i near wave theory for the co n d i t i o n s of a b r e a k w a t e r at

Europoor t Ro t te rdam.

An i m p o r ta n t a sp e c t of the eva lua t ion of the des ign p rocess is the co m p a r i so n

o f Goda 's wave p ressure fo rmu lae w i th the l i near wave theory and w i t h the

resu l ts of a hyd rau l i c mode l s tu dy .

The l i near wave theory is based upon the c o n c e p t t h a t w a v e s can be

ch a ra c te r i se d as l inear , s inusoida l waves [ref 1], The resu l t ing s imple

m a th e m a t i ca l r e p re se n ta t i o n is easy to app ly and g i ves a g o o d a p p ro x i m a t i o n of

wave behav iou r .

R e s t r i c t i o n s of t h i s s t u d y

The fo l l ow ing l im i t i ng cond i t i ons are

app l ied :

• A quas i -s ta t i c approach is used to

ana lyse the w a ve fo r ce s on the

m o n o l i t h i c b re a kw a te r .

• The w a v e s are cons ide red not to

break as breaking is not taken in to

a c c o u n t

in the

ca l cu la t ion

of the

w a ve p re ssu re s w i th the l inear

w a ve th e o ry .

• The ve ry com p lex p rob lems of the

founda t ion fa l l ou ts ide the scope

o f th i s s tudy .

•

The

hydrau l ic design data

of a

Europoort

Rotterdam

Figure 1.6

The Netherlands

b r e a k w a t e r at Eu ropoor t Ro t te rdam w i l l be used for the ca l cu l a t i o n s , see

f i gu re 1.6 and f igure 1.7.

The d i rec t ion of

t h e w a ve c re s ts is

a s s u m e d

to be

c o n s t a n t and

n o rm a l to the

breakwate r ax i s .

Th e b re a kw a te r is

co n s i d e re d to

have a ce r ta in

l e n g t h , see f igure

1.8.

The

i n te ra c t i o n w i th

th e S id e s ( O th e r

Figure 1.7

ca issons) is

n e g l e c te d .

Situation approach channel Europoort Rotterdam

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O u t l i n e o f c o n t e n t s

Figure 1.8

Length of caisson

In order to analyse the design process of a

ca isson b reakwate r the des ign ph i losophy i s f i r s t d i scussed in chap te r 2 . Th is

inc ludes the fa i l u re mechan isms, the de te rmina t ion o f the des ign pa ramete rs by

means o f an economic dec is ion p rob lem and Goda 's des ign p r inc ip les .

Pr ior to the actua l descr ip t ion of the design wave pressure formulae in chapter

4 ( l inear wave theory) and 5 (Goda), chapter 3 summar ises the hydrau l ic design

cond i t i ons fo r Europoor t Ro t te rdam. The hydrau l i c cond i t i ons a t deep wa te r a re

t rans fo rmed dur ing the p ropaga t ion in to sha l lower wa te r wh ich in f l uences the

probab i l i t y dens i ty func t ion o f the wave he igh t .

In chap te r 6 the exper imen ta l s tudy i s descr ibed . Hydrau l i c mode l tes ts were

car r ied ou t to compare the theore t i ca l resu l ts w i th exper imen ta l da ta . A mode l

ca isson was p laced in a wave channe l and a t tacked w i th regu la r waves . The

hor i zon ta l dynamic wave fo rce has been measured and the u l t ima te fa i l u re

mechan ism have been cons ide red .

The resu l ts o f the p robab i l i s t i c des ign method w i th the l i near wave theory , the

formula o f Goda and the exper imenta l data are d iscussed in chapter 7 .

Recommenda t ions fo r fu r the r research a re g i ven a lso .

The f ina l conclus ions are g iven in chapter 8 .

I t i s no ted tha t de ta i l ed background in fo rmat ion abou t the exper imen ts a re

g iven in the append ices .

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introduction

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design principles

D E S I G N P R I N C I P L E S

Goda developed an easy to apply pract ica l design method [ re f 4 ] . In order to be

ab le to ana lyse h i s des ign ph i losophy , th i s chap te r dea ls w i th the ques t ion :

W he n is a b reakw ate r d es ign a t an op t imu m fo r a pa r t i cu la r s i te ? The answ er

is:

The best design is defined as the structure that fulfils the requirements

at minimum total costs.

An inves tmen t i n a b reakwate r s t ruc tu re i s an economic dec is ion p rob lem and

depends on a l l harbour re la ted act iv i t ies. For a breakwater protects the harbour

aga ins t waves in such a way tha t the ha rbour bas in i s su f f i c ien t t ranqu i l fo r the

sh ips to nav iga te and moor . Wi thou t the b reakwate r the sh ips wou ld on ly be

ab le to use the ha rbour w i th reasonab le good wea ther cond i t i ons . Whether the

breakwate r can improve the ea rn ing capac i ty o f the ha rbour depends fo r

instance on the type and number o f sh ips us ing the harbour , the needed quay

fac i l i t i es , the meteoro log ica l da ta , the hydrograph ica l da ta , the ha rbour re la ted

economic sys tems, the fu tu re deve lopment o f the ha rbour , e tc . Th is imp l ies

tha t the func t iona l requ i rements o f a b reakwate r a re re la ted to the economy o f

the harbour .

The to ta l cos ts o f the des ign depend on the inves tmen t and ma in tenance o f the

s t ru c tu re . The inves tm en t w i l l be h igh fo r a s t ruc tu re w i th a h igh s t re ng th . The

strength o f the structure wi l l be designed so as to res is t the extreme design

cond i t i ons . These ex t reme des ign cond i t i ons resu l t f rom the accep ted

probab i l i t y o f u l t ima te fa i l u re fo r the l i fe t ime o f the s t ruc tu re , wh i le the

probabi l i ty o f fa i lure due to the amount o f over topping is dependent on the

cres t e leva t ion o f the s t ruc tu re .

In order to determine the probabi l i ty o f fa i lure , i t is f i rs t necessary to def ine

when fa i l u re o f a ca isson b reakwate r occurs . Th is i s descr ibed in sec t ion 1 . The

second sec t io n p resen ts subseq uen t l y a theo ry o f ho w to f ind the o p t im um

design load in re la t ion to the costs. Goda 's design pr inc ip les concern ing the

design parameters and accord ing ly fa i lure o f the structure are g iven in the th i rd

s ec t i on .

2 .1

Fa i lu re me chan isms

A b reakwate r fa i l s when i t does no t fu l f i l i t s ma in func t ion : p ro tec t the ha rbour

aga ins t waves . Fo r examp le , when a c r i t i ca l va lue o f wave d is tu rbance in the

harbour bas in i s exceeded , the sh ip hand l ing has to s top , wh ich reduces the

earn ing capac i ty o f the ha rbour . The des ign o f a b reakwate r depends the re fo re

on the requi red degree of protect ion of the harbour against the waves. Th is

degree o f p ro tec t ion i s de f ined by the layou t o f the s t ruc tu re , the pe rmeab i l i t y ,

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design principles

the c res t l eve l (amoun t o f ove r topp ing ) and the energy absorp t ion (pe rcen tage

o f re f l ec t ion o f the incoming waves) . The de f in i t i on o f fa i l u re o f a b reakwate r

is:

The b reakwate r fa i l s when the waves in the ha rbour a re h igher than a l l owed

accord ing to the des ign c r i te r ia .

A l though fa i l u re can occur du r ing bo th cons t ruc t ion and opera t ion , i n th i s s tudy

only fa i lure dur ing operat ion of the breakwater has been considered because the

cons t ruc t ion can be ca r r ied ou t du r ing good wea ther cond i t i ons . The p robab i l i t y

of fa i lure represents the probabi l i t ies o f exceeding a g iven l imi t s ta te . The two

d i f fe ren t s ta tes a re the se rv i ceab i l i t y l im i t s ta te and the u l t ima te l im i t s ta te

T h e S e r v i c e a b i l i t y L i m i t S t a t e ( S L S )

The serv ic eab i l i ty l imi t s ta te is the sta te o f the bre ak wa ter dur ing norm al

load ing cond i t i ons . Fo r th i s s ta te the pe r fo rmance o f the b reakwate r i s

eva lua ted under the ' no rma l ' o r da i l y cond i t i ons to wh ich the s t ruc tu re w i l l be

exposed dur ing most o f i t s l i fe t ime .

Fai lure is def ined in th is s ta te as: the breakwater does not fu l f i l i ts funct ion

because the am oun t o f ove r topp ing i s too h igh , wh ich d i s tu rbs the ha rbour

ac t i v i t i es . The harbour has to be c losed down. Th is can happen regu la r l y and

there fo re the cos ts , whereas the losses due to the c los ing o f the ha rbour fo r a

ce r ta in pe r iod can be subs tan t ia l , shou ld be taken in to accoun t . The h igher the

cres t he igh t o f the b reakwate r the less over topp ing w i l l occur .

An o the r s ta te o f fa i lu re i s , tha t the b re akw ate r does no t fu l f i l i ts requ i rem ents

any more a f te r a few years , wh ich i s de te r io ra t ion o f s t ruc tu ra l res i s tance over

t i m e .

Th is means fo r i ns tance tha t the s t ruc tu re has moved th rough the years

o r tha t the scour p ro tec t ion i s d i smant led . Th is type o f fa i l u re can be p reven ted

by:

• Inc reas ing the des ign res is tance in o rder to guaran tee su f f i c ien t s t reng th

dur ing the serv ice l i fe .

• Con t ro l l i ng the de te r io ra t ion th rou gh inspec t ion and ma in tenan ce

procedures .

Bo th me thod s o f imp rove me nt shou ld be take n in to acco un t a t the des ign s tage

and w i l l a f fec t the cos ts o f the des ign .

T h e Ultimate

L i m i t S t a t e ( U L S )

The u l t ima te l im i t s ta te occurs du r ing ex t rem e cond i t i ons and has a ve ry sma l l

p robab i l i t y o f occur rence . The b reakwate r fa i l s when the ex t reme hydrau l i c

load ing is h igher than the res is tance of the structure. By eva luat ing a l l the

fa i l u re mechan isms tha t a re l i ke l y to occur under spec i f i ed ex t reme cond i t i ons ,

the ab i l i t y o f the s t ruc tu re to su rv i ve ex t reme cond i t i ons i s checked .

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design principles

The th ree most impor tan t fa i l u re mechan isms of the u l t ima te l im i t s ta te are

s l i d ing , o ve r tu rn i n g and fa i lu re of the f o u n d a t i o n , see f ig u r e 2 . 1 , 2.2 and 2.3.

Ul t ima te fa i l u re w i l l be cons ide red to h a ve o ccu r re d w h e n :

•

A

d i sp l a ce m e n t

is

caused

by the

ho r i zon ta l wave fo rce exceed ing

the

hor i zon ta l f r i c t i on fo rce ,

see

se c t i o n

2 . 1 . 1 .

• An ove r tu rn ing is caused by the ho r i zon ta l and u p l i f t w a ve fo r c e s , see

se c t i o n 2 .1 .2 .

harbour

harbour

harbour

Figure 2.1

Sliding

Figure 2.2

Overturning

Figure 2.3

Failure of the

foundation

Fai lure of the f o u n d a t i o n can be ca u se d by se ve ra l p h e n o m e n a . Wa ve i m p a c t

fo r ce s

for

i ns tance

are

re la t i ve l y h igh because the y resu l t f ro m w av e b reak ing .

These fo rces can cause the f o r m a t i o n of q u i cksa n d due to r o ck i n g m o t i o n s of

the ca isson . These p rob lems fa l l ou ts ide the s co p e of t h i s s tu d y but in order to

avo id tha t p rob lems w i th the f o u n d a t i o n w i l l o ccu r , a reasonab ly th i ck po rous

fi l ter layer has to be p laced on the sa n d b o t to m to prevent h igh pore pressures.

2.1.1 B r e a k w a t e r s l i d i n g

Breakwate r s l i d ing

is the

ho r i zon ta l t rans la t ion

of the

ca i sso n , w h i ch o ccu rs

w h e n the ho r i zon ta l wave load is h igher than the ho r i zon ta l f r i c t i on f o rc e .

s e a

harbour

X

' v hydrostatic

/

_X

s

pressure

W

s e a

Fw

c

Figure 2.5 Wave pressure on caisson for <

wave crest

h a r b o u r

flu

Ff

N

N

Figure 2.4

Th e w a te r l e ve l at the harbour s ide is

a s s u m e d to be the sa m e as the

w a t e r l e v e l at the seas ide . So the same

hydros ta t i c p ressure ac ts on bo th

s ides

of the

ca i sso n ,

see

f igure

2.5.

The resu l t i ng fo rce is co n se q u e n t l y the ho r i zon ta l dynamic w av e fo rce

F

w

.

Disp lacement w i l l occur when th i s fo rce

F

w

e xce e d s

the

ho r i zon ta l fou nda t ion

Schematization of forces under a

wave crest

page

9


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design principles

f r i c t i on fo rce F

f

, see f i gu re 2 .4 . As an approx imat ion i t i s assumed tha t the

ca isson fa i l s when the f r i c t i on fo rce i s exceeded and any d i sp lacement occurs .

The hor i zon ta l dynamic wave fo rce

F

w

is:

F

w

=F

w

smut

[ 2 1 ]

i n w h i ch cu is the angular f req uen cy =2n/T

T is the w av e per iod

r is the t im e

The load f requency w is far less than the natura l f requency of the structure

the re fo re the fo rces can be cons ide red as s ta t i c .

When the re i s no ve r t i ca l mo t ion , the re i s ve r t i ca l equ i l i b r ium:

N - W - U - N '

[ 2 2

is the resu l t i ng upw ard no rma l fo rce

is the weight o f the ca isson

is the bu oya nt forc e of the ca iss on

is the ins tan taneo us resu l tan t ve r t i ca l dynam ic fo rce

caused by propagat ion of wave pressures under the

s t ruc tu re

The ins tan taneous resu l tan t ve r t i ca l dynamic fo rce N' can be expressed in

te rms o f the ho r i zon ta l wave fo rce

F

w

:

N'

= e

F

w

= e F

w

sinca

t

[ 2

-

c

i n w h i ch e i s a co e f f i c ien t , wh ich can be foun d f rom a foun da t io n

mode l

in w hi ch A/

W

U

AT

The hor i zon ta l f r i c t i on fo rce

F

f

\s

depend ing on the no rma l fo rce

N

and on the

f r i c t i on be tween concre te and rubb le mound . The fo rmu la i s :

F

f

)

l

2

-

5 ]

3

i n w h i ch 6 is the angle o f f r ic t ion be tw ee n so i l and co ncr ete

0 is th e angle of intern al fr i ct ion of the soi l

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design principles

For 0 = 4 5 ° : fJ = t a n ( 2 /3 -4 5 ° ) = t a n 3 0 ° = 0 .5 8

This is actua l ly an in terna l fa i lure mechanism and is considered as ' fa i lu re o f the

f o u n d a t i o n ' .

The ac tua l f r i c t i on be tween the top laye r o f the founda t ion and the base o f the

caisson has to be empi r ica l ly der ived. For a r ibbed ca isson base the design

va lue fo r th i s f r i c t i on fac to r i s approx imate ly 0 .5 . Th is decreases to

f j =

0 .4 for

a f la t base [ re f 11] .

I f the re i s ho r i zon ta l mo t ion , i t fo l l ows tha t :

1 2 6 1

i n w h i ch

m

b

is the v i r tua l ma ss of the ca isso n

dv/dt

is the acc e lera t ion of the ca isson

The water and so i l mass surrounding the ca isson wi l l in f luence the iner t ia

charac te r i s t i cs o f the ca isson by tak ing pa r t i n the movement as w e l l . Th e re fo re

the v i r tua l mass i s de f ined as an equ iva len t mass wh ich wou ld beg in to move

wh en a d i sp lacem ent o f the ca isson occu rs . Mo t ion s ta r ts wh en the s ta t i c

f r i c t i on fo rce i s f i r s t exceeded . The ex t ra v i r tua l mass wh ich has to s ta r t

acce le ra t ing i s cons ide red to con t r ibu te ex t ra res is tance aga ins t any m ov em ent

o f the s t ruc tu re .

2 . 1 . 2 B re a kw a te r o ve r tu rn i n g

The hor i zon ta l dynamic wave fo rce F

v

and the ve r t i ca l dynamic wave fo rce

AT ten d to ro ta te the ca is son to the

harbour s ide, see f igure 2 .6 . The

coun te r moment i s p rov ided by the

tu rn ing moment o r ig ina t ing f rom the

we igh t o f the s t ruc tu re .

width o f c a i s s o n

ru b b le mou nd

s e a

w-u

h a r b o u r

F w

1

N'

Figure 2.6

Turning moments

Figure 2.7

Dynamic wave pressure

Th e ve r t i ca l d yn a m i c w a ve fo r ce

N'

is assu me d to have a t r iangu lar d is tr ibut ion

over the base, see f igure 2 .7 . However, the pressure at the heel o f the ca isson

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design principles

does not have to be zero. I t depends on the character is t ics o f the rubble mound

founda t ion because the dynamic up l i f t p ressure depends on the ve loc i ty

d is tr ibut ion of the water par t ic les underneath the ca isson. The la t ter is in i ts

tu rn dependen t on the pe rmeab i l i t y o f the rubb le mound founda t ion . The

ve loc i ty w i l l decrease due to f r i c t i on and tu rbu lence . The ve loc i ty d i s t r i bu t ion i s

assumed to be l inear but the curve in the pressure d is tr ibut ion wi l l p robably be

mu ch m ore gen t le . So a t r i angu la r p ressure d i s t r i bu t ion fo r the ve r t i ca l d ynam ic

fo rce AT seem s a conse rva t i ve assu mp t ion .

Equ i l i b r ium o f moments a round the hee l , see f igure 2 .6 , y ie lds:

(F

w

• arm,) • (N


22/200

design principles

fa i l u re o f the s t ruc tu re t imes the p robab i l i t y o f fa i l u re . The income f rom the

harbour ac t i v i t i es and the to the ha rbour re la ted economic sys tems on accoun t

o f the b reakwate r shou ld compensa te the to ta l cos ts .

Every design wave or design load has a

p robab i l i t y o f exceedance . I f the des ign wave i s

s m a l l , the cos ts o f cons t ruc t ion w i l l be re la t i ve l y

low but the r isk wi l l be re la t ive ly great . As the

magni tude of the design load increases, the r isk

w i l l decrease , due to the decreas ing p robab i l i t y

tha t the des ign cond i t i ons w i l l be exceeded . Th is

imp l ies tha t the des ign load must be such tha t the

to ta l cos ts a re min im ized , see f i gu re 2 .8 .

C o s t

l-AV+B

Fo

D e s ig n w ave o r l oad

Figure 2.8 Determine

optimum design

wave

I f design load F

0

i s exceeded then d isp lacement x

wi l l occur (s l id ing is considered as the f i rs t

occurr ing fa i lure mechanism). Fa i lure wi l l be

de f ined as d i sp lacem ent x = x

0

, the load is then

F..

Thus the probabi l i ty o f co l lapse is the

probabi l i ty o f force F

1

be ing exceeded (P(F

7

) ) . The d imensions of the ca isson

are known i f the design load F

0

i s chosen . Hence the cos t o f cons t ruc t ion ,

inves tmen t / , i s a func t ion o f F

0

:

i - m [ 2 . 8 ]

Dam age i s de f ined as a ce r ta in change in the s ta te o f the s t ruc tu re , wh ich does

no t i n f l uence the func t ion ing o f the b reakwate r . Damage to a mono l i th i c

b reakwate r i s o f ten p rogress ive . However , i t i s assumed tha t a second s to rm o f

a g i ven in tens i ty causes jus t as much d isp lacement as the f i r s t one . Repa i r

work wi l l on ly be carr ied out in the ca lm season once per year . The annual

chanc e o f damage repa ir cos ts is the n independen t o f the numbe r o f d ama ge

occur rences in tha t year . Because i f a b reakwate r moves tw ice as much as a

resu l t o f a second storm, i t w i l l cost as much to jack i t in to p lace again .

To de te rmine the amoun t o f r i sk , o r the so ca l l ed an t i c ipa ted damage, i t i s

assumed tha t an insu rance company i s w i l l i ng to i nsu re a ca isson aga ins t

damage . I f the theore t i ca l annua l p remium i s s :

s = (proba bi l i ty o f dam age) • ( the cos t of repai r ing dam age)

The ant ic ipated damage per year is :

s = P ( F

1

) W

[ 2 . 9 ]

in w h ic h P(F

?

) is the prob abi l i ty tha t forc e F

1

i s exceeded , wh ich i s the

probabi l i ty o f u l t imate fa i lure

W is the cos t o f repai r ing w he n a fa i lure o f the struc tur e

occurs

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design principles

The capi ta l ized va lue of the sum of the 'premiums' s depends on the l i fe o f the

s t ruc tu re . I f i t s l i fe i s 100 years , the cap i ta l i zed an t i c ipa ted damage D is [ref

16 ] :

D = M

( 1

1 ) . s [2.10]

i n w h i ch 6 is the rate of inte res t in % per year

The to ta l cos ts C are de f ined as the cos ts o f cons t ruc t ion / and the cap i ta l i zed

a n t i c i p a te d d a m a g e D. Hence C = l + D. I t is assumed that the cost o f

construct ion / is a l inear funct ion of the requi red vo lume of the ca isson per

met re o f exposed f ron t V, so

l - A V + B

[ 2

'

1 1 ]

i n w h i ch A is the pr ice per cub ic me tre vo lum e of the ca iss on

B is the addi t iona l pr ice per cub ic me tre leng th o f the

caisson (cost o f toe protect ion etc . )

The re la t ion between the requi red vo lume of the ca isson per metre o f exposed

f ron t Vand the des ign load F

0

resu l ts f rom the fa i l u re mechan isms.

2 . 2 . 2 D i m e n s i o n s o f c a i s s o n

Fai lure wi l l be considered to have occurred when the ca isson is t ransla ted

(s l id ing) or ro ta ted (over turn ing) . Both fa i lure mechanism are re la ted to the

we igh t and the geometry o f the s t ruc tu re . Fo r a p r i smat i c ca isson the he igh t

and the w id th have to be op t im ized .

Th e m i n i m u m re q u i r e d w i d th b is re la ted to the s l id ing mechanism because for a

ce r ta in c res t he igh t ( resu l t i ng f rom the over topp ing c r i te r ion ) the w id th

de te rmines the we igh t o f the ca isson wh ich p rov ides the s tab i l i t y . A Sa fe ty

Factor i l lustra tes the stab i l i ty through a ra t io o f the force of res is tance and the

dr iv ing force. The Safety Factor o f s l id ing is expressed as:

S-F;

Mng

- 5-

*

constant,

> 1

[ 2 . 1 2 ]

The va lue o f constant

1

depends on the uncer ta in t i es in the assu mp t ions and

fo rm u lae bu t shou ld be a t l eas t more than 1 to assure s tab i l i t y . W he n the inpu t

data are not very re l iab le , the va lue of the safety factor should be made h igher

than 1 .

Ano ther impor tan t sa fe ty requ i rement fo r the s tab i l i t y i s tha t the en t i re base

shou ld con t r ibu te to the upward no rma l p ressure . The moment a rm o f

N

is in

that case equal to 1 /3 o f the width . A t r iangular pressure d is tr ibut ion for N is

a s s u m e d ,

see f i gu re 2 .9 . The max imum bear ing p ressure p

max

ac ts a t the heel

o f the s t ruc tu re fo r a wa ve c res t . Th is max imu m bear ing p ressure depend s on

page 14


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25/200

design principles

Th is p rocedure has to be execu ted fo r severa l he igh ts i f the min imum cres t

he igh t can be taken h igher as the op t imum he igh t . Accord ing to the theory

op t imum des ign fo l l ows f rom an economic ana lys i s .

2 . 3 D e s i g n a c c o r d i n g t o G o d a

The fo rm u la o f Goda [ re f 4 ] i s a wo r ldw ide used des ign metho d fo r ve r t i ca l

b reakwate rs based on the quas i -s ta t i c approach . See chap te r 5 fo r the wave

pressure form ula e. Goda made h is form ula a f te r a lo t o f hydra u l ic sca le m ode l

stud ies He judg ed the re l iab i l ity o f h is form ula by pred ic t in g the acc ura cy of

b reakwate r s tab i l i t y w i th the a id o f the pe r fo rmance o f p ro to type b reakwate rs .

His formula is va l id for breaking and nonbreaking waves. The basic source of

th i s sec t ion is God a 's 'Random seas and des ign o f ma r i t ime s t ruc tu re s (19 85 ) .

2 . 3 . 1 D e s i g n p a r a m e t e r s

The des ign pa ramete rs a re the des ign wave , charac te r i zed by i ts wave he igh t

and wave per iod and the des ign wa te r dep th .

D e s i g n

w a v e h e i g h t

Goda s ta tes tha t the h ighes t wave in the des ign sea s ta te mus t be emp loyed .

This is based on the pr inc ip le that a breakwater should be designed in order to

be safe against the s ing le wave that has the largest pressure among storm

w a ve s . Acco rd i n g to Go d a th e d e s i g n w a ve

H

d

is the h ighes t wav e ou t of 25 0

wa ve s Th is wa ve has a p robab i l i t y of exceed ance o f 0 .4 % se awa rd o f the

sur fzone wh erea s w i th in the su r fzone the he igh t i s take n as the h ighes t o f

random break ing waves a t a d i s tance 5 - /V

s

(H

s

is the mean of one- th i rd o f the

h ighes t waves , see chap te r 3 ) seaward o f the b reakwate r . Goda de f ines the

sur fzone or breaking zone as: 'A re la t ive ly wide zone of var iab le water depth in

wh ich wave b reak ing takes p lace . Concern ing the b reak ing o f random sea

w av es , the b reak ing po in t as we l l as the b reak ing wa ve h e igh t cann o t be

de f ined c lea r l y , i n con t ras t to the case o f regu la r waves ' . So H

d

- H

max

-H

0A

%-

Goda s ta tes tha t the ra t io H

m

JH

s

i s a f fe c te d by the numb er o f wa ve s in a

r ec o r d . The va lue of H

max

shou ld the re fo re be es t im a te d , based upon the

dura t ion o f the s to rm and the number o f waves . He wr i tes tha t the p red ic t i on

genera l ly fa l ls in the range:

H

d

= (1.6 ... 2.0) H

s

I

2

14 1

To avoid possib le confus ion in the design, a def in i te va lue of H

d

= 1.8 H

s

is

recommended in cons ide ra t ion o f the pe r fo rmance o f many p ro to type

breakwate rs as we l l as w i th regard to the accuracy o f the wave p ressure

es t ima t ion . Cer ta in l y the re rema ins the poss ib i l i t y tha t some waves exceed ing

1 8 H w i l l h i t the s i te of the b reak wate r w he n s to rm w av es equ iva len t to the

design condi t ion a t tack. But the d is tance of s l id ing of an upr ight sect ion, i f i t

were to s l i de , wou ld be ve ry sma l l . I t shou ld be remarked , however , tha t the

prescr ip t i on

H

d

= 1.8-H

S

is a reco mm end a t ion and no t a ru le .

page 16


26/200

design principles

D e s i g n w a ve h e i g h t H

d

is the sm al ler one of 1.8-H

s

or H

b

. H

b

is the l im i t ing

wave he igh t o f a b roken wave a t 5-H

s

d i s ta n ce o f f sh o re .

D e s i g n w a v e p e r io d

The per iod o f the h ighes t wa ve i s take n as tha t o f the s ign i f i can t w av e ,

T

d

= T

s

. The wave per iod does no t exh ib i t a un ive rsa l d i s t r i bu t ion law such as

the Ray le igh d i s t r i bu t ion fo r wave he igh ts accord ing to Goda . Never the less

found emp i r i ca l l y tha t the rep resen ta t i ve pe r iod pa ramete rs a re in te r re la ted .

Go d a fo u n d th a t T

d

l i es i n the range (0 .6 . . .1 .3 )7 ; and takes T

d

in the midd le o f

tha t range .

D e s i g n

w a t e r d e p t h

The recommended des ign wa te r dep th i s based on the fac t tha t the g rea tes t

wave p ressure i s exe r ted no t by waves jus t b reak ing a t the s i te , bu t by waves

wh ich have a l ready begun to b reak a t a d i s tance . Fo r the sake o f conven ience ,

the d i s tance 5-H

s

f rom the b reakwate r w i l l be used fo r the des ign wa te r dep th .

Goda der i ved th i s f rom labora to ry da ta on b reak ing wave p ressures .

2 . 3 . 2 R e s i s t a n c e a ga i n s t f a i l u re

The ca isson must be sa fe aga ins t s l i d ing and over tu rn ing . The sa fe ty fac to r

against s l id ing of Goda is def ined as:


27/200

design principles

M

i

s

t h e tu rn i n g m o m e n t d u e to t h e d yn a m i ca l w a ve

'Fw_goda .

fo rce accord ing to Goda

The sa fe ty fac to rs shou ld no t be less than 1 .2 accord ing to Goda . He takes the

coe f f i c ien t o f f r i c t i on

f j

as 0 .6 ( f r i c t i on coe f f i c ien t be tween concre te and rubb le

s tones ) .

The bear ing capac i ty o f the founda t ion shou ld be ana lysed . Goda assumes tha t

a t rapez oida l d is tr ibut ion of bear ing pressure ex is ts bene ath the bas e. The

largest bear ing pressure at the heel p

m a x g o d a

is ca lc u la te d as:

2 W 1 u

/ W * -

T i ^

1

^

3

'

[ 2 1 7 ]

i n w h i ch N

o d a

is the upw ard no rma l fo rce accord ing to Goda

a

3

°f is the lever arm of the up wa rd norm al forc e

M

goda

accord ing to Goda :

_ _

M

N

g o d

. [ 2 . 1 8 ]

goda f j

' goda

i n w h i ch M

N o d a

is the tu rn ing m om ent o f the upw ard no rma l fo rce

~

9

°

3

acc ord in g to Goda around the heel o f the str uct ure

P r e c a u t i o n s a g a i n s t I m p u l s i v e B r e a k i n g W a v e p r e s s u r e

The p ressure due to b reak ing waves may r i se to more than ten t imes the

hydros ta t i c p ressure co r respond ing to the wave he igh t , though i ts du ra t ion w i l l

be very shor t . Goda exp la ins that i t would be ra ther foo l ish to design a ver t ica l

b reakwate r tha t i s d i rec t l y exposed to impu ls i ve b reak ing wave p ressures . A

rubb le moun d b rea kwa te r w ou ld be the na tu ra l cho ice in such a s i tua t ion . I t i s

no t the magn i tude o f the g rea tes t p ressure bu t , ra the r , the occur rence o f the

impu ls i ve b reak ing wave p ressure tha t i s mos t impor tan t . Tab le 2 .1 i s a gu ide

for judg ing the possib le danger o f impuls ive breaking wave pressure.

2 . 3 . 3 D i m e n s i o n s o f

c a i s s o n

The w id th sa t i s fy ing the cond i t i ons o f the sa fe ty fac to rs and the max imum

bear ing pressure is the m in im um requi red wi d th in re la t ion to a cer ta in c res t

e l e v a t i o n . The cr i ter ion of the crest e levat ion in Japan is a t a he ight o f 0 .6

t ime s the s ign i f i can t w av e he igh t above des ign wa te r l eve l . Th is c r i te r ion is

used in s i tua t ions w her e a sma l l amoun t o f wa ve o ver topp ing and res u l tan t

w a ve t r a n sm i ss i o n i s t o l e ra te d .

Goda 's exper imen ts showed tha t the requ i red ca isson w id th depends on the

wave per iod . The w id th has to i nc rease w i th an inc reas ing wave per iod (see

page 8


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design principles

Table 2.1

Questionnaire for judging the danger of impulsive breaking wave pressure

A - l Is the angle between the wave direction

and the line normal to the breakwater

less than

20°

?

1

Y e s

A-2

Is the rubble mound sufficiently small

to be considered negligible?

1

Y e s

A-3 Is the sea bottom slope steeper than 1/50?

I Yes

A - 4 Is the

steepness

of the equivalent

deep water wave less than about

0.03?

1

Y e s

A-5

Is the breaking point of a progressive

wave (in the absence of a structure)

located only

slightly

in front of the

breakwater?

1

Y e s

A-6

Is the

crest

elevation so high as

not to allow much overtopping?

I Yes

No

L i t t l e

Danger

No

Go to B-l

No

Li t t l e Danger

No

• L i t t l e Danger

No

- > L i t t l e Danger

No

— L i t t l e

Danger

Danger of Impulsive Pressure Exists

B - l

Is the combined sloping section and

top berm of the rubble mound broad

enough (refer to Fig. 4.20)?

1

Y e s

B-2 Is the mound so high that the wave

height

becomes

nearly equal to or

greater than the water depth above

the mound (refer to Fig. 4.21)?

1

Y e s

B-3 Is the

crest

elevation so high as not to

cause

much overtopping?

I Yes

No

— > - L i t t l e Danger

No

• Li t t l e Danger

No

— •

L i t t l e

Danger

Danger of Impulsive Pressure Exists

ch a p te r 5 for explanation and appendix F for the

ca l cu la t ions) .

page 19


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design principles

2 . 3 . 4 R u b b l e m o u n d fo u n d a t io n

Berm he i gh t

" It is best to set the he ight o f the rubble mound foundat ion as low as possib le

to p reven t the genera t ion o f l a rge wave p ressure . Bu t the func t ion o f a rubb le

mo und - to sp read the ve r t i ca l load due to the we ig h t o f the up r igh t se c t ion

a n d

the wave fo rce over a w ide a rea o f the seabed- necess i ta tes a m in imum he igh t ,

which is requi red not to be less than 1.5 m in Japan. Fur ther more the t o p

should not be too deep, in order to fac i l i ta te underwater operat ions of d ivers in

leve l l i ng the su r face o f the rubb le mound fo r even se t t i ng o f the up r igh t sec t ion .

A cos t ana lys i s w i l l y ie ld the op t imum he igh t . "

1

Berm width

" I f the seabed is sof t , the d imensions of the rubble mound should be

de te rmined by sa fe ty cons ide ra t ions aga ins t c i rcu la r s l i p o f the g round . The

berm in f ron t o f an up r igh t sec t ion func t ions to p rov ide p ro tec t ion aga ins t

possib le scour ing of the seabed. A wide berm is desi rab le in th is respect , but

the cost and the danger o f inducing impuls ive breaking wave pressure prec ludes

the des ign o f too g rea t a be rm w id th . The p rac t i ce in Japan i s fo r a m in imum o f

5 m under no rma l cond i t i ons and abou t 10 m in a reas a t tacked by la rge s to rm

waves . The berm to the rea r o f an up r igh t sec t ion has the func t ion o f sa fe l y

t ransmi t t i ng the ve r t i ca l l oad to the seabed . I t a l so p rov ides an a l l owance o f

some d is tance i f s l id ing should occur .The grad ient o f the s lope of the rub le

mound is usual ly set a t 1 :2 or 1 :3 for the seaward s ide and 1:1 .5 to 1 :2 for the

h a r b o u r "

2

F o o t - p r o t e c t i o n b l o c k s

" In b reakw ate r con s t ruc t ion in Jap an , i t is cus tom ary to p rov ide a fe w r ow s o f

foo t -p r o te c t io n conc re te b locks a t the f ron t and rea r o f the up r igh t sec t ion see

f i g u re 2 .1 2 .

Cres t E leva t ion

v iO.O

C o n c r e t e C r o w n

v iO.O

Upr igh t Sec t ion

oot - P ro tec t i on

Concre te B locks

A r m o r S t o n e s ( B l o c k s )

N X

\

X

Upr igh t Sec t ion

Foot P ro tec t i on

Concre te B locks

Upr igh t Sec t ion

3 Rubble Mnund Foun dat ion

Figure 2.12 Idealized typical section [ref 4]

The foo t p ro tec t ion usua l l y cons is ts o f rec tangu la r b locks we igh ing f rom 10 to

40 tons , depend ing on the des ign wave he igh t . Foo t -p ro tec t ion b locks a re

Goda, V., Random seas and design of maritime structures. University of Tokyo Press, 1985. page 139

Goda, Y., Random seas and design of maritime structures. University of Tokyo Press, 1985. page 139

page 20


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31/200

design principles

p a g e 2 2


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hydraulic design conditions

H Y D R A U L I C D E S I G N

C O N D I T I O N S

To be ab le to say someth ing about the probabi l i ty o f fa i lure o f the breakwater ,

the p robab i l i t y o f the wave load shou ld be de te rmined wh i le the s t reng th o f the

s t ruc tu re i s cons ide red to be cons tan t . A b reakwate r fa i l s i n the u l t ima te l im i t

s ta te (ULS) when the ex t reme wave load i s h igher than the res is tance o f the

s t ruc tu re . The p robab i l i t y o f exceedance o f th i s ex t reme wave load de te rmines

the re fo re the res is tance and the d imens ions o f the s t ruc tu re (chap te r 2 ) .

The wave load i s a func t ion o f the wa te r dep th h, t h e w a ve h e i g h t H and the

wave per iod T. The re fo re the jo in t p robab i l i t y dens i ty func t ion o f the h igh w a te r

leve l ( i nc lud ing t i des and meteo ro log ie e f f ec ts ) , the wa ve h e igh t and the wa ve

per iod is needed.

The ex t reme wave load resu l ts f rom a s ing le wave in a s to rm. The p robab i l i t y o f

exceedance of th is s ing le wave cannot be re la ted to the probabi l i ty o f

exceedance o f a charac te r i s t i c wave he igh t . There fo re the f requency o f

exceedance o f i nd iv idua l des ign wave he igh ts i s needed . The chance tha t a

ce r ta in des ign wave he igh t H

d

is exceeded dur ing the l i fe t ime / o f the structure

needs to be found . When the jo in t p robab i l i t y dens i ty func t ion o f

H

and

T

is

g i ven fo r a con s tan t wa te r de p th , the p robab i l i ty o f exceedan ce o f H and 7 can

be de te rmined .

The var ia t ion in the seabed leve l has a s ign i f icant in f luence on the hydrau l ic

load ings because a change in the wa te r dep th w i l l a f fec t the charac te r i s t i cs o f

the waves . Th is imp l ies tha t fo r a cons tan t des ign wa te r dep th , shoa l ing and

break ing o f waves in f l uence the jo in t p robab i l i t y dens i ty func t ion o f the wave

he igh t and the wave per iod . The p robab i l i t y o f the wave loads changes

accord ing ly .

In the f i r s t sec t ion the p robab i l i ty dens i ty func t ion s o f the wa te r d ep th , the

wave he igh t and the wave per iod a re g i ven fo r the Nor th Sea (deep wa te r ) . In

the second sec t ion the t rans fo rmat ion o f waves en te r ing f rom deep wa te r i n to

sha l lowe r w a te r i s descr ibed . The chance tha t a ce r ta in des ign wa ve he igh t i s

exceeded i s de te rmined in the th i rd sec t ion .

W a v e

s t a t i s t i c s

i n o p e n s e a

A wave can be charac te r i zed by i ts wave he igh t and wave per iod . There fo re the

dis tr ibut ion of wave he ights and wave per iods g ive the probabi l i ty o f

occur re nce o f a s ing le wa ve fo r a con s tan t wa te r d ep th .

In deep wa te r a s to rm can be charac te r i zed by the s ign i f i can t wave he igh t H

s

and the peak per iod

T ,

assuming a s ta t i s t i ca l l y s ta t iona ry sea s ta te du r ing the

s t o r m .

Fo r sha l low wa te r cond i t i ons th i s assumpt ion can be unrea l i s t i c due to

page 23


33/200


var ia t i ons

in

wa te r l eve l caused

by

t ida l and/or

set up

e f f e c t s .

See

f i gu re

3.1 for

the or ig in

of the

p robab i l i t y dens i ty func t io ns (p .d . f . ) .

In f igure

3.1 the

fo l l ow ing express ions

are

used :

• m e a n

sea

l eve l (MSL) , w h ic h

is the

re fe rence wa t e r l eve l . A rise

o f

the MSL due to

l ong te rm c l ima t i c va r ia t i ons (usua l l y taken

as

0 .1

- 0 . 1 5 m) is not

t a ke n i n to a cco u n t ;

• ve r t i ca l t i de , wh ich

is

because

of the

as t ron om ica l d r i ven fo rc e

en t i re l y de te rmin is t i c ;

astronomic forces

T I D E

Mean sea level

Tide

(vertical)

Wind set up

' J

I p.d.f.. storm surge level h

X

global climatic conditions

meteorological conditions

W I N D

Waves

Wind

waves

I

p.d.f. wave steepnes s j p.d.f. wave height H

1 L

J

JOINT P R O B A B I L I T Y

D E N S I T Y

F U N C T I O N

H,TJ

A )

Figure

3.1

Hydraulic design conditions

for

deep water

• w i n d

set up,

w h i c h

is

ca u se d

by

shear s t ress , exe r ted

by

w i n d ,

on

the

wa te r su r fac e causes

a

s lope

in the

w a te r su r fa ce

as a

resu l t

of

w h i c h w i n d

set up and set

d o w n o c c ur

at

d o w n

and up

wind shore l i nes ;

• s to rm su rge leve l , wh ich

is the

h ighe st s t i l l wa te r leve l dur ing

a

s t o r m ;

I n d i v i d u a l s e a s t a t e s

A n ac tua l wave record f rom

a

w a ve g a u g e

in the sea

g i ves

an

i r regular w av e

pro f i l e .

For an

e xa m p l e

see

f igure

3.2. On the

hor izonta l ax is

the

t i m e

is

g i ven

page

24


34/200


>-

Figure 3.2

Typical wave record Iref 5]

and the wa te r su r face e leva t ion

n

is g iven on the ver t ica l ax is . To be ab le to

ana lyse the waves , the mean wa te r su r face leve l i s de f ined as the ze ro l i ne .

A wa ve i s de f ined as a wa te r m ove me nt b e tw een a po in t where the s u r face

prof i le crosses the zero l ine upward and the next zero-up-cross ing po in t . So the

hor i zon ta l d i s tance be tween two ad jacen t ze ro -up-c ross ing po in ts de f ines the

wave per iod T. The ve r t i ca l d i s tance b e tw een the h ighes t and low es t po in ts i n

a wave i s de f ined as the wave he igh t

H.

When the waves a re l i s ted in

inc reas ing o rder o f the wave he igh t , a rep resen ta t i ve wave he igh t can be

d e f i n e d .

Of ten the s ign i f i can t wave he igh t H

s

i s used , wh ich i s the mean o f

one- th i rd o f the h ighes t waves .

The s tandard record ing pe r iod o f a wa ve record i s 20 m inu tes and rep resen ts

wave cond i t i ons over a 3 hour pe r iod , du r ing wh ich the cond i t i ons a re assumed

to be ' s ta t i on ary ' . Dur ing each s ta t iona ry sea s ta te a sho r t - te rm w av e he igh t

d i s t r i bu t ion app l ies .

3 .1 .1 D is t r i bu t ion o f w a ve h e i g h ts

When the charac te r i s t i c va lues , the s ign i f i can t wave he igh ts H

s

o f e a ch s to rm ,

a re p lo t ted , a s ign i f i can t re la t i on i s found : the long te rm d is t r i bu t ion .

Frequen t i e

X

Figure 3.3 Histogram of wave heights /ref 15]

A smooth d i s t r i bu t ion o f wave he igh ts i s ob ta ined by us ing many wave records ,

page 25


35/200


in which the ord inate is then the re la t ive f requency so that the area under the

h is tog ra m is equa l to 1 , see f i gu re 3 .3 . No rma l l y the wa ve he igh ts a re

norma l i sed by the s ign i f i can t wave he igh t wh ich can be the Ray le igh

d is t r i bu t ion fo r the d i s t r i bu t ion o f i nd iv idua l wave he igh ts (see f i gu re 3 .4 ) :

[ 3 . 1 ]

H In m

Figure 3.4 Rayleigh distribution for H

so

= 9 m (R=100 years)

Tab le 3 .1 g i ves the long te rm d is t r i bu t ion o f s ign i f i can t wave he igh ts fo r deep

wate r (Nor th Sea) .

. 2 D i s t r i b u t io n o f w a v e p e r i o d s

The wave per iod does no t exh ib i t an un ive rsa l d i s t r i bu t ion law . The range o f the

per iods depends on the or ig in o f the waves. In some cases, the per iod

d is t r i bu t ion i s even b i -moda l , w i th 2 peaks co r respond ing to the mean per iods

o f the w ind waves and swe l l (waves genera ted in ano ther w ind f i e ld a rea ) .

The wave per iods can be ana lysed by assuming tha t sea waves cons is t o f an

in f in i te number o f waves w i th d i f fe ren t f requenc ies , see chap te r 4 .

To de te rmine the p robab i l i t y dens i ty func t ion o f the wave per iod , the p robab i l i t y

dens i ty func t ion o f the wave s teepness s i s used . The re la t i on be tween the

wave per iod and the wave s teepness i s :

[ 3 . 2 ]

2n

i n w h i ch s

p

i s the wa ve s teepness w i th pe r iod T

p

H

s

i s the s ign i f i can t wa ve he igh t

L

p

i s the wa ve leng th o f the wa ve w i th pe r iod T

p

T

p

is the peak per iod (around w hi ch the w av e ene rgy is

co n ce n t ra te d )

page 26


36/200

' hydraulic design conditions

H and s a re assum ed to be independen t s toc has ts . Th is ass um pt ion i s fa i r l y

0 . 1 2 i " 1

0.1¬

0.08¬

0.06¬

0.04¬

0.02-

O-l r

«(%)

Figure 3.5 Probability dens ity function of the wave steepness in the North Sea [ref 15]

conserva t i ve because the re i s a re la t i on be tween H

s

and s

p

: h i g h s te e p w a ve s

are more l ike ly to occur than smal l s teep waves. Figure 3 .5 g ives the probabi l i ty

dens i ty func t ion o f the wave s teepness o f the Nor th Sea .

Table 3.1 Hydraulic design conditions [ref 13]

Hydrau l i c des ign cond i t i ons

At deep water

At the site

R (years)

Hso (m)

T (s)

h (m)

0.1

4 .5

7 .4

1 2 .8

0 .5

5.5

9 .0

1 3 .0

1

6.0

1 0 .0

1 3 . 2

5

7.0

1 1 .0

1 3 .7

10 7.5 1 1 .5 1 3 .9

2 0

8.0

1 2 .0

1 4 .2

5 0

8.5

1 2 .5

1 4 .4

1 0 0

9.0

1 3 .0

1 4 .6

5 0 0

1 0 .0

1 4 .0

15 .1

1 0 0 0

10 .5

1 5 .0

1 5 .3

5 0 0 0

1 1 .5

1 6 .0

1 5 .8

page 27


37/200


I t is a normal d is tr ibut ion g iven by:

1 1 _

( s - O

2

,

/2rc

°

ex p [ -

[ 3 . 3 ]

2 o

s

i n w h i ch s i s the wa ve s teepne ss

f i s the mea n = 3 . 7 %

a i s the s tandard dev ia t ion = 0 .5 %

When the two p robab i l i t y dens i ty func t ions a re comb ined the boundar ies o f the

jo in t p robab i l i t y dens i ty func t ion o f the wave he igh t and the wave per iod i s

c

Figure 3.6 Boundaries of the joint probability density function of H

s

and T

p

and the given values from

Table 3.1 Iref 13)

f o u n d , see f igure 3 .6 and appendix A for the der ivat ion.

As t ronomic t i des and meteoro log ie e f fec ts g i ve the s ta t i s t i cs o f h igh wa te r

leve ls .

I t i s assumed tha t bo th the wa te r l eve ls and the s to rm waves occur

s imu l taneous ly and tha t one s to rm las ts 6 hours . Wave cond i t i ons measured

and extrapola ted at deep water near the design locat ion are g iven in Table 3 .1

i n w h i ch

R

H

s

T

h

i s the re turn per iod •

i s the s ign i f i can t wave he igh t a t deep wa te r

is the wave per iod

is the water depth a t the s i te

The chance o f occur rence in te rms o f the re tu rn pe r iod can be t rans fo rmed by

assuming that a year can be d iv ided in a number o f s torm in terva ls . Th is is a

con serva t i ve as sum pt ion because a s to rm does no t occur i n every in te rva l i n

page 28


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rea l i ty . Th is t rans fo rmat ion i s on ly va l i d when the p robab i l i t y o f exceedance i s

expressed in years , wh ich i s ca l l ed the long te rm d is t r i bu t ion . Fo r such a

d is t r i bu t ion ma ny wa ve records over a ce r ta in pe r iod o f t ime a re needed .

Th e f r e q u e n cy f= 1/R is the num ber o f s torm in terv a ls in a year m ul t ip l ied by

the chance tha t the wave he igh t H i s exceeded dur ing a s to rm. A s to rm

durat ion is taken as 6 hours. So

f

_ 3 65 d a y s * 2 4 h o u r s

# m

_ 365 * 24

p (

^

= 1 4 6 Q p ( f y ) [ 3 4 ]

s t o rm d u ra t i o n

h o u r s

6

Which imp l ies tha t 1460 s to rms per year occur . Fo r examp le :

R = 1 year f=\ -» ?(H) = 1/1460 = 6 . 8 5 * 1 0 "

4

R=

10 years -*

f=0A

-»

?{H) =

0 . 1 / 1 4 6 0 = 6 . 8 5 * 1 0 ~

5

T r a n s f o r m a t i o n o f d e e p w a t e r d a t a t o d a t a a t t h e s i t e

The t rans fo rmat ion o f waves p ropaga t ing f rom deep wa te r i n to sha l lower wa te r

can be schemat i sed as i l l us t ra ted in f i gu re 3 .7 . Waves can be t rans fo rmed due

to shoa l ing , b reak ing , d i f f rac t ion and re f rac t ion .

• Shoa l ing is a change in wa ve he igh t wh en wa ve s p ropaga te in

vary ing water depths as a resu l t o f the change in the ra te o f

energy f l ux .

• W ave b reak ing occurs because o f the l im i ta t i on o f the wa ve

he igh t i n re la t i on to the wa te r dep th and the wave s teepness .

• D i f f ra c t ion is the t ran s fo rm at ion of the wa ve s due to the

in te r fe rence o f the waves w i th the s t ruc tu res they mee t . The

resu l t i ng wave f i e ld a round a b reakwate r i s d i f fe ren t f rom the

u n d i s tu rb e d w a ve f ie ld . The wave d i rec t ion i s neg lec ted in th i s

s tudy the re fo re on ly the in f l uence o f waves re f lec ted by the

s t ruc tu re w i l l be take n in to acco un t (see chap te r 4 fo r the t heo ry

o n s ta n d i n g w a ve s ) .

• Re f rac t ion is the change in the wa ve p ropaga t ion ve lo c i ty and in

the d i rec t ion o f wave p ropaga t ion when waves p ropaga te in

va ry i n g w a te r d e p th .

When waves approach sha l lower wa te r w i th the i r c res ts a t an ang le to the

dep th con tours , the wave c res ts appear to cu rve in a way tha t the ang le w i th

the dep th con tours decreases . The wave ce le r i ty decreases as the wa te r dep th

decreases . Fo r s imp l i c i ty , re f rac t ion in f l uences w i l l be neg lec ted .

W a v e s h o a l i n g

The var ia t ion in wave he ight due to var ia t ion in the speed of energy

propagat ion, i .e . the group ve loc i ty is g iven by [ re f 1 ] :

page 29


39/200


joint probability

density

function H.TJi)

shoaling

breaking

limit f h) limit f L )

censored probability density function

H,T)

P(failure)

Figure 3.7 Transformation deep water data

i n w h i ch

H

Ho

Cg

( C J

k

h

g'o

H

Ho

N °

0

2kh

[ 3 . 5 ]

„ [ 1 +

N s i n h ( 2 / c / j )

] tanh/V/j

i s the shoa l ing coe f f i c ien t

is the wave he ight a t the s i te

i s the wave he igh t a t deep wa te r

i s the g roup ve loc i ty o f the waves

is the g roup ve loc i ty o f the waves in deep wa te r

i s the wave number

(2nlL)

i s the water depth a t the s i te

The phenomenon o f shoa l ing can no t be neg lec ted because K

s h

is purely a

fu n c t i o n o f h/L. The shoa l ing coe f f i c ien t fo r Europoort R o t te rda m i s i nd ica ted in

Tab le 3 .2 .

Table 3.2

Transformed hydraulic design conditions

R

( y r s ) H s o (m) h ( m ) H m a x l

=0.5h

L ( m ) K s h = H / H o H m a x 2 = H s o * K s h

H s

(m)

H x P(H a)

N

0.1

4.5

12.8

6.4

70

0.9133

4.1

4.1

6.2

0.00685

3000

0.5 5.5

13

6.5

91 0.9308

5.1

5.1

7

0.00137

2500

1 6

13.2

6.6

104

0.9487

5.7

5.7

7.5

0.000685

2000

5

7

13.7

6.9

118

0.9667

6.8

6.8

8

0.000137

1000

10

7.5

13.9

7

125

0.9766

7.3

7

8.2

6 .85E-05

1000

20 8

14.2

7.1

132

0.9858

7.9

7.1

8.5

3 . 4 2 E - 0 5

1000

50 8.5

14.4

7.2

139

0.9958

8.5

7.2

8.7

1 .37E-05

900

100

9

14.6

7.3

147

1.006

9.1

7.3

8.9

6 .85E-06

800

500 10

15.1

7.6

162

1.025

10.3

7.6

9.3

1.37E-06

600

1000

10.5

15.3

7.7

175

1.048

11 7.7

9.5

6 .85E-07

500

5000

11.5

15.8

7.9

191

1.066

12.3

7.9

9.9

1 .37E-07

500

page 30


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The ra t io

H/H

0

is obta ined f rom the va lue of

h/L

0

us ing appen d ix C f ro m the

Shore Protection Manu al [ ref 14 ] . The deep wa te r wa ve leng th is L

0

=gT

2

/2n.

W av e break in g . .. . . .

Break ing o f waves can occur fo r two reasons . The f i r s t reason i s the l im i ta t i on

o f wave he igh t due to the wa te r dep th . Second ly , the wave s teepness i s

l im i t ed .

The breaking cr i ter ion due to the water depth is normal ly g iven by the breaker

index

(y

br

)

de f ined as the ra t io o f the max imu m wa ve he igh t to wa te r de p th

rat io (H/h):

2

* r - i - y „

[ 3

-

6 1

.

s

l . J max •

b r

h h

For regular waves

y

br

has a theore t i ca l va lue o f 0 . 78 . W h i le fo r ir regu la r wa ve s

( rep resen ted by H

s

) va l u e s a re fo u n d fo r ^ = 0 . 5 - 0 . 6 [ re f 5 ] . Th e a c tu a l li m i t in g

w a ve h e i g h t r a t io ^ d e p e n d s m a i n l y o n th e be d s lo p e m and the wave

s teepness s . In th i s s tudy y

br

=0.b is tak en for i r regular w av es .

Waves in deep wa te r b reak when a ce r ta in l im i t i ng wave s teepness s i s

exceeded . M iche [ re f 5 ] s ta tes tha t the max imum s teepness o f a nonbreak ing

wave i s 0 .142 = 1 /7 . The wave s teepness i s de f ined as the ra t io o f wave he igh t

to w a ve l e n g th H/L.

However the b reak ing c r i te r ium fo r s tand ing waves d i f fe rs f rom those fo r

t rave l l i ng waves accord ing to Wiege l [ re f 13 ] :

H

x

= 0.109

L

tanh

kh

[ 3

-

7 1

i n w h i ch H

x

is the ma x im um progress ive wa ve he igh t

k is the wa ve number (2nlL)

h i s the wa te r dep th

L i s the wa ve leng th

As ind ica ted in Tab le 3 .2 , the s tand ing wave b reak ing c r i te r ium i s never a

govern ing fac to r fo r the s ign i f i can t wave , s ince the h igher o f these b reak long

be fo re reach ing the b reakwate r .

The numer ica l model ENDEC [re f 5 ] g ives design graphs in which the in f luence

of both shoal ing and wave breaking is inc luded. See f igure 3 .8 .

The inpu t pa ramete rs fo r Europoor t a re :

1 .

Loca l re la t ive wa te r dep th

i n w h i c h L

op

h

i s the deep wa te r w av e leng th (w i th

peak per iod 7^ = 1 0 s ) = qT

p

2

l2n =

( 9 .8 1 * 1 0

2

) / 2 r r = 56 m

is the de sign w at er de pth (R = 50

years has been taken) = 14 .4 m

= 1 4 . 4 / 1 5 6 = 0 . 0 9

page 31


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3 . Deep wa te r wa ve s teepness s

op

= H

so

IL

op

i n w h i ch H

so

= 9 m (R = 10 0 years has been tak en)

s

op

= 9 / 1 5 6 = 0 . 0 6

The max imum s teepness ava i lab le in the des ign g raphs i s 0 .05 . Fo r 0 .05 the

ou tpu t i s :

HJh =

0 . 4 5 :

H

s

= 0.

45-14.4 = 6. 5 m

So H

s

= 0.5-h seem s to be a good ap prox ima t ion .

W h e n h = 14 .4 m the des ign s ign i f i can t wave he igh t H

s

is:

/V - = 0 . 5 - 14 .4 = 7 .2 m

C h a n c e

that

d e s i g n w a v e h e i g ht

H

d

i s e x c e e d e d

Each storm can be character ized by a g iven va lue of

H

s

,

t h e s i g n i f ica n t w a v e

he igh t . Assume th i s s to rm cons is ts o f n w a ve s , w h i ch a re d i s t r i b u te d a cco rd i n g

to a Ray le igh d i s t r i bu t ion .

The chance tha t an a rb i t ra ry chosen des ign wave he igh t H

d

is exc eed ed by any

g iven wave i s :

- z &

2

[ 3 . 8 ]

f \ H = e

The chance that th is wave is not exceeded in a ser ies o f

n

w a ve s i s :

page 32


42/200


[ - F X H ^ Y

1

I

3

-

9

So the chance tha t

H

d

is exceeded at least once in a s ing le s torm conta in ing

n

w a ve s i s :

E j -

1 - [1 - F \ H } \

n

[ 3 -

1

° ]

Th is chance has to be comb ined w i th the chance tha t H

s

o cc u rs , w h i ch m u s t

com e f rom a long te rm d is t r i bu t ion o f s ign i f i can t wav e he igh ts , p ^ ) can be

de te rmined as the chance tha t some wave he igh t H

S

-A H

S

i s excee ded minus the

chance tha t the he igh t H

S

+ AH

S

i s exce eded . p(H

s

) i s the chance tha t H

s

fa l ls in

the in te rva l hav ing a w id t h o f 2AA/

S

. Take as an approx imat ion

AH

S

is 0 .5 m

t h e n E

1

i s no t changed s ign i f i can t l y . So the chance tha t H

d

occurs du r ing any

sing le s torm per iod is :

E

Z

=

P H J E , [ 3 . 1 1 ]

I t is s t i l l possib le that the chosen

deisgn of caisson

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