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STATISTICALDISTRIBUTION

OF

HORIZONTALWA V EFORCESON

VERTICALBREAKWATERS

JaniceMcKENNA

1

an dWilliam ALLSOP

2

ABSTRACT

This

paper

discusses

th e

statistical

distribution

of

wave

impact

forces

on

vertical

wall

structures.

tescribesewnalysis

arried

ut

ased

n

heistribution

ype

o

identify

the

parameter

combinations

that

lead

to

wave

impacts.

Comprehensive2-dimensionalrandomwavemodeltestswerecarried

out

tomeasure

wavepressures

n

a

range

of

structure

types.nitialnalysisof

these

testshowed

thatth eWeibull

distribution

could

beused

todescribe

non-breaking

waveforces.

Forsomestructuralconfigurationshowever

th e

wave

forceswere

found

to

give

a

poor

fi t

with

he

Weibull

istribution.

hese

ata

ad

een

xcluded

ro m

he

nitial

analysis.

he

newanalysisdescribedin

thispaper

has

resulted

in

a

revised

parameter

map

to

summarisethe

risk

of

wave

impact,

derived

from th e

full

data

set

and

based

on

th e

distributiontype.

1 .

INTRODUCTION

Vertical

wall

structures

werewidely

used

in

the

UK

as

seawalls

andbreakwatersbefore

rubble

mound

breakwaters

nd

rip-rap

revetmentsgained

popularity

inth e900's.n

Japannd

taly,

aissonsontinue

o

e

avoured

or

reakwateronstruction.

thorough

understanding

of

the

relationship

etween

wave

forces

n

vertical

walls

nd

overallstructuregeometry

isnecessary

to

enable

th e

design

of

asuitable,

ost-effective

structure,

with

anacceptably

low

risk

of

failure.

t

isalso

desirable

to

minimise

the

risk

of

severexposure

to

arge

breaking

wave

mpactorces,o

hatuture

maintenance

costs

are

not

undulyhigh.

Verticalwallsin

the

marineenvironmentaresubjected

to

highlyvariablewave

loads,

yet

existing

esign

methods

reeterministic.

hemain

esign

method

or

aissons,

developed

by

Goda(1974,985),

provides

a

good

estimate

of

non-breakingwaveforces

(McKenna,

997).

For

wave

mpacts

owever,

he

redicted

orces

sing

Goda's

Assistant

Engineer,

BabtieGroupLtd.,

9 5Bothwell

Street,

Glasgow,

G 27HX,UK

VisitingProfessor,

TheQueen'sUniversity

of Belfast,UniversityRoad,Belfast,

B T7

INN,

UK

Manager

Coastal

Structures,H R

Wallingford,

HowberyPark,

Wallingford,Oxon

OX10

8BA,

UK

2082

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COASTALENGINEERING1 9 9 8 083

method

areeffective'alues,amped

by

the

response

of

the

structureandfoundation,

ratherthan

actual

impact

loads.

hese

wave

impact

loads

were

previously

thought

to

be

unimportantforthestabilityof

massivestructuressuchascaissons,ut

O umeraci

t

al

(1995)

have

shown

that

repeated

wave

impacts

can

cause

incremental

displacements.

It

s

ecessary

o

ecognise

he

wideariationnorce

ype

ndmagnitude,

ndo

incorporateameasureoftheir

probabilityofoccurrence,nordertodevelopimproved

designmethods.

Work

withinM A S TI I - P R O V E R B Sasbeenirectedtowardsthis

Identification

of

a

uitable

tatistical

istribution

or

wave

orces

nerticalwall

structures

is

particularlymportant

fo r

the

evelopment

of probabilistic

design

tools.

Previousworknth e

tatistics

ofwave

orces

asoncentrated

n

stablishinghe

extreme

istribution

f

eries

of

(theoretically)

egular

waves,

ee

or

nstance

Kirkgoz

(1995),

but

these

results

cannotbe

applied

to

real

(random)

seas.

The

Weibullistribution

canbe

used

toescribepulsating

wave

forcesmeasuredn

model

tests

on

caisson

breakwaters

using

randomwaves.

llsopt

l(1996a)

av e

shown

that

the

onset

of

wave

mpacts

can

be

defined

as

a

change

n

gradient

of

the

probability

plot

where

wave

forcesstart

to

increaserapidlyabovethose

predicted

by

th e

simple

Weibull

distribution.

The

design

load

fo r

many

structures

will

however

be

du e

to

wave

impact

forces

rather

than

on-breaking

aves.

t

s

herefore

mportant

o

e

ble

o

stablish

he

statisticaldistributionofwaveimpactforces,

nd

to

knowth erelativeproportionsof

pulsatingan dimpactforces

fo r

agivenstructural

configuration.

2.ODELTESTS

Comprehensive

arametric

model

ests

were

onductedn

andom

wave

lume

t

H R Wallingford

during

994.

hese

tests

were

designed

to

nvestigate

th e

influence

of

structure

geometry

on

wave

pressures

and

forces,

using

random

waves.

ver

200

tests

ere

arried

ut

n

ll,

o

xplore

he

ffects

f

arying

he

ollowing

parameters:

a)

ignificantoffshoreandinshorewave

heights,

H

s0

and

H

s

; ;

b)

aterdepth

in

front

of

structure,

h

s

;andcrestfreeboard

R ^;

c)

ave

steepness,s

m0

,

and

peak

wavelength

atstructuretoe,

L

p

j ;

d)

ater

depth

over

moundinfront

ofwall,

d;

an d

bermheight,

h i , ;

e)

erm width,

Bt,;

f)

ront

slope

ofmound,

a;

g)

epthofembedmentofcaissonintomound,lvh

c

;

Thecaisson

was

instrumented

with

8

pressure

transducers

on

th e

front

face

and

4

n

the

undersides

hownnFigure.ressure

datawere

acquired

nll

ransducers

simultaneously

at

4 0 0 H z

fo r

500wavesin

each

test.

hese

testshave

been

described

previously

by

Allsopet

al(1996b).

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Figure

1 InstrumentedModelCaisson

3.

ATA

ANALYSIS

A

data

analysisprogramwasdeveloped

to

carry

outtheinitial

analysis

of

th e

pressure

dataacquiredfromthesemodeltests.hepressure

data

were

spatially

integratedat

eachtime-stepto

give

horizontal

anduplift

forcetime

histories.

n

orderto

studyth e

statistics

of

th eataet,nformationegarding

he

orce

maxima

wasequiredor

eachwave

event.nvent

was

efinedrom

thebeginning

of

each

rapidressure

riseon

th e

transducer

at

still

water

level,

nd

th e

analysis

program

was

configured

to

searchfo rth erequiredinformationwithineach'event'.

As

horizontalandupliftpressureandforcemaximadonotnecessarilyoccuratth e

sametime,anumberofoutputfileswerewritten,sothateachphenomenoncouldbe

studiedseparately.

Thenext

stage

of

th e

data

analysis

was

to

rank

thepressure

and

forcemaxima

containedin

th e

output

filesin

orderofmagnitudetoenabletheir

statistics

to

be

explored.hispaperconcentrates

on

the

results

ofthefurtheranalysis

of

the

horizontal

forces.

4.

TATISTICSOFW A V E

FORCES

Wavempact

orces

n

oastal

tructures

re

xtremely

ariable,

nd

re

herefore

better

described

by

their

statistics

than

by

any

single

deterministic

value.

n

selecting

an

appropriate

tatistical

istribution

fo r

wave

orces,

t

s

n

dvantage

to

maintain

anyassociationbetweenwaveheightsandwaveforces.f anarrowbandprocessis

assumed,waveheightscan

be

approximatedbyth eR ayleighdistribution,whichisa

special

form

of

the

more

generic

3-parameterWeibull

distribution.

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2085

Analysis

of

th e

pressure

datafrom

these

model

tests

by

Allsop

etal

(1996a,b)

showed

that

th e

Weibulldistribution

could

be

used

to

give

a

good

description

ofwave

forces

from

on-breakingwaves

referred

o

ere

sulsatingwaveorces),s

hownn

Figure

.t

was

lso

ound

hat

the

nset

ofwave

mpacts

ould

e

efined

y

change

n

he

radient

of

th e

Weibull

lot,

where

he

wave

orces

tart

ncreasing

morerapidly

withnon-exceedance

level,

Figure

3 .

f

Non-exceedance

probability

( )

30.8

3 .2

3 .4

m

H si

0.08m,

snii

0.03

Structure9

sw l

1.70m

-1

ln(-ln(l-P))

2.72

^

Figure

2 Weibull

distribution

ofpu lsatingwaveforces

Non-exceedanceprobability

(% )

12.7

0.83.23.49.9

K 0

lib

0.367m

hb

0.457m

--

7.40

Hso0.25msmo

0.04

swl1.61m

0

ln -ln l-P))

8.37

Figure3 Transition

from

pulsatingforcesto

impactforces

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Thismethodofestimatingth e

percentage

ofimpacts

was

appliedtoeach

test,and

was

found

o

ive

ood

greement

withbservationsromhelume

estingndwith

analytical

considerations

of

the

physical

processes

involved.

hisanalysis

led

to

the

developmentof

a

parameter

map,

Allsop

et

al(1996a,b)that

could

beusedto

estimate

th e

risk

of

wave

impacts

on

a

particular

structure.

Furtheranalysisby

McKenna(1997)revealedthatth eWeibull

distribution

couldalso

be

used

to

describe

wave

impact

forces

in

some

instances,

Figure

4.twasalsonoted

that

fo r

other

casesth epercentage

ofimpactsould

be

difficultto

determine,

s

th e

change

in

the

gradient

of

th e

distribution

was

not

distinct.

hese

cases

were

selected

fo r

urther

nvestigation,

ndt

was

foundthat

th e

ata

oints

lotted

s radual

curve

n

Weibullxes,atherthan traightinewith

harp

hangenradient,

Figure

5.

The

tructural

onfigurations

orresponding

o

hese

urved

istributions

ere

investigated

nd

acommon

linkwasound

n

th e

relationship

between

theffshore

waveconditions

an d

th e

localwater

depth

at th e

structure.

imiting

valuesofH

so

/h

s

=

0.425

nd

mo

/h

s

17were

dentified,

eyondwhichwaveorcesoongerithe

Weibull

distribution,

Figure

6.

he

distortion

ofth edistributioniscaused

by

shallow

water ave

ransformations.he

mallestaves

each

he

tructure

elatively

unchanged,

but

there

isa

gradual

increase

in

th eamount

of

modification

of

the

wave

shape

as

th ewave

heights

and

lengths

increase.The

incidentwaveheight

distribution

was

found

to

be

non-R ayleigh

for

these

cases,

Figure7.

Non-exceedance

probability

(% )

12.7

0.8

3 .2

3 .4

9.9

f

0

a

Hsi

0.15m,

sini

0.04

V

1.72

Structure3

swl

1.52m

ln -ln l-P))

Figure

4

Weibull

Distribution

of

horizontal

waveimpact

forces

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1998

2087

N on-exceedanceprobability( )

0.02

2.7

3 .2

2

t e s t _ 1 0 0 2 < r ]

-

-2

ln(-In(l-P))

Figure

5 Non-Weibull

distribution

z

0.14

Figure

6 ParameterrangesforWeibulldistributedforces

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offshoreH nshoreH

Spectrum

H2 5S 6E D ,

S W L

1.34m

1

- ln 1-p)

Figure7

Modifiedinshore

wave

height

distribution

5.

EVELOPMENT

OF

PARAMETERM AP

Attempts

to

analyse

th e

full

data

se tby

Allsop

et

al

(1996b)

had

highlighted

th e

need

to

eparate

he

ata

nto

egions

f

imilar

esponse

haracteristics.

he

dimensionlessparameters

that

influenced

waveforces

were

identified

as:

elative

mound

height,

hb/h

s

;

elativewave

height,

H

s

;/d;

elative

bermlength,B

eq

/L

P

i .

The

effectsof

these

parameterswere

summarised

inth eparametermappresented

by

Allsop

t

al

(1996a,b).

hederivation

ofthat

parameter

map

ad

concentrated

ona

central

ore

of

data,

where

he

waterepth

ver

he

ubblemound

erm,

, as

greater

than

twice

he

ignificant

nshore

waveheight,

H;.

ases

where

he

water

level

wascloseto

or

below

thetop

of

th erubble

moundwereexcluded.

Potentialweaknesseshad

been

recognised

in

that

derivation,

both

inexcluding

of

a

se t

of

data

fromth eanalysis,andinusingth edimensionlessparameterH

s

;/d,whichtends

toinfinityasth ewaterdepthoverth erubble

mound

tendsto

zero.

twasconcluded

that,

fpossible,

his

arameterhouldot

e

sedn

urther

work

oxtend

he

parametermap

to

include

th e

fulldataset.

The

approach

used

n

this

urther

work

to

include

th e

ull

data

et

in

the

parameter

map

onsidered

he

ffects

fhe

on-Weibull

istributed

orces,

nd

ncluded

additional

parameters

to

describe

th eeffectsoftransformationfrom

the

offshorewave

climate

to

th e

inshore

wave

climate.he

dimensionless

parameters

used

nth e

new

parametermap

(Figure

8)

are:

elative

offshore

wave

height,H

S 0

/h

s

;

elativeoffshore

wave

length,

L

m o

/h

s

;

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a

o

a

s

n

J

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elativeinshore

waveheight,H

s

j/h

s

;

elative

berm

height,hb/h

s

;

elativeberm

length,

B

eq

/L

p

j .

The

significance

ofeach

ofthese

parameters,

andtheircontribution

to

th e

description

of

the

overall

physical

processes

aredescribed

below.

5.1

egree

of

Wave

Breaking H

so

/h

s

andL

mo

/h

s

)

Inorder

to

represent

th e

effectsofstructural

geometry

properly,it

is

first

necessaryto

identify

hoseests

whereth eeabedauses

wavebreakingnthe

pproach

o

he

structure.heseestsmust

e

dentified

t

he

eginningofth enalysis,

s

hey

would

distort

anyconclusionsaboutth e

overall

response

to

th e

structure

itself.

hese

tests

were

easily

identified

by

th e

curved

distribution

of

data

when

plotted

on

Weibull

axes,asdescribedpreviously.

Further

work

is

required

to

identify

th e

parameter

influences

n

ases

or

whichthe

data

are

not

Weibulldistributed. suitable

statistical

distributionmust

be

established

fo r

thesetests

andth eresponse

to

the

structuregeometry

may

then

be

identified

from

this

new

analysis.

Itis

not

possible

at this

stageto

speculateon

effectsof

individual

parametersin

these

cases,

but

it

is

certain

that

structures

in

shallow

water

are

at

risk

of

exposure

to

breakingwave

forces.

he

overall

levelofforces

on

thestructuremayhoweverbe

low,as

th e

incident

wavesmay

be

substantially

broken

(and

aerated)

before

reaching

th estructure.

lackmore

and

H ewson

(1984)haveshownthat

wave

forcesdueto

highlyaerated

waves

are

significantly

lower

than

corresponding

'deep

water'

waves.

Configurationsthat

fallinto

this

category

should

be treatedasthough

they

will

be

subjected to

high

impact

forces,anddesignedaccordingly,until

further

work

has

been

carried

out

in

this

region

ofth e

parameter

map.his

isparticularly

important

for

structures

in

areas

with

high

tidalranges.

5 .2

averegime

at

structure

H

S

i/h

s

)

The

most

ignificantarameterffecting

he

nset

fwave

mpacts

or

Weibull

distributeddata

wasfound

to

be

th erelative

incidentwave

height,

H i/hs.

his

parameter

representsheikelihood

fwave

reaking

t

heoe

f

the

tructure,efore

ny

significant

interaction

with

the

structurehastaken

place.

hecriticalvalueofHsi/h

s

to

cause

th e

onsetofimpacts

was

investigatedbyplotting

the

percentage

ofimpacts,P i% ,

for

each

testagainst

H

S

j/h

s

.

Initial

investigations

usingWeibulldistributeddatafromstructures0,,and2

suggested

that

if

th e

valueofH

s

j/h

s

was

lessthan

a

criticalvalueof

0.2,

there

would

be

nowave

breaking,and

th e

resulting

forces

would

therefore

be

pulsating,see

Figure

9.

Extension

of

thisnalysis

o

nclude

ll

tructureshowedthatmpacts

id

ccur

or

some

configurations

where

H

i/h

s

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2091

verticalwall

tructures

and

structures

with

lowrubble

mounds.

second

limitofH si/h

s

=

0.3

wasidentifiedasthe

point

beyondwhichomeimpactswererecorded

in

all

tests,

alsoillustratedinFigure

10 .

Figure

9 Variation

in

percentage

of

impactswithH

si

/h

structures

0,1and2)

,

0

0 . 1

^11H-S^

0 .2

. 3

Hsi/hs

Figure 0

Variation

in

percentageofimpactswith

H

si

/h

s

all

structures)

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Inthe

zonebetweenthese

limits,ie

0.2