wave forces on square caissons_mogridge_jamieson

8/13/2019 Wave Forces on Square Caissons_Mogridge_Jamieson

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CHAPTER 1 3 3WAVEFORCESONSQUARECAISSONS

byG.R.Mogridge*andW.W.Jamieson*

ABSTRACT

Theforcesandoverturningmomentsexertedbywavesonlargeverticalsquare-sectioncaissionshavebeenmeasuredinthelaboratory. Eachmodelcaissonextendedfromthebottomofawaveflumethroughthewatersurfaceandwasorientedeitherwithonesideperpendiculartothedirectionofwavepropagationorturnedthroughanangleofforty-fivedegreestothisposition. Foragivenorientation,eachmodelwastestedforarangeofwaveheights(uptothepointofbreaking)forvariouswaveperiodsandwaterdepths. Adigitalcomputerwasusedfortheacquisition,processing,plottingandstorageoftheexperimentaldata.

Inadditiontotheexperimentalwork,anapproxi-matetheoreticalmethodispresentedwhichallowsthewaveloadingsonasquarecaissontobeestimatedbymeansofasimpledeskcalculation. Theexperimentaldatashowsthatthissimplemethodofcalculationisreasonablyaccurateoverawiderangeofwaveconditionsandcaissonsizes.

INTRODUCTION

Inrecentyears,considerableresearchhasbeencarriedouttoestimatethewaveloadsonvariousshapesofmonolithicoffshorestructures;however,verylittleinforma-tionisavailableintheliteratureconcerningwaveloadingsonlargeverticalsquare-sectioncaissonsrestingontheoceanbottomandextendingthroughthewatersurface. HogbenandStanding 3 )describeanumericalmethodwhichcanbeusedforthesolutionofwaveloadsonlargebodiesincludingsquarecaissons. ThesamemethodwaspreviouslyusedbyGarrisonandChow(2),MilgramandHalkyard(7),andnumerousothers. Althoughnumericalmethodscanbeusedforshapes*AssistantResearchOfficers,HydraulicsLaboratory,

NationalResearchCouncilofCanada,Ottawa,Ontario,Canada,K1A0R6.

2 2 7 1


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2272OASTALENGINEERING-1976forwhichclosedformsolutionsdonotexist,programmingistimeconsumingandcomputationcostscanbehigh. HogbenandStand-ingpresentedafewresultsofwaveloadsmeasuredonamodelofasquarecaisson. Theexperimentaldatawascomparedwiththenumericalsolution 4 )andalsoanapproximatesolution

3 )basedonthediffractiontheoryofMacCamyandFuchs(6).Formostoftherangeofrelativecaissonsizestested,theresultsfittedbothsolutionsreasonablywell. Ijima,ChouandYumura 5 )havepresentedanumericalmethodforcalcula^tionofwavescatteringbyisolatedbreakwatersofarbitraryshape. Waveheightdistributionsaboutrectangularbreak-waterswerecalculatedandwerefoundtocomparefavourablywithdistributionsmeasuredinthelaboratory. Althoughwaveforceswerenotmeasuredonthemodelbreakwaters,theywerecalculatedusingtheknownvelocitypotentialsattheboun-dariesofthebreakwaters.Inthispresentationitisassumedthatingeneralterms,thewaveforcesonasquarecaissonorcylinderoccurinasimilarmannertothewaveforcesonacircularcylinder.Thatis,ifthesizeofthesquarecylinderissmallrela-tivetothewavelength,thecylinderdoesnotdeformtheincidentwaveandthewaveforceonthecylinderconsistsofthesumoftheinertialandviscousforces. However,ifthesizeofthecylinderorcaissonislargerelativetotheincidentwavelength,itcausesreflectionanddiffractionoftheincidentwavesandtheviscousdragforcesareneglig-ibleincomparisontotheinertialforces.Thispaperpresentsanapproximatetheorywhichcanbeusedtoestimatethewaveforcesandmomentsonlargevert-icalsquarecaissons. Thetheory,basedonthelineardif-fractiontheoryofMacCamyandFuchs(6),hasbeensimplifiedsothatonlythreegraphsneedbeusedtoobtainthecompletesolution. Forceandmomentmeasurementsonmodelsofsquarecaissonsshowthattheapproximatetheorygivesasatisfac-torysolutionoveralargerangeofcaissonsizesandwaveconditions.

THEORETICALMETHODItisassumedthatthewaveforceonalargesquare-sectioncylinderorcaissoncanbeexpressedasaninertialforceifthecoefficientofmassusedincludestheeffectsofwavereflectionanddiffraction. Thusthehori-zontalforceperunitlengthofthecaissonisexpressedas

fx =Cmspb2du'dt 1)whereC isthecoefficientofmassforthesquare-sectioncaisson,pisthemassdensityofthefluid,bisthesidelengthofthecaissonanddu/dtisthehorizontalcomponent


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FORCES ON CAISSONS 2273ofthewaterparticleacceleration. SubstitutingintoEq.1theexpressionfordu/dtfromlinearwavetheorygives

7 r2pb2H coshk( y+d )ms T2 sinhkd cos(kx-at) 2 )whereHistheincidentwaveheight,Tisthewaveperiod,disthewaterdepth,aisthewavefrequency,a=2 i r / T ,kisthewavenumber, e =2 i r / L ,Listhewavelengthandtisthetime. Thecoordinatesarechosensothatxispositiveinthedirectionofwavepropagationandyispositiveintheupwarddirection. Theoriginofthesecoordinatesisatthestillwaterlevelwherethewatersurfaceelevationplottedagainstxhasamaximumnegativeslope(Fig.4). Byequatingthein-ertialforceperunitlengthonasquarecaissontothatonacircularcaissonofdiameterD

C pb2du/dt=0.25Cp i r D2du/dtmsitisfoundthatthediameterofanequivalentcircularcylinderis

3 )

D =2 b C AC 2e ms' m 4 )IfitisassumedthatC C then,m

D =2 b / i re 5 )

TheforceperunitlengthontheequivalentcircularcylinderisobtainedusingthediffractiontheoryofMacCamyandFuchs(6):2pgHcoshk( y+d)k coshkd A(ka cos(at-a ) 6 )

where

and

A(kae = j j |2( kae +Y|2( kae

a=tanJ^(kae)

gistheaccelerationduetogravity,aeistheradiusoftheequivalentcircularcylinder,andJ^andY^areBesselfunc-tionsofthefirstandsecondkindrespectively,bothofthefirstorder. Eq.6isequatedtoEq.2forx=0togivetheexpressionforthecoefficientofmassC forasquarecaisson:

- i T 5 -A


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2274OASTAL ENGINEERING-1976AmodifiedcoefficientofmassC *sdefinedasms

T C *= yrrrA(ka 8 )ms i


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FORCESONCAISSONS 2 7 5

EXPERIMENTALMETHOD

Thesquarecaisson(Fig,1 )usedintheexperimentswas12in.by12in.incrosssectionandwasconstructedof1/4in.thickplexiglass. Itwassupported1/8in.abovetheflumebottombyarigid3in.diametersteeltubeclampedtoasteelframeabovethewaveflume. Aforcemeterwascon-tainedwithinthecaissonandconsistedoftwo3 /4in.dia-meterstainlesssteelstrainrods12in.long.hewaveforceonthecaissonwastransferredtothetopstrainrodthroughahorizontalsteelbarandtothebottomstrainrodthroughasteelbaseplate(Pig.1). Foilstraingaugesgluedonthestrainrodswerealignedsothattotalhorizon-talforcescouldbemeasuredinthedirectionofwavepropa-gationandnormaltothewavedirectiontogivelongitudinalandtransverseforces,respectively. ThephotographinFig.1showstwobrassbushingsonthestrainrodswhichmakethefixedconnectiontothesupporting3in.diametersteeltube.Thesteelplateconnectsthefreeendofthebottomstrainrodtothecaisson,andthesteelbarwhichpassesthroughaperturesinthesupportingtubeconnectsthefreeendoftheupperstrainrodtothecaisson. Usingthetwobridgeout-putsfromthestrainrods,itwaspossibletomeasuretotalforcesandtocalculatethecorrespondingtotaloverturningmoments. A moredetaileddescriptionofthewaveforcemeteranditscalibrationisgivenbyPratteetal.9). Fromthecalibrationcurvesoftheforcemeter,itserrorbandwasestimatedtobelessthan2 %overtherangeofforcesmea-sured. Thenaturalfrequencyofvibrationofthecaissoninthemaximumdepthofwaterwasapproximately11Hz.

Waveprofilesintheflumeweremeasuredusinganon-contactingcapacitivewavetransducersuspendedabovethewatersurfacemidwaybetweenthecylinderandtheflumewall.Thewaveflumewasapproximately6ft.wide,4 .5ft.deepand2 2 0ft.long.Alimitednumberoftestswereconductedusingasquarecaisson2ft.by2ft.incrosssectiontoobtainforceandmomentdataforlargervaluesoftherelativesizeb/L. Themodelcaissonwassuspendedfromaforcemeterinawaveflume12ft.wide,4 .5ft.deepand16 2ft.long. Theforcemeter(Fig.2 )consistedofthreealuminumstrainmem-bers3in.indiameter. ThestrainsinthesememberscausedbywaveloadsweremeasuredusingsemiconductorstraingaugesformingsixWheatstonebridges. Threebridgesmeasuredstrainduetobendingandgaveoutputsproportionaltolongi-tudinal,transverseandverticalforces. Threebridgesmea-suredstrainduetotorqueandgaveoutputsproportionaltothemomentsaboutthethreecoordinateaxes. Adetaileddes-criptionofthisforcemeterisgivenbyFunke(1).Preliminarytestsinthe12ft,waveflumewerecarriedoutwitha12in.square-sectioncaisson(Fig.2 )toconfirmexperimentsalreadyconductedinthe6ft.flume.


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2276OASTAL ENGINEERING-1976Waveheightsweremeasuredusingacapacitiverodwavegaugeatthelocationofthemodelbutwithoutthemodelintheflume. Thus,adifferentmethodofmeasuringincidentwaveheightswasanadditionalcheckonthe6ft,flumeresults.

Monochromaticwavesweregeneratedinfivewaterdepthsrangingfrom9.7to29.0in.andninewaveperiodsweretestedfrom0.77to2.58sec. Foreachwaterdepthandperiodtested,anumberofwaveheightsweregeneratedrang-ingfromsmallamplitudewavestothosethatwereonthepointofbreaking. Eachmodelcaissonwastestedwithtwofacesparalleltotheincidentwavefronts g=0)andalsoat45tothisposition g=45) . Adigitalcomputerwasusedfortheacquisition,processing,plottingandstorageofdata. Samplingofthedatawascommencedafterthewavespassingthemodelreachedsteadystateconditions. Foreachtest,thewaveprofileandthecorrespondingwaveforcesweresampledeveryonehundredthofasecondforatotalofsixseconds. Totalforcesandoverturningmomentswerecomputedandautomaticallyplottedalongwiththemeasuredwavepro-file. Experimentalresultswerealsoprintedanddatawasstoredonmagnetictape. Althoughithasnotbeenpossibletoincludealltheexperimentaldatainthispaper,there-sultspresentedarerepresentativeofthecompletetestingprogram. AcomprehensivepresentationoftheexperimentalresultsmaybefoundinMogridgeandJamieson(8).

EXPERIMENTALRESULTS

Thenumberofwavesineachtestrecordvarieddependingonthewaveperiodbecauseofthefixedsamplingtimeofsixseconds. Anytestwithavariationinwaveheightormaximumforcemeasurementwithintherecordofmorethan5%wasdiscarded. Fromthetestrecords,totalforcesandoverturningmomentsweretakenasaverageabsolutemaximumvaluesofpositiveandnegativemeasurements,forwhichthevariationwasnormallylessthan5%. Theforcesandcorrespondingmomentsmeasuredinthetransversedirec-tionwerenegligibleandarenotincludedinthepresentationofexperimentalresults.

Usingthemeasuredabsolutemaximumforceswithknownvaluesofwaveheight,length,periodandcaissonsize,C n f ewascalculatedforvaryingvaluesofb/LandthenplottedonthetheoreticalcurveinFig.3 . DimensionlessmomentswerealsocalculatedusingtheknownvaluesofCjfsandwereplottedonthetheoreticalcurveinFig.5 . Theexperimentaldatashowsgoodagreementwiththetheoryforb/Lbetween0.092and0.399andalsoford/Lbetween0.063and0.78 6 .Theresultsshownforb/Landd/Lgreaterthan0.09wereobtainedbyaveragingallthetestresultswithwavesteep-nesseslessthan0.09. Forb/Landd/Llessthan0.09,onlydataforwavesteepnessesoflessthan0.01wereused. Evenwiththisrestrictiononthedataused,thereisstillsome


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FORCESON CAISSONS 277deviationoftheC sdatainFig,3,butthemomentdatainFig.5showsgoodagreement. Itisconcludedthattheapprox-imatetheorycanbeusedforsituationswhereb/Landd/Larebothgreaterthanapproximately0.09.lthoughthereisnodatapresentedinthispaperdefiningthed/Llimit,experi-mentswereconductedford/Lapproximatelyequalto0,09andb/Lgreaterthan0.09forwhichtheexperimentaldataagreedwiththelineartheory. Additionaldataford/Lequalto0.08andb/Lgreaterthan0.09,didnotfitthetheoreticalcurve,thusdefiningd/Lofapproximately0.09asthelimitforthetheory.

Toexaminethevariationoftheforcesandmomentswithwavesteepness,theyhavebeenexpressednon-dimension-allyandplottedagainstH/LinFigs.6to8 . Theexperimen-talresultsforb/Landd/Lgreaterthan0.09(Figs.6and7 )showreasonableagreementwiththetheoreticalcurvesforwavesteepnessupto0.09. Fig.8showsdataforwhichthereisareasonablecomparisonwiththetheoreticalcurvesonlyforverylowwavesteepnessbecauseb/Lislessthan0.09.Forcloseagreementathighwavesteepness,itisnecessaryforbothb/Landd/Ltobegreaterthan0,09. Thelargestdeviationbetweenthetheoryandtheexperimentalresultsisapproximately85%andoccursforthemomentmeasurementforB=45,b/L=0.047andH/L=0.0 3 2. Thelargedifferencesfromthelineartheoryforb/Landd/Llessthan0.09andlargeH/Larebelievedtobeduetonon-linearityofthewavesandtheeffectofviscosityintroducingdragforcesofconsiderablemagnitude. Forb/Landd/Llargerthan0.09,theseeffectsareapparentlynegligiblealthoughatvaluesofd/Lapproaching0.09,thewaveswereobviouslynon-linearandtherewasflowseparationatthecornersofthecaissonunderallwaveconditions.

TherearetwofactorswhichareevidentintheexperimentalresultsfromwhichthedatainFig.8hasbeenobtained. Firstly,theabsolutevaluesofpositiveandnega-tivewaveloadsarenolongerapproximatelyequal. Theydifferfromtheiraveragevaluesbyasmuchas3 0%forthedatainFig.8forwhichb/L=0.04 7and =45. Secondly,thephaseanglesbetweenwaveloadrecordsandwaveprofilesdonotcorrespondtothetheoreticalvaluesforagivenbyEq.6 . Toillustratethis,resultswhichcomparewellwiththetheoryhavebeenplottedinFig.9andresultsforb/Lequalto0.047whichdonotcomparewellwiththetheoryhavebeenplottedinFig.10. InFig,9 ,positiveandnega-tiveforcesareapproximatelyequalandforallwavesteep-nessesthemeasuredphasedifferencebetweentheforcere-cordsandthewaveprofilesareapproximatelyequaltothetheoreticalvalueofa=8.8. AlthoughthetheoreticalphasedifferencefortheresultsinFig.10isa=1.3,themeasuredphasedifferenceincreasestoalmost50withincreasingwavesteepness. Thisischaracteristicofanincreasingdragforce. However,theincreasingphaseangle


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2278OASTAL ENGINEERING-1976canalsobeduetothenon-linearityofthewavesbecausethemaximumaccelerationofthefluidnolongeroccursatthestillwaterlevel.

Althoughmostoftheexperimentalresultsdescribedabovewereobtainedina6ft,wideflume,someexperimentswereconductedina12ft.flumeusingdifferentequipmentandtechniques. Sincetheresultsareintermixed,itisnecessarytoshowhowthemeasurementsinthetwowaveflumescompared. Thecomparisonismadebyrepeatingthesametestsineachwaveflumeusing12in.squarecaissons. ThefirstcomparisonshowninFig.11isforb/L=0.184,d/L=0.4 45andB=0. Thesecondcomparison(Fig.11)isforaseriesoftestswheretherearelargedeviationsfromthetheory,thatis,forb/L=0.047,d/L=0.115and3=0, Inbothcasesthedataobtainedinthetwoflumesagreeclosely,givinganaddeddegreeofconfidencetotheexperimentalresults.

CONCLUSIONS

Anapproximatemethod(Figs,3 ,4and5 )basedonthediffractiontheoryofMacCamyandFuchs 6 )hasbeendevelopedtogivethesolutionforthewaveloadsonasquarecaissonpiercingthewatersurface. Themethodhasbeenfoundtobesatisfactoryforrelativecaissonsizesofb/Lfrom0.09to0.40,forrelativewaterdepthsofd/Lfrom0.09to0.79andwavesteepnessesupto0.09. Forb/Landd/Llessthan0.09,thetheoreticalmethodcannotbeusedexceptforwavesofverylowsteepness,becauseviscousdragforcesandnon-linearityofthewavesbecomeimportant. Theabovelimitsofapplicabilityforthetheoryarethesamewhetherthealignmentofthesquarecaissonisbeamontothewaves,8=0,orturnedthroughanangleof45tothisposition.

ACKNOWLEDGEMENT

ThisprojectwasfundedinpartbytheDepartmentofPublicWorks,Canada.

REFERENCES

Funke,E.R., ASixDegreeofFreedomDynamometerfortheMeasurementofWaveForcesonModelsofOffshoreStruc-tures ,NationalResearchCouncilofCanada,HydraulicsLaboratory,LaboratoryTechnicalReportNo.LTR-HY-54,1976.


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FORCES ONCAISSONS 279

2 .arrison,C.J,andP.Y,Chow, WaveForcesonSubmergedBodies ,Proc,ASCE,Vol.98,No.WW3,August,1972,pp.375-392,3 .ogben,N.andR.G.Standing, WaveLoadsonLargeBodies ,InternationalSymposiumontheDynamicsof

MarineVehiclesandStructuresinWaves,editedbyR.E.D,BishopandW.G.Price,MechanicalEngineeringPublicationsLimited,London,1975,pp.2 58- 2 7 7.4 .ogben,N.andR.G.Standing, ExperienceinComputingWaveLoadsonLargeBodies ,SeventhOffshoreTechnologyConference,Houston,PaperNo.OTC2189,May,1975,pp.413-431.5 .jima,T. ,C.R.ChouandY.Yumura, WaveScatteringbyPermeableandImpermeableBreakwaterofArbitraryShape ,Proc.FourteenthCoastalEngineeringConference,Copen-hagen,Vol.3 ,Chap.110,June,1974,pp.1886-1905.6 .acCamy,R.C.andR.A.Fuchs, WaveForcesonPiles: ADiffractionTheory ,U.S.Army,BeachErosionBoard,TechnicalMemorandumNo.69,December,1954,17pp..7 .ilgram,J.H.andJ.E.Halkyard, WaveForcesonLargeObjectsintheSea ,JournalofShipResearch,Vol.15,No.2 ,June,1971,pp.115-124.8 .ogridge,G.R.andW.W.Jamieson, ADesignMethodfor

theEstimationofWaveLoadsonSquareCaissons ,NationalResearchCouncilofCanada,HydraulicsLabora-tory,LaboratoryTechnicalReportNo.LTR-HY-57,197 6 .9 .ratte,B.D.,E.R.Funke,G.R.MogridgeandW.W.Jamieson,WaveForcesonaModelPile ,FirstCanadianHydraulicsConference,Edmonton,May,1973,pp.523-543.


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2280 COASTAL ENG INEERING 197 6

TPSRNR If*

It J

f t _ _ m .

LU

>


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FORCESON CAISSONS 2281

FIG.2 MODEL CAISSONAND FORCE METER (I2FTWAVE FLUME)


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2282 COASTALENG INE ERING 1976

Fmax y2THEORY 10-0EXPERIMENT1 45

*>< >

^B

I46769b/L

FIG.3 C^sAS FUNCTIONOF b/L

(DEGREES)

+204 -0

0-0-20-30-40-50-60

_\ >y H ys r

COORDINAn kxt)rE SYSTEMJHL I' ' ^m Ts '

O-1-2-3 0-4 0'5 0-6 0-7 0-8 0-9-0b/L

F I G 4 PHASEANGLE AS AFUNCTIONOFb/L


13/19


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2284 COASTALENG INEERING 197 6EXPERIMENTAL 45.MACCAMYAND FUCH5 Fmax.MmaX'AVERAGEOFMAX IMUMFORCESANDMOMENTSNPOSITIVENDNEGATIVEIRECTIONS

=0 168 =0 137

FIG.6 DIMENSIONLESSMAXIMUM FORCES AND OVERTURNINGMOMENTSONA SQUARECAISSON


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FORCES ON CAISSONS 2285 EXPERIMENTAL j8.o-J45 O WACCAMYAN DFUCHSFmax .Mmax 'AVERAGEOFMAX IMUMFORCESAND MOMENTSNPOSITIVEANDNEGATIVEDIRECTIONS

0-7 1 1 1

0-6 70-5 0-4 0-3 0-2 U 9* ~ 0-

1 1 00306C 1-2 1 i 1 1-00-8 _ Mmax 0-6 0-4 0-2 y 1 00306H/L

pit?

1 2

10

o0 8 0 6 - 1 ?

0my O n

0-4

0-2 ~ ~yj

004-0608FIG.7 DIMENSIONLESS MAXIMUM FORCES AND OVERTURNING MOMENTS

ONAQUARECAISSON


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2286 COASTALENG INEERING 197 6EXPERIM ENTAL 0 = 45 0MACCAMYNDFUCHSFmax .Mmax .AVERAGEOFMAX IMUMFORCESAN D MOMENTSNPOSITIVEANDNEGATIVEDIRECTIONS

0-7 ~ I0-60-5 ~ 0 4

0O -

0-3

0-2 o0-t -

002-04H/L Mmax pgb4 t-2 I -0-80-6 0-4 o0-2-- 073 . - O-0 8 0 70 60 5

_ 1O

0 ~ 0

~ ~ a

,0010203H/L 1-2

1 1 1

10 o0-8

0

O

0-6 0 ~

0-4OB

< f l0-2

002H/L

00304 FIG.8 DIMENSIONLESS MAXIMUM FORCES AND OVERTURNINGMOMENTSONAQUARECAISSON


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FORCESON CAISSONS 2287

[IIy 'TIME\ST IMETIMEH/L =0-010/L 0-020 H/L0-032/L0-059 H/L -0075

0-7

Fmax^9b3

0-60-50-40-30 2 0 1

EXPERIMENTAL DMACCAMYANDUCHS

0-03 006 009H/L

FIG.9 PHASING BETWEEN WAVE PROFILESNDORCERECORDS FORNCREASING WAVETEEPNESS=0-128 j =0-246 3 0


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2288 COASTAL ENG INEERING 197 6

TIMETIMEH/L =0004 H/L =0010 H/L =0021 H/L0032

FmaxPQb

1 00 9080 70 60 504

0 30 20

0

Fmo*.MA X I MUM FORCEEXPERIMENTAL( Fmo,NPOSITIVEDIRECTION . A FmgiINNEGATIVEDIRECTION O ~ * MacCAMYAN DFUCHS G

0A

o ^ ^^ A _ o A A

~ _

10-01 002

H/L003 004

FIG.10 PHASING BETWEENWAVE PROFILESNDORCE RECORDS FORNCREASING WAVETEEPNESS

~0047, -|-0- 5 y 8=45


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FORCESON CAISSONS 22896FTWIDEFLUME3 I2FTWIDEFLUME

Fmax.Mmax'AVERAGEOFMAX IMUM FORCESANDMOMENTS INPOSITIVEANDNEGATIVEDIRECTIONS

0-6 0-5 _ . .0-4 _ G0 _

Fmax pQb 0-3

0-20- _

o

B

0(3

O

O

I00408

H /L ~ =0-184 , -y- -445 =0

0-8 ^ ~ 10-7 0

0-6 o-5 a

0-4 O

0-3 ~ o0-2 O l

f i

MmaxpQ bA

1-21 1 1

o1-0

Oa 0-8

G 0-6 0-4

a

E

0-2 < * > % 0020304H/L

0115 f-0

FIG.II COMPARISONOFTESTESULTS USING DIFFERENTEXPERIMENTALMETHODS

wave forces on square caissons_mogridge_jamieson

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