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‘%‘~ Paperst.be$m6ent.datExtremeLoadsResponseSy!npasiutn

: : Arlin@an,VA,October1>20,1981~-.O #$,*,.,.,s+’

Predicting Hull Bending Moments for Design—

Edward V. Lewis, Consultant

Robert B. Zubaly, Webb Institute of Naval Architecture,

Glen Cove, NY and SUNY Maritime College, Bronx, NY

ABSTRACT

This paper surveys current knowledge ofthe various hul1 longitudinal bending loadsacting on seagoing ships, with particularattention to new developments in the last fiveyears. These “demands” on the structure includestill water bending, low and high-frequencywavebending and thermal effects. A probabilisticapprOach is followed wherever possible, and anattempt is made to generalize on the acceptedbasic approach to predicting a 1ong-term cumu-lative distribution of wave-induced bendingoment, showing that many apparently diversemethods are closely related, The problem ofcombining all loads to obtain a completeprobabilistic picture of loads is discussed inrelation to the probability of either ultimatefailure or propagation of fatig”c cracking tothe point where repair is required. Consider-ation is also given to unexpectedly high loadsthat may be encountered under unusual circum-stances at sea.

The paper concludes with a discussion ofthe application of the expanding understandingof demand to the problems of design. It is shownthat the U1timate goal is to be able to matchstructural “capability” to demand by predictingfailure probability, and finally to establishacceptable levels of such probability. Theprincipal gaps in the ideal procedure that mustbe filled before this is possibleare listed,Meanwhile, it is shown that the partial appli-cation of the probabilistic approach has alreadyresulted in more rational longitudinal strengthstandards.

NOMENCLATURE

c anticipated cost of damageF anticipated total cost of failure;1J~ significant wave height

initial cost of shipy nth moment of spectrum

total expected 1ifetime costp)q,r probabi1itiesP,Q,R cumulative probabilitiesR Rayleigh parameter

standard deviation; spectrum shape parameter$B bending moment spectrvros< wave spectrumT period

T. averaae DeriodT: peri02 of encounter

variables;Iy’z response amplitude operatorE spectrum broadness factorP angle, wave component to dominant wave

standard deviation, or rms value: ship-to-wave heading angle0! circular frequencywe encounter frequency

INTRODUCTION

Background

For many years after the introduction ofsteel into ship construction, the design of themain hull girder was based on a nominal standardbending moment in association with an allowablestress that varied with ship length. As shipsbecame larger and faster, and novel seagoingvehicles were developed, a more rational approachwas needed. This led, over the past decade, toa great increase in knowledge of all types ofloads acting on ships’ hulls. At the same time,understanding of the nature of hull girderfailure, as well as sophisticated new techniquesof structural analysis, were developed.

It has become clear that the loads--partic-ularly those caused by waves at sea--are highlyvariable in nature and ca” best be analyzed bystatistics and described in probability terms.Furthermore, the strength of the hull is alsovariable, as a result of variations in scant-lings, steel quality, workmanship, and otherfactors. Hence, the principles of reliabilitytheory, developed in the field of civil engi-neering, are peculiarly adapted to the problemOf ship longitudinal strength. The ideal goalhas become to develop methods of expressing boththe loads, or’’demand~on the hull and itsstructural strength, or’’capability:in terms ofprobabilities, so that the probability of failurecan be calculated. A satisfactory and economicaldesign will then be one in which failure proba-bility has been reduced to an acceptably lowvalue.

Meanwhile, the reliability approach hasalready clarified the longitudinal strengthproblem and has provided a rational basis forextrapolating the design loads for larger a“dlarger ships. Until the ideal, fully proba-bilistic methods become available, partial

31

probabilistic approaches are being usedbyallclassificationsocietiesin evolving and uP-dating their standards of strength for routineuseindesign--particularlyforwavebendingmoments.

ItisthestatedpurposeofthisSymposiumtoconsiderthestate-of-the-artinallaspectsofstructuralresponsetoextremeloadings,includingattentiontothereliabilityapproachtodesign.Hence,itseemsappropriatethatthispapershouldreviewcurrentknowledgeofhullloads,particularlylongitudinalbendingmoments.Itmaybeconsidereda sequeltoa surveypaperon“DynamicLoadingsJuetoWavesandShipMotions,”bythepresentauthors,whichwaspresentedatthejointSSC-SNAMEShipStructureSymposiumheldfiveyearsago[1].

SincethelastSymposium,thebasicprinciplesdiscussedintheabovepaperhavenotchangedandthereforeneedonlybriefsummaryatthistime.Principalattentioninthepresentpaperwillbegiventooutstandingnewdevelopmentssince1975.FormoredetailedsurveysreferenceismadetoreportsofCormnittees1.2and1.3totheISSCof1976and1979[2][3][4][5].Butsincethe1975paperdidnotdealwithspecificproceduresforcalculatingextremewave-inducedbendingmoments,anattemptwillalsobemadeheretogeneralizethebasicprinciplesunderlyingvariousavailableproceduresthatmayappearmoredifferentthantheyreallyarebecauseofdifferencesinnotation,emphasisandappli-cation.Attentionwillbedirectedtogapsinourunderstandinganduncertaintiesthatatthepresenttimestillpreventfulladoptionofthereliabilityapproachtothedesignofshipsandothermarinestructures.Theproblemofcombiningloadsina formsuitablefordesignandthepossibilityofunusuallyhighbendingmomentsunderabnormalconditions,suchasshoalingwater,opposingcurrents,etc.,willalsobediscussed,

ModesofFailure

Beforeattemptingtoconsidermethodsofevaluatingallthelongitudinalbendingloadsandthencombiningthem,itiswel~todiscussthedifferentmodesofhullfailurethatthedesignermustguardagainst.Thisshouldclarifytheformofloadinformationthatisneededandwhichloadsarecritical.Thefollowingstatemen~largelyfromthe}975paper,stillstands.

Caldwell[6]considersultimatefailureasthecompletecollapsebybucklingofthecompressionflangeandsimultaneoustensilefailureofthetensionflange.However,itisclearthata considerablylessseveredamagewouldbea seriousmatter,asindicatedbysuchfactorsasnecessityformajorrepairs,interferencewithnormalshipoperationandnon-watertightness.Hence,forourpurposewemaydefinedamageasa structuraloccurrencethatinterfereswiththeoperationoftheshiptotheextentthatwithdrawalfromserviceforrepairisrequired,suchas:

- Localhul,ldeflection:bucklingand/orpermanentset.

- Fatiguecracking.- Brittlefracture,minor.

Failureisa severedamagethatendangersthesafetyoftheship:- Collapseofthehullgirder,bybuckling

and/orpermanentset.- Extensivebrittlefracture.

Consideringthevarioustypesofdamage(orfailure)inmoredetail,excessivehulldeflec-tionisa rareoccurrence,exceptlocally,andcompletefailureorcollapseisevenrarer.Thissuggeststhatconventionalstandardsoflongitudinalstrengtharegenerallyadequate--infact,theymaybeexcessive.Loadsthatcancombinetothreatenhullfailurearestillwaterbendingmoments,wave-inducedbendingmoments(quasi-static),vibratory(highfrequency)loads,andthermaleffects[7].

Secondisthepossibilityoffatiguecracking,whichseldomconstitutesfailurebutisimportantfortworeasons:Fatiguecracks,whicharefairlyfrequent,cangrowtothepointthattheymustberepaired,andfatiguecracksarenotchesthatundercertaincircumstancescantriggerrapidpropagationasbrittlefracture.Nibberingnotes[8],“Itisa favorablecircumstancethatfatiguecrackspropagateveryslowlyinship’sstructures.” Cyclicloadstobeconsideredincludethesameloadsasmentionedabove,withwidelyvaryingperiodicitiesandmeanvalues.

Brittlefracture,whichwasa seriousproblemwithearlyweldedshipsduringWorldWar11,waslongagobroughtundercontrolbyinsuringsatisfactory“notch-toughness”ofshipbuildingsteel,aswellasbyeliminatingseveredesignstressconcentrationsandbyimprovingtechniquesforweldingandinspection.However,brittlefracturecananddoesoccur,andthereforethephilosophyhasbeenoneof“fail-safe”design.Crackarresters,consistingofrivettedseamsorstrakesofsteelhavinglowertransitiontemperatureareprovidedasstandardpractice.Thesehaveproveneffectiveinlimitingcrackpropagationandtherebyrestrictingbrittlefracturetoaminordamageratherthana hullfailureproblem.

Itcanbearguedthen,thatsincefatiguecrackingdoesnotnormallythreatenthelifeoftheshipandbrittlefracturecanbecontrolled,theprimarycriterionofrationalshipstructuraldesignshouldbeoneofultimatestrength--reducingtheprobabilityofexcessivedeflectionthroughbucklingorplasticflowtoanaccept-ablelevel,asdefinedsubsequently.Thesecondarycriterionofrationaldesignisfatiguestrength--reducingthenumberofcasesofcracksthatpropagatetoa sizerequiringrepairtoa costeffectivelevel,alsodiscussedsubsequently.Accordingly,theextremeandcyclic~oadsthataffectbothultimateandfatiguestrengthwillbegivenparticularattentionhere.!4smentionedabove,theseconsistof:

32

.

wil’

Stillwaterbendingmoments.Wave-inducedbendingmoments(quasi-static).Vibratorybendingmoments.Thermaleffects,

Inthenextsectionseparateconsiderationbeaiventoeachofthesebendinamoment

componen&,andhowtheycanbepredi;ted.

BENDINGMOMENTCOMPONENTS

StillWaterLoads

Thelongitudinalbendingmomentsexperiencedbya shipinportcanvarywidely,particularlywiththeamountanddistributionofcargo.Hence,theyrepresentanimportantload,bothasanadditiontowavebendingmomentsthatcouldcauseultimatefailureandasameanvalueaboutwhichthecyclicloadingoscillates.Althoughstillwaterbendingmomentsareeasilycalculatedwhenthedistributionclfcargoandotherweightsisknown,ithasbeenfoundthatrecordsareseldomkeptand,therefore,itisdifficulttoassemblestatisticaldataonactualvaluesinservice[7][9].

However,itappearstentativelythatstillwaterbendingmomentscanbedescribedbytwo(ormore)normalprobabilitydensityfunctions,onerepresentingoutboundandtheotherinboundconditions[7].SeeFig.1. ThevalidityofthisapproachhasbeenconfirmedbyrecentworkofIvanov[10]andSeeding[11].Typicalconditions,suchasfullload,ballast,lightload,etc.arefirstestablished.Cal-culationsarethenmade--whichcanbeverifiedbyservicerecords--ofboththeaveragevalueofbendingmomentandthestandarddeviationforeachbasiccondition,assumingnormaldistributionfunctions.Thisapproachisdesirablebecauseshipsareneverloadedexactlyinaccordancewiththedesigner’sloadingmanuals,anditisconsistentwithanoverallprobabilisticapproachtodesign.However,ManiarpointsoutindiscussionofCommittee1.3report,ISSC1979[5]thatoperatorswillavoidextremeconditions,andthereforethesedistributionfunctionsmaybetruncated,asshowninFig.1. Seealso[9].

Thebendingmomentcreatedbya ship’sownwavewhenmovingathighspeed(aboveFroudeNo.of0,20)canbedeterminedbymodeltest,orfrompublishedmodeltestresults[12].

Concernisoftenexpressedregardingresidualstressesinsteelplatesandshapesandstresses“locked-in”bytheweldingprocess.Itispossiblethat,wheresuchstressesexistincombinationwithotherstressesatawelddefectornotch,theymightundercertainconditionscontributetotheinceptionofa brittlefracture.However,asnotedin[7],“Forothertypesoffailureitseemsreasonabletoconsiderthemtobeofminorsignificancetolongitudinalstrength,sincetheytendtobeeliminatedby‘shakedown’oradjustmentinservice.Thatis,anoccasionalhighlongitudinalwavebendingload--incombinationwithotherloads--maycauselocalyieldingintihehighresidualstressregion.

Uponterminationofthishighwaveload,thestructurewilltendtoreturntoa conditionofreducedresidualstress.”

nn590uT00uND voYAGe5 p;

( I I, L Io 20 40 60 60 1005T3LL WATERB.M.(H06GIN6), 1000 FT-TONS

Fig.1 HistogramofStillWaterBendingMomentsforContainershipNewOrleans[7]

QuasiStaticWaveBending

Whena shipencountersirregularwavesorswellswithwavecomponentsintheranaeof0.5to2.0timesthe’lengthoftheshi~(shorteratobliqueheadings),significantbendingmomentsaredeveloped,actingatthecomparativelylowfrequencyofwaveencounter.Thesewave-inducedbendingmomentswerefirstdeterminedbymodeltestsinwaves[12][13].Subsequently,thedevelopmentofthestriptheoryapproachtothecalculationofshipmotionsbyKorvin-Kroukovsky[14]ledtomethodsforcalculatingstressandbendingmomentinregularwaves[15][16][17].

Theextensionofregularwaveresultstopredicting’variousshipresponsestoshort-crestedirregularseaswasaccomplishedbySt.DenisandPierson[18],ontheassumptionthatboththeirregularwavesandtheshipshort-termresponsesarestationarystochasticprocesses,wherestochasticmeansrandombutfollowingdefinitestatisticallaws.Byshort-termismeantperiodsoftimeoftypicallyafewhoursduringwhichseaconditionsremainessentiallyconstant,statisticallyspeaking,sothatthecharacterofboththeseaandtheship’sresponsecanbeassumedtobe“stationary”.Hence,undertheseassumptions,thebendingmomentresponsecanbepredictedforanyshipforwhichresponseamplitudeoperators(RAOS)areavailableinanyknownorassumedseaspectrum.Fig.2 showshowthedirectionalwavespectraaremultipliedbytheappropriateRAOStoproducethedirectionalresponsespectra,

SB((JJ,LI)=SIC(W,ll)IY(U,d12, (1)

33

where sB is the bending n!nmentSPeCtrUm: S< ISthe wave spectrum, Y2 is the RAO, u is c1rcularfrequency and v is the angular direction of awave component relative to the dominantdirection. When these components are integratedover wave direction a single response spectrumis obtained, whose area and shape define thebending moment response,

SB(0)= /sBhw)du. (2)

~(~), FT’-5Ec

100IIF POINT sPEcT$?uM(INTE13RATEP)

5:E-.0 .5 /.’0 1:5

6), l/5EC

/Y@,pf@T-TOWFO=

LJ!i$L

3xk?W“ UEADIN6

2%109 150”

1200IK109

00 Lo L

-!&(1+) , (FT-TONS\z

klow18

MTE6RATE ~ 180”tfD&

COMPONENTS :-180 *

50$d 450°

,120°

00 .5 1.0 1:5

Fig. 2 Short-Crested Sea and Typical BendingMoment Response

34

Short-term statistics, applicable to periodsof time from i hour to several hours (while seaconditions remain relatively unchanged), can bederived from the response spectrum by taking thevarious moments of SB(W),

(3)

First, the area under the spectrum, ~, definesoz. the variance of the short-term process, i.e.the mean-square value of the deviations fromthe mean at equal (or random) intervals, anda is the root-mean-squarevalue. Thus,

2a = m.

a .@

Furthermore, it is wel1 known that if theresponse spectrum is narrow-band, the short-tenmpeak-to-mean amplitudes are Rayleigh distributed,

p(x) . # ~-x2/R

In this case the parameter of the Rayleighdistribution, R, is given by,

R = 2mo = 202

(4)

Various statistical properties of the peak-to-mean amplitudes can then be determined. Forexample,

Significant amplitude(average of highest 1/3) = 2.0 ~

Average “apparent” period, basedon zero crossings = 2,-

For a broad spectrum, of course, theRayleigh distribution is not applicable, anda more generalized distribution is required,involving the spectrum broadness parameter,

I m? I*

“b-w(5)

This generalized distribution function .isacomplicated expression [9] which reduces tothe Rayle<gh form when c = O.

An important study of the applicabi1ity ofthe Rayleigh distribution was made by Oalzell,Maniar and Iisu[93 on midship stress (bendingmoment) records. They found that for an ocean-going dry bulk carrier (Fotini L, CB = 0.B4),“the assumption that the~erm wave-inducedmoment fits the Rayleigh distribution would bereasonable, if not always true. The magnitudesof compression and tension are symnetri~al.”For a lar e, high-speed container ship (SL-7,

?CB = 0.53 the bending moment maxima “at slowspeed fit the Rayleigh distribution reasonablywell.” But at normal operating speeds, “arelatively small portion of the data fits theRayleigh distribution.” A much better fit wasobtained with the generalized Rayleigh distribu-tion, with broadness parameter,s ‘approachingunity for over half the data” (i.e. approximating

——

a normal distribution), particularly in quarterinand following seas. “The ratio of compress?onto tension of wave-induced bending momentmaxima is approximately 1:1.2 for extremes and1:1.06 for rms.” It is concluded that “if itis assumed for the simplification of analysesthat the short-term wave-induced bending momentmaxima are always Rayleigh distributed, theresulting long-term prediction is 1ikely to beconservative to a degree which is not possibleto judge based on the work performed.” Thismeans that we may accept the Rayleigh assump-tion at this time, considering the manyuncertainties that still remain, but in thelong-run further refinement wil1 be needed,particularly for high-speed ships.

Combined Vertical and Lateral Nave Bending

A ship sailing obliquely into a train ofwaves will be subject to unsymmetrical bendingabout a neutral axis that continually shifts itsangular position. This can be accounted for byconsidering that there is a lateral (horizontal)as well as a vertical component of longitudinalbending moment. (Note that the directions arenot really horizontal and vertical when theship rolls.) RAOS for lateral bending momentcan be determined either by experiment [19]or by theoretical calculation [17]. Lateralbending moment is mainly a dynamic effect.since the only gravity force~ are proportionalto the sine of the roll angle. If one isconcerned only with extreme stresses--as occurat the deck-sheer strake intersection and atthe bilge--an effective vertical bending momentcan be determined [7] [20]. This effectivebending moment is that simple vertical momentthat would produce the same stress all acrossthe deck as the maximum deck edge stressresulting from combined vertical and horizontalbending. It depends on the ratio of the sectionmoduli for vertical and lateral bending.

However, for ultimate strength we are notinterested i“ the stress at the deck edge(or bilge), since this structure is not likelyto fail under compressive loading. We aremore concerned with the behavior of deck (orbottom) plating panels, and the above effectivebending moment is unrealistic because it assumesthat the deck edge stress continues uniformlyacross the deck. The panel loads will begreater or less than those calculated on theassumption of vertical bending alone, and, ofcourse, the loads will have a non-uniformdistribution across the section. It isdifficult to deal with this problem withoutconsidering the local longitudinal stresses atcritical points in the section, and thjs canbe done for a specific midship section bycalculating RAOS for stresses at the criticalpoints, using the RAOS for vertical and lateralbending and their phase angles. Then short andlong-term probabilities can be calculated inthe same manner as for vertical bending moments,as discussed in subsequent sections. Anotherapproach described by Oalzell [21] is to makeuse of so-called cross spectra. tlegives anexample of combining the effects of verticalbending moment, horizontal bending moment,torsional moment, shearing forces and axialforce by this method. See also [20] and [22].

,9 The above procedures, however, can onlyDrovide orobabilitv information on stresses atieparate~ specific-points, not the simultaneousstresses at different points. It is clear,therefore, that this problem of providingdetailed combined bending moment information inD?Obabilitv format for use of the structuraldesigner r;quires further research.

Meanwhile, for the present we have twopractical alternatives: 1) to confine themajor calculation efforts to vertical bending,with an empirical allowance for the effects ofcombined vertical and lateral bending; 2) tosubstitute a single extreme regular wave (atdifferent headings) for the irregular seaway,which can provide an approximate deterministicsolution.

Torsional moments are also of interest,nninly in respect to their effect on localstresses, such as at hatch corners. RAOS canbe calculated or obtained experimentally and,hence, probabilities of stresses at specificpoints can be calculated. It is not expectedthat torsional loads will have a significanteffect on ultimate strength of the hull girder,however.

Oynamic Vibratory Bendin$

The vibratory loads excited by waves atsea are of relatively high frequency, corres-ponding to the natural frequencies of hullvibration. As indicated in our 1975 paper [1],these loads are either transient or cyclic innature. The former category is generallydescribed by the terms slamming and whipping,where slamming refers to the initial effect ofa wave-ship impact and whipping to the conse-quent vertical hull vibration in one or moremodes. Nave-excited cyclic responses aregenerally referred to as springing. Both thetransient and cyclic hull responses can, inprinciple, be handled by the theory of vibra-tion of a free-free beam. Since fairly com-plete discussions of the nature of thesevibratory loads and problems of dealing withthem in design are given in [1] and [7], onlysome highlights will be considered here.Examples of actual stress records showing bothlow and hiah-freouencv effects (as shown inFig. 3), with ex&si~e anaiysei for a drybulk carrier and a high-speed container ship,are given in [9]. Mention should also be madeof the work of Professors Bishop and Price[23] [24] in developing an integrated theory ofship dynamic response, covering both low andhigh frequencies, that clarifies short-termbehavior. Since the high-frequency effectsare not always present, it is desirable hereto consider them separately from low-frequencyresponses for long-term predictions. SeeFig. 3.

Sm-inaina can be treated as a stochasticprocess ov~r ihe short-term, and methods areavailable for calculating the RAOS, assumfngthat only the fundamental [2-noded) verticalmode need be considered. A method of calcu-lation was first presented and applied byGoodman [25] and later extended--particularlyfor Great Lakes bulk carriers--by Hoffman andvan Hooff [26] and Stiansen, et al [27]. !--

35

_ 17,9W,s!

I,)

Ill1+

h)%“..,.dvc.d5,”s,‘$.,,.,,.,,(F,.,,...,APO,,..0.1“z,

,)%,,.M.’(s,,,,,,.,)S,,,,,VM,,,ons[Fr,qwqA,w.x,8.70“z)

Fig. 3 Typical Record of Midship StressVariation, M.v. Fotini L, Showing Filtered Lowand High-Frequency Stresses [7]

Having the RAOS, response spectra can becalculated in the normal manner, and hence along-term cumulative distribution can bedetermined. One difficulty is that wave spectraldata may not be sufficiently well defined i“ therange of encounter frequencies corresponding tothe natural frequency of hull vibration. Anotheris the abundant evidence [28] [29] that springingis also excited by longer waves--at encounterfrequencies that are whole fractions (sub-har-monics) of the natural frequency--as well as byshort waves at resonant frequencies. Pendingthe development and confirmation of a suitablenon-linear theory to explain these effects, itmay be possible to make an approximate allowancefor them.

Most difficult of all to deal with is thetransient phenomenon of slamming--eitherasso-ciated with bottom impact or flare immersion.A great deal of excellent work has been done onbottom impact slarmningby Ochi [30] [31] andothers, which shows that it is possible inprinciple to predict the frequency of occurrenceof slams under specified conditions of sea state,ship drafts, speed and heading. Similarly,Kaplan [32] and others have dealt with flareentry slarmning. Following the slam, vibratorywhipping usually occurs which decays in accord-ance with the damping characteristics of theship. A study by van Hooff in [7] confirms thegeneral opinion that slams do not occur at ran-dom but within a fairly narrow range of phaseangles to the wave excitation. Typically, thewhiDPing that adds to the first wave-inducedhogging-moment peak after the slam will producea higher total bending moment than the slamitsel?. See Fig. 4.

Full-scale strain recording on board S.S.Wolverine State has captured hull girder slamand whipping stress variations experienced duringrough sea operations of the vessel. Evaluationo? data 1333 obtained in various sea states hasshown that the severity of slam response--whenit occurs--tends to be somewhat insensitive tosea condition. Fig. 5 shows histograms ofmidship slam stress variation for the Wolverine-when exposed to different Beaufort Nos.The similarity of the high-stress portions ofthese histograms can be explained by noting thatthough slamming frequency and severity areincreasing functions of sea state, and vary withship speed and heading, the latter two areunder the control of the ship master who willtend to alter course and speed so as to keepthe frequency and severity of such slam loadingswithin reasonable bounds, according to hisexperience.

Presumably, continued research and studymay eventually permit us to predict accuratelythe frequency of slamming and the resultingadditional bending moments expected underdifferent conditions. However, the factor thatmakes this goal not only elusive but also in asense unnecessary of attainment is the abovepractice of the master to alter course or speedto prevent slatnningimpacts exceeding a certainlevel of subjective severity. These changes cmbe quantified only by obtaining statistical dataon attainable speed o“ different types of shipat different headings to a wide range of seaconditions. However, what is really needed issimply full-scale data on the additional bendingmoment resulting from the most severe slammingand whipping that are _ to happen. Thesecan be obtained by taking midship stress recordsand translating them into bending moments, aswas done for the Wolverine State in [7]. Ofcourse, the severity of slamming that is permittedwill vary among individual captains and with shiptype, hull form, etc. Hence, data should becollected from several ships of various types,but not necessarily over long periods of time.Records that include several storms in whichslatmningoccurs should suffice.

The problem may be different for navalships that may be called upon to maintain highspeed into head seas in spite of local or hullgirder damage. Here the problem may be topredict the highest impact loading to be expectedat a certain maximum speed in a particularlimiting sea state, in order that the structurecan be designed accordingly.

Some data on maximum whipping stress inrelation to quasi-static bending stress werepresented in the report of Corwnittee3 to theISSC 1973 [34] and quoted in [1]. Full-scaledata on four different ships, varying in lengthfrom 130 - 230 m. (426 - 754 ft.) are given byAertssen [35J.

Thermal Effects

Records of midship stress obtained onfive bulk carriers [36] [9] indicated surprisinglyhigh thermal effects. These showed a consistentdiurnal variation, with magnitudes of 3-5 kpsiin some cases. The temperature gradients that

1“

36

Fig. 4 Definition of Stresses and Phase Angles Involved in Sla!mning[7]

Fig. 5 Peak-to-peak Slam Stress Distributions in Different Weather Conditions,S.S. Wolverine State [33]

produce such thermal stresses may not be, strictlyspeaking, loads but they are considered as loadshere because they have similar effects.

Although it often happened that high thermalstresses occurred at times of low wave bendingstresses (sunny weather), and vice versa (stormyand cloudy weather), this was not always thecase [36]. The exceptions are presumably timeswhen a heavy swel1 was running while the weatherwas clear.

It should be noted that the thermal stresschanges recorded here were overal1 averages,Since they were based on combined p~rtand star.board readings. Because of the effect of 1ocalshading, it can be expected that even largerthermal stresses would be experienced. HOWWP,

it can be assumed that such local high thermalstresses can be ignored when considering ultimatestrength.

In order to include overall thermal effectsin design calculations, two distinct steps arerequired: estimating the magnitude of the effectunder different conditions of sun exposure (Fig.6) and estimating the frequency of occurrence ofthese different conditions in se?vice. A pro-cedure for dealing with this problem was pre-sented by Lewis, et al [7] and sunrnarizedinthe 1975 paper [1]. Little further work hasbeen done on the subject.

I

L.. _37

————— —___ ~

T“cnhlu..- I

LONG-TERM PROBABILITIES

Evidence From Full-Scale Tests

7[’’’”””

L-l-J-4I

Fig. 6 Calculated Thermal Stresses,Wolverine State [7]

Local Loads

S.s.

Finally, consideration must be given tolocal loads, caused by cargo and internal or ex-ternal fluid pressures, that would be appliedto main hul1 girder components such as thebottom shel1. Insofar as the primary loadcriterion is concerned, the principal local loadto be considered is that of hydrostatic pressureon the double bottom. AS discussed by Evans [37],this secondary loading results in a bendingmoment at the middle of each hold and a largerone in way of each bulkhead. The former causessignificant compression in the bottom platingand tensile stress in the inner bottom, both ofwhich would be superimposed on the longitudinalbending stresses. These local stresses arehigher in the vicinity of longitudinal girdersin transversely framed bottoms, but would bemore uniform across the ship in the case of themore common longitudinal double bottoms. Sincethe bottom pressure is higher when wave crest isamidships than when wave tt.oughis amidships,this effect is greatev i“ hogging than in sag.glng and would increase the compressive stresses.Pressures can be approximated on the basis ofstatic head for the present purpose, althoughmethods have been developed for taking intoaccount the dynamic effects of ship motions.See Hoffman [3B],

Although local stresses can be added tolongitudinal bending stresses, the true effectof combined local and longitudinal bending onultimate strength is a complex matter, to bediscussed in a later secticm.

A sound approach to the application ofprobability methods to predicting extreme hullloads--particularly those due to waves--requiresthe study of statistical data on actual loadsmeasured in service. Fortunately, considerableful1-scale stress data have been collected inrecent years under projects sponsored by SSC,ABS and others [36] [39]. A sunrnaryof shipsinvolved in such tests is given in Appendix 2.

Various analyses and studies of the abovefull-scale results [40] [41] have revealed anumber of basic facts:

- During a short-term observation (i.e.periods of time in which statistical measures ofresponse do not vary significantly) the responsescan be considered to be stochastic precesse$,treated statistically in a similar manner asocean waves.

- Consequently, short-term responses canbe described approximately by Rayleigh probabilitydensity functions having the form shown in Fig.7. (Such functions are handy because they arecompletely described by a single parameter, aspreviously explained.)

P(x)(KPS1)-l

.4

k.3 I

I

/’.2

1’,1

/

/00il-\\\,

2

1

hb-,i

\\

\\

\

34s6STRESS,X,KPSI

Fig. 7 Typical Fit of R.ayleighOensity Functjo”to Wave StressOata [41]

- Bending moment response varies with thefollowing factors, in order of decreasingimportance: significant wave height, shipheading to waves, wave spectral shape or appa-rent period, ship loading, ship speed.

- Hence, statistical analysis is facilitatedby separating data in accordance with thesefaCtOrS, particularly significant wave height(or other measure of sea severity, such as theindirect measure of Beaufort No. or wind speed).

- The statistical long-term distributionof bending moments (derived from stresses bymeans of static “calibrations” in port) can be

36

,.0

,,.5

,.”

,.. ”.0,, w,. ”,=.,,

Fig. 8 RMS Stresses fpom Short-Term Records plotted vs. Beaufort No.,S.S. Wolverine State in North Atlantic Service [41].

interpreted as a summation of many short-termdistributions.

The accompanying Fig. 8 presents samplesof ful1-scale results obtained by TeledyneMateriak Research for the S.S. Wolverine State[41]. The plot shows Rayleigh parameters (rms-values) divided into different Beaufort Nos.(wind speeds), each point representing one 20-minute stress record. The nns data show roughlynormal distributions of the parameters withineach Beaufort No. (except above Beaufort 8, wheredata are relatively scarce). The scatter is d“emainly to the factors previously mentioned--variation in spectral shape, ship heading, shiploading, and ship speed. In addition, thevariability in wave height vs. 8eaufort No. isan important factor here (See Compton [42]).

Fig. 9 shows the result of a similar analysisof available records for the SL-7 containership,divided into 8eaufort No. groups. Visually thefit of the Rayleigh parameters to a normal dis-tribution seems reasonably good. (These resultsare from an unpublished Webb Institute reportto A8S)

On the basis of the preceding discussion ofobserved loads, we can now proceed to considersuitable ways to describe the wave loads fordesign. It has been shown that damage can occureither on the basis of a single very large bendingmoment causing failure or as a result of cumu-lative damage resulting from many CyCleS ofrepeated loads. To predict the first tYpe ofultimate failure, we m“ make “se of either ~cumulative distribution of all cycles of load,as discussed above, or we can focus attentionon only the highest loads by mea”s of ~xtyemevalue theory or order statistics [Ochi, 43].FOP the second type of damage, fatigue, only

. WEATHER &?D”P E912 REC0ZD5

o- 5 . c, KP51 15

N

Fig. 9 Comparison of Stress Histograms WithNormal Distributions, SL-7 Container Ship.

39

a complete picture of all load cycles willsuffice, as provided by a long-term cumulativedistribution.

Statements to the effect that we & useorder statistics to find the highest expectedload are not correct, since a cumulative dis-tribution of loads will tell us the value thatis expected to be exceeded once in a ship’slifetime--or the lifetimes of 100 ships. Thisis as good a measure of extreme load as theexpected highest value. As a matter of fact,the two values should be approximately equalThe former method has been in use for compat’a-tive sutdies for many years by all classificationsocieties and many researchers. It can, inprinciple, provide complete loading informationfor both ultimate strength and fatigue.

The extreme value approach has been applied,particular y in recent years, to offshorestructures when fatigue is not under considera-tion. It has been developed into a rigorousdesign method by Ochi, who has found that itis a simple and useful method when only theultimate or most severe 1ifetime load isrequired [43] [44]. He has shown that whenproperly carried out, the two approaches leadto comparable results [43].

This paper wil1 deal mainly with the long-term cumulative distribution method, whichinvolves first the problem of short-termprobabilities previously discussed.

Cumulative Probabilities

The long-term cumulative approach wasfirst developed by Bennet [45], Band [46],Lewis [40], and Nordenstrdm [47] as a meansof analyzing ful1-scale stress data obtainedover periods of one to three years andextrapolating to longer periods, correspondingto the lifetime of a ship--or of many ships.The approach was then applied to calculatingpredicted long-term probabilities of exceed-ance for bending moment (or stress), fordesign use.

Different writers have presented manyvariations of the basic long-term predictionprocedure, including Compton [42], Lewis,et al [7], Nordenstr6m [47], Fukuda [48],Seeding [49], Goodman [50], Oalzell, et al [9]and Ochi [43]. They are all based on the ideaof predicting short-term probabilities andthen combining them on the basis of assumed1ifetime service profiles to obtain long-termprobabilities. Some variations in thevarious methods:

- Choice of wave spectra.

- Sequence of dealing with various factors.

- Whether or not component and finaldistributions are fitted to specific mathematicalformulations.

The first step in all methods is theselection of suitable sea spectra covering awide range of both severity (i.e. significantheight or spectral area) and spectral shape.

‘la

The simplest and most common method of doing thisis to adopt a mathematical spectral formulation--such as the 2-parameter ISSC formula [51]--whichfor each significant height (spectral area)allows for a wide variation in location of thespectrum peak. See Fig. 10. This approachleaves something to be desired in the investiga-tion of wave loads, where the extreme ratherthan average conditions are most important,because actual sea spectra show far more varietyin shape than this idealized family. A methodthat allows for shape variation is to use arandom sample of wave spectra, grouped by theirareas, so that a variety of realistic shapesare utilized [40]. Fig. 11. A family of 8 to10 spectra having the same significant heightwill provide a means of determining a mean andstandard deviation of rms response (for fixedshiD SDeed and headinu). This aDDYOaCh toselectin spectra has-been made more explicit

?by Ochl 52], who developed a formulation ofdouble-peaked spectra, established descriptiveparameters, and then used statistical wavedata to arrive at suitable families or groupsof spectra to use in the calculations. Thismethod may be more complicated than resultsjustify. In any case, five or six differentsignificant heights are needed. Ochi [43]has also developed an interesting weightedsample technique for selecting spectra.

The next step is to obtain RAOS for bendingmoment by either model test or calculation.Model experiments in regular waves involve anextensive program to cover al1 wave lengthsand headings, Plus several speeds and at leasttwo conditions of loading. Hence, theoreticalcomputer calculations are preferable for con-ventional ship forms. However, the need forcontinuing verification of theory by modeltesting, especially for new or unconventionalforms, must not be overlooked. A number ofcomputer programs are available for calculatingthe RAOS at all headings, typically in incrementsof 30 or 45” [53] [17].

Having the RAOS, the bending moment responsespectra can be calculated by superposition forall of the selected wave spectra. Sometimes thespectra are assumed to represent long-crestedseas for simplicity. But this results in

L!!0,

-i

0,

I;.=,~,‘“ t“

4,, !., ,,, ,,.

Fig. 10 ISSC Ideal Wave Spectrum Family

_—.

*?5K

,.+Ez–.T._.-...,..:-----.T .-1 I

—-—... .. .. . . . . . ..— ~~

0,.,

H,,,C, ..-.

ONE PAQAMETE,05P.CIZ.F.IIQ,.ridi

.c.

5

e0 02 o.~ 0:6 O.’a 1.0 ,,2 !.’4 1:6 ,:.5 7.$

UFig. 11 Example of 10 Sample Wave Spectra Having

a Significant Height of 30 ft. [95]

consistent over-estimates, and therefore it ispreferable to introduce short-crestedness (theresult of differences in direction of componentwaves in a storm sea) by means of a suitable“spreading function”, as discussed subsequent y.

Almost all methods in general use for pre-dicting a long-term cumulative distribution ofbending moment amplitudes begin with theassumption that a Rayleigh distribution is anadequate statistical description of the short-term response--either the single (peak-to-mean)or double (peak-to-trough)amplitudes of bendingmoment. For greater accuracy, the previouslymentioned generalized distribution involvingthe band-width parameter, . , could be used butwould add greatly to the complexity of thecalculations. Until all other aspects ofdesign calculations are brought to a higherlevel of precision, this refinement is con-sidered unnecessary.

The Rayleigh probability density functionfor single amplitudes, x, as derived from (4)is,

p(x) = x/m. e-xL12mo (6)

and integrating, .

P(xi) = prob (x ~xi) =I

p(X) dx =

xi

~-x~/2mo . ~-x~/2 02 (7)

where .2 is the mean-square value (variance) ofdeviations from the mean and

20 = m.

where m. is the area under the response spectrum.

In order to predict the long-term distribu-tion, we must combine many conditional distrib”-tions--assumed to be Rayleigh--~epresentativeof all the situations expected to be encounteredin a ship’s lifetime. These different situationscan be described by a number of parameters, firstthose that refer to wave conditions:

Significant height,Spectral shape.Directional chavacterjstics,of

- one wave system, or- combinations of two or more storm seas

or a sea and swell.

In order to simplify the problem, it is customaryto neglect wave systems coming from more than onedirection and to assume one particular form ofspreading function to describe the short-crested-ness of a single storm sea,

f(u) = : COS2 M (8)

where u is the angle between an angular wavecomponent and the dominant wave direction. Ifthe one-dimensional or point spectrum is S< (.),the directional spectrum is assumed to be,Sc( w, u) = St (.) f(u).

Variations in significant height aredefined by variations in the area under thespectrum. Variations in spectrum shape can bedefined either by use of families of randomlyselected measured spectra or by using a fornlu-lation that specifies families on the basis ofcharacteristic periods, as previously explained.

The following parameters refer to shipconditions:

Loading (drafts).Speed.Heading to the seaway,

The most important of these variables toconsider is the ship heading relative to domi-nant ~ave directjon. Ship speed, which has arelatively small effect on wave bending moment,can be eliminated as a variable by assumingeither the design sea speed or the highestpracticable speed for the particular seacondition and ship heading under consideration.The effect of amounts and distribution ofcargo and other weights, which in turn affectdraft and trim, transverse stability, longitu-dinal radius of gyration, etc., can be a,-~~pli~~t~dp~rhlm.Usually, however, it canbe simplified by assuming two representativeconditions of loading, such as normal ful1load and ballasted, or typical outbound andjnbou”d. Then completely independent shortand long-term calculations can be carried outfor both, or attention can be concentrated onthe condition giving the higher values ofbending moment.

With the above simplifications,we are leftwith the following variables, or parameters,to be considered In any one probabilIty calcu-lation: significant wave height, H1ls; spectrumshape, S; and ship heading to the seaway, 4’.S indicates a generalized quantity which can bedefined in dif~erent ways.

Resoonse sw?ctra are calculated then forone or two conditions of loading, for al1 shipheadings and for all sea spectra, using an

each. Each responseting

appropriate ship speed forspectrum is then integrated and the resulirms values, a, are the parameters thatcharacterize all of the short-term responses

The final step in the calculations combinesal1 of the above Rayleigh distributions,weighted by the frequency of occurrence of thedifferent spectrum shapes, ship headings, andsignificant wave heights. An attempt will bemade to describe the basic vrinciDies in themost general terms possible’,so that thevarious procedures now in use can be understoodas var-i.ationsof the general approach. Firstof all, it must be clearly stated that in whatfollows the variable under consideration, x,is the peak-to-mean amplitude of bendingmoment, i.e. the deviation of successive peaks(usually the highest maxima between zerocrossings) from the mean value. (It is usuallyassumed that the response is symmetric aboutthe mean and therefore that the statisticsof sagging and hogging are the same). It shouldbe noted that in some cases the double amplitude,peak-to-trough values have been considered theprime variable.

The long-term probability density functionis obtained as a sumnation of many short-termdensity functions, weighted by the frequencyOf occurrence of each. Initially we assumedthat each short-term sample has the same num-ber of amplitudes or peaks. The short-termfunctions are conditional probabilities, sinceit is assumed that each applies while the shipis operating in a particular steady sea conditionat constant heading to the sea (as well ~5 atconstant loading and speed). Each fu”’tion isdefined by its Rayleigh parameter, .2= mo, orc .@, determined by the variables H

1/3, s

and $, which are assumed to be mutually indepen-dent. The long-term formulation may be expressedin many ways, but it is basically a joint proba-bility of x and O, expressed as,

q(x, u) = P(xla) P(.), (9)

where P(x(0), the probability of x for a giveno, is the conditional density function of xwith respect to o, assumed to be Rayleigh. Thus

-X2J2.2P(xIO) = (x/a2) e (lo)

and p(a) is a complicated function--or group offunctions--that defines the probabilities of aon the basis of different combinations ofthe variables, H113, s, $; i.e. o is a functionof H1,3> s, $.

The density function of x is q(x) =/q~x,o) do , and the ‘wnulative long-termdistribution, which is of particular interestto us, is obtained by a second integration, sothat, 9’?

,[x>xi)‘/xi/ ,(x, o)d. d.0

Since one of the integrals, the cumulativeRayleigh distribution, is

) xi

the above becomes,.

/

2

Q(x>xi) = e-xi2’20 p(c) do (12)

o

whereo= 0(H1,3, S, y).

The problem is to define the complicatedmanner in which o varies with the parametersH113s s, $. i.@. tO determine P(.). one approachIs to assume that p(u) can be replaced by ajoint density function of H1,3, S, b all assumedto be independent,

p(H1,3, S, $) = P(H1,3) P(S) P($) (13)

This is equivalent to the othe~ expression,P(U), since for each combination of values ofthe parameters there wil1 be a specific value ofo to be incorporated into the integration. 1“either case, three separate steps are required,and different researchers have made differentassumptions and simplifications in carrying outthese steps. (Appendix 1 summarizes theassumptions made by various classificationsocieties in 1973). They all arrive at weightingfunctions or probabilities indicating the con-tribution of each combination of parametersto the total probability density function of .,defining the Rayleigh distributions. (Initially

42

each of these is assumed to cover the samenumber of peaks, or x-values.)

In practice, the above short and long-termprobabilities are calculated by numerical sum.nkationinstead of by integration. For example,one might make use of 5 significant wave heightbands, 10 different spectral shapes or meanperiods and 12 bands of ship headings.

The procedure in use at ABS and WebbInstitute is based on the study of full-scaledata which suggested that if data are stratifiedinto different sea (or wind) groups, the Rayleighparameters of the response within each groupfollow approximately a normal density function(truncated at 0), as shown in Figs. 8 and 9.Hence, Band [46] showed that the singlefunction, p(o), can be separated into theproduct of two functions, one being the proba-bility density function of o within individualwave height groups and the other the probability~~~~ewhich the ship encounters each wave group.

P(.) = P(H1,3) - P’(o) (14)

where c in p’(o) depends on S and $, and P*(u)is assumed to be a normal distribution on thebasis of ful1-scale observations previouslydiscussed. Spectrum shape is accounted for byusing families of measured wave spectra havingequal areas, and equal probability of allheadings is assumed. See Lewis [40].

Then, for the assumed normal distributionof o in each wave group and ship heading,

p’(c) = (l/n ~-(~ - m)*/2s2, (15)

where m is the mean value and s is the standarddeviation of os for each combination of S and$. Hence, for all conditions combined,

-X,2,202

Q(x, xi]=z Xe 1 P(H1,3) ,?/3 i

(1/JZ7) e-[” - ‘)2/2s2 (16).

Fig. 12

,,, ,

The summation with respect to HI/j is done last,so that a serie$ of long-term CW-VM for differentwave groups is an intermediate step (Fig. 12).The final IonQ-term result. determined bv numeri-Cal sunnnation~can be presented graphically(Fig. 12), although in some procedures itfitted to some specific fumti o”, swch as aWeibull distribution. (The 3-parameter Weibulldensity function, which includes the Rayleighas a special case, has a wide range of applica-tions, both short and long-term. Fitting tothis function can be done by plotting onspecial Weibul1 probability paper.) However, itshould be emphasized that it is unnecessary tofit the final calculated long-term probabilitiesto any explicit distribution function.

Another approach is that of Nordenstrbm [47],who left the summation over ship heading to thelast step and expressed all other probabilitiesin the Weibull format.

Fukuda [48] also carried out the sumnationover shiP heading as the last step, but heretained the Rayleigh distribution to describethe short-term respons~ with parameters HI/3,S and $ constant. He made use of an idealizedwave spectrum formulation in which Ta, theaverage period, is the shape parameter, and atabulation of observed wave heights and periodsdefined the probabilities, from which the jointprobability distribution of the Rayleigh para-

~:;:d”;ugr:c~;;:io” ‘f Hi/3 and Ta’ ‘as

Another difference among procedures isthat, whereas most methods assume that allshort-term samples cover the sank?number ofcycles of load, Ochi [43] points out that thisassumption is not correct. Hence. he has shownthat ‘itis important to allow fot-”thefait thatdifferent sea spectra, ship headings and shipspeeds will lead to different numbers of cycles(OV peaks) per unit time. This correction hasalso been introduced by Seeding [49].

A few simple calculations for a ship of500-ft length in regular waves of its ownlength (which give near-maximum bending moment)wil1 show the magnitude of speed and headingeffects on number of cycles. See Table 1.

,—

o(x>xi),FUaABILTY OF EXCEEDINGxiPredicted Long-Term Piobability of Exceeding Peak-to-Mean Stress ValuesDifferent Weather Groups, S.S. Wolverine State and Hoosier State [41]

in

I

43

Table I Effects of Speed and Heading on C clesof Bending Moment (Regular NavesY

16 Knots

Te sec. n = cycles in 20 reins.

Head SeaS 6.7 179Beam Seas 10 120Following Seas 20 60

20 Knots

Te sec. n = cycles in zo mjns.

Head Seas 6.0 200Beam Seas 10 120Following Seas 28.5 42

Suppose account is taken of variations inthe nwnber of amplitudes in a given period oftfme, dS proposed by Ochi. It is necessarythen to compute, in addition to the areas ofresponse spectra, mo, the second moments, m2,as well The number of crests per unit time(i.e. maxima between O-crossings) is equal tothe number of zero up-crossings, or

+rm (for each short-term case). A

weighting function must then be applied wherebyeach n is multiplied by the ratio,

NO. of crests per unit time, short-term .Ave. no. crests/unit time, long-term, n

1

-r

‘227n ~ “

The average number of crestsfor all as,

per unit time

and the corrected cumulativedistribution becomes,

m

do (17)

probability

Expressed as a summation for calculation pur-poses,

(19)

Since this more accurate expression involvesa great deal of calculation, it was of interestto determine whether or not it would have asignificant effect on the long-term prediction.Accordingly, computer calculations were carriedout of wave bending moment amidships for theSL-7 container ship at all headings to one familyof short-crested seas. Assumed conditions:

Ship Speed 25 knotsSignificant wave height 24.5 feetEqual probability of all headingsRAOS from Oavidson Lab. model tests

The results showed a wide range of n-values(from 210 cycles in 20 minutes in head seas to28 cycles in following seas). However, the meanvalue of bending moment came out to be almostidentical (0.4% diffet-ence). The reason for thissmal1 effect appeared to be that the bendingmoment is almost the same for head and followingseas (although lower for other headings). Hence,the difference in weighting of head and bow seashad 1ittle effect on the average. The actualcalculations are summarized in Appendix 3.

It is believed that this result is typicalof what would be expected for any ship whenbending moment only is considered. In the caseof pitching motion or vertical acceleration atbon, both of which show much higher response inhead than in following seas, it would be expected,however, that this correction would have asignificant effect. It is concluded then thatfor calculating long-term bending moments, theintroduction of number of cycles can be omitted.

Lifetime Probabi1ities

Up to this point, the variable x beingconsidered is the amplitude of a peak or half-cycle of bending moment, hog or sag, betweenO-crossings. Hence, the probabilities arrivedat may be termed probabilities per cycle.However, ultimately we are concerned with prob-abilities to be related to the probabilitiesof strength (capability). In the latter, consi-deration is given to the effect of materialquality, scantlings, workmanship, etc., on thevariability of load-carrying ability of thestructure, i.e. the variability in strength ofmany similar ships. Neglecting the effects offatigue and corrosion for the present, thestrength probabi1ity has no relation to thenumber of cycles of load, and in fact, representsa fixed characteristic of the structure that isindependent of time. In other words, neglectingcorrosion and fatigue, it represents the probablefailure load at any time in the ship’s lifetime.Hence, a compatible form for the probabi1ity ofloads (demand) would be to consider many similarships and to determine the density function forthe bending moment expected to be exceeded oncein the lifetime of any one ship.

Of course, on the average the bendingmoment value to be exceeded once in a ship’s1ifetime can be read from the cumulative distri-bution at the value of NL= l/Q, where Q is theprobability per cycle, and NLis the number ofcycles in a lifetime. To obtain the number ofcyc”lesexpected in a ship’s 1ifetime, a roughestimate may suffice. But an accurate value may

44

be determined from the previously given equation(17) for n.and total time t (sec.),since NL= n t.

But when many ship$ are considered, the1ifetime exceedance value would be higher fot-some and less for others. In fact, there istheoretically a probability of 0.67 that thisaverage value wil1 be exceeded by any one ship.The meaning of this probability is that 67% ofa large fleet of similar ships would be expectedto experience a bending moment greater than theabove average in their 1ifetimes. Hence, thebending moment corresponding to N.is not asatisfactory design value, although it is usefulfor comparative purposes.

To obtain a meaningful design value forthe extreme bending mment to be expected inservice, we must consider what may be termedthe “lifetime probability.~t This is theprobability that a given level of bending momentwill be exceeded by any one of a large numberof ships (e.g., a probability of .01 means oneship in a hundred, .001 means 1 in a thousand,etc.). The 1ifetime Drobabilitv can be deter-mined in several ways; but the host convenientis simply to extend the 1ong-term cumulativedistribution (Fig. 16) to much lower probabilities(larger N) than would otherwise be required.Karst [54] has shown that for many load cycles,

Lifetime probability of exceedance =

(cumulative prob.fcycle) (number of cyclesin lifetime) = Q . N. (20)

For example, if there are 108 cycles in a ship’slifetime, and we read a certain bending momentvalue at Q = 10-10, then the lifetime probabilityof exceedinq that value is

~o-lO-x ,.8 .10-2

(equivalent to Ochi’s risk parameter, a = 0.01)

Similarly, lifetime probabilities can bedetermined for other values of bending moment,as shown by the upper scale of Fig. 24, whichis discussed in a later section.

However, it should be noted that theabove is an approximation which breaks dew” whencumulative probability per cycle equals thereciprocal of the number of cycles. At thispoint, the correct value for 1ifetime probabilityis 0.67 instead of 1.0. The value approaches1.0 as load approaches zero, but this range isof no interest to us.

Lifetime Cyclic Loading (based on [7])

From the fatigue viewpoint the type cfloading is one of irregular cyclic load reversal,usually with fluctuating mean load and possibleoccasional overload at points of stress concen-tration. It is further complicated by diurnalthermal stress variations. These loads aretabulated below along with the estimated cyclesof load reversal for each in a typical ship’slifetime:

Still water 340Wave bending 10’ - 10’Oynamic 106Thermal 7000

The fluctuating mean load is the so-calledstill water bending moment, discussed in a pre-vious section. In general the specificationof two probability curves, one for outbound (A)and the other for inbound (B) conditions, willprovide the information needed for fatiguedesign. However, one additional item is needed:the time that the ship operates in condition Abefore changing to B, time operating in condition8, etc. In general both times will be equalsimply to one-half the total round voyage timeand will be measured in weeks. To be moreaccurate the effects of consumption of fuel andadditions of salt water ballast should beincluded.

The cyclic loading consists of the low-frequency wave-induced bending moments a“d thehigh-frequency dynamic bending moments previouslydiscussed. Their phase relationship is perhapsof less significance for fatigue than forbrittle fracture. At any i-ate,long-term cumula-tive distributions of low-frequency loads shouldbe available as part of the load determinationfor ultimate loading. However, the earlierdiscussion of dynamic loads was directed mainlytoward the problem of their possible contributionto ultimate failure, To determine the cyclicload spectra of dynamic loads, additional infor-mation is needed. One must estimate or predict:

a) The frequency of occurrence during aship’s lifetime of episodes of slamming andspringing.

b) The number of cycles of whipping andrate of decay following a slam, or

c) The average number of cycles ofspringing i“ one episode a“d the correspondingdistribution function.

In principle, it should be possible some timein the future to make predictions of the abovefor a specific ship on a particular trade route.For the present, estimates on the basis ofmeasured data on similar ships should suffice,since there are so many uncertainties indetermining fatigue strength. Hence, a long-tevmdistribution of each high-frequency load canbe constructed.

From each of these distributions one canobtain a cyclic load spectrum in the followingmanner. The reciprocal of the probability isthe number of cycles, N. For a ship’s lifetimeof NL cycles, a scale of NF = NL - N is thenconstructed on the distribution plot. Then NFgives the number of cycles expected in theship’s lifetime of any desired level of bendingmoment. See Figure 13, which deals with wavebending effects only [55].

Renort SSC-240 r71 oives sample c.vclicloading”“spectra” fo~ ~h~ s.S. !401ver{neState,as shown in Fig. 14. The curves for low-frequency wave bending were obtained from thelong-term cumulative distribution of bendingmoment by converting to stress and reversingthe scale, as previously explained. The curvefor dynamic loading (slamming and whipping) wasobtained by first estimating a lifetime histogramof stress cycles, using available voyage recordsas a guide, then integrating.

,.I

!.

I

L-45

[ \10-010+ @ 0+ 10+ me 10+ 10* 10- (0. I‘moklLIrY w EXCEEDANCE

) J10 10= 10, !o~ 10, ,0. w ,0$M.WBEROF CYCLES

Fig. 13 Long-Term Distribution of BendingMoment or Stress, with Reversed Seale ShowingNumber of Cycles of Different Stress Levelsin One Ship Lieftime (lOa Cycles) [7, 55].

Finally, information should be provided onthe expected diurnal variation of thermal effects,as previously noted.

The above information should provide thedata needed by the stress analyst to evaluatethe different types of cyclic loadings, as wel1as the variation in mean stress, as discussedsubsequently.

COMBINING LOADS

One of the most difficult problems--orseries of problems--to be solved in attaininga rational approach to hul1 structural designis that of combining the different loadspreviously discussed into one complete picturein probability format. As pointed out before,the various loads that are superimposed havewidely varying frequencies. Furthermore, itis often difficult to separate loads fromstresses. H~gh-frequency loads usually evincethemselves as stress (or deflection) wavesthroughout the length of the ship, originatingin local hydrodynamic pressures. Local loads(forces and moments) on panels and structuralcomponents can best be determined from thestresses acting around the boundaries. Hence,many schemes for combining loads actually dealwith combining stresses. This problem was

discussed previously in connection with theconsideration of combining lateral and vertical1ongitudinal bending components, where the twocomponents are at the same frequency.

Unfortunately one simple, practical solutionis not available. Methods of combining loadsor stresses based on statistical methods ofcombining variances are inadequate unless theform of the correspond ng probabi1ity densityfunction is known. Hence, for practical purposes,it seems best to consider different combinationsof loads separately, and a review of the situationwill show where further research is needed.

The combination of 1ow-frequency wave-induced bending moments with still water momentsis discussed by Lewis, et al [7]. Here it isassumed that for most ships the stil1 waterbending moment can be described by two normaldistributions, one corresponding to a full-loadcondition and the other a ballast or 1ight-loadcondition, corresponding to (A) outbound and(B) inbound, or vice versa, voyages. Fig. 15,It is then recommended that complete short andlong-term calculations be carried out for bothconditions A and B. To obtain single long-termcurves for hogging and sagging--including stillwater bending--requires that the wave bendingmoments be first expressed as a probabilitydensit function (instead of a cumulative distri-

7bution The functions for stilI water.a“dbending moments can then be combined on thebasis of joint probability, since the twophenomena are essentially independent [7] [34’J.

Let x be a random variable describing thewave-induced bending moment (hog or sag). Thedensity function of x will be called pw(x).Let y be a random variable describing the stillwater bending moment, with density functionPS(Y), which will be assumed to be normal. Weare interested in the cumulative distributionfunction for the random variable z = (x + y),which is given by the convolution integral [7],

.

/P(z) = PW(X) P*(Z - X) dx (21)

-m

Since pw(x) is not known in explicit form, theabove integral cannot be evaluated analytically.However, it can be determined numerically forany specific case. Fig. 16.

1 I I I

Lm ., NWi. m., ,,.,s .,v,mm

Fig. 14 Estimated Cyclic Loading Spectra for S.S. Wolverine State [7]

46

“’t,“00..,(,,.”, ..,,,,

-w w m ,m .

Fig. 15 Estimated Probabi1ity Density Functions of Still Water Bending Momentfor S.S. Wolverine State [7]

We come now to combining the thermal effects,which can be interpreted in terms of a zerobending moment at night when thermal gradientsare small and an effective sagging bendingmoment in daytime--especiallyif the sun isshining. To simplify this problem of combiningloads, we can make the conservative assumptionthat the thermal effects are always pt-ese”ti“day time and then have a constant value, Al1wave data can be roughly divided into twoclassifications--thosethat QCCW at night withn? thermal effects and those that occur.in day-tlme, with thermal effects sumri !wmsed. Thiswould lead to two long-term ciIr-ves~m show”j“Fig. 1~. We can conclude the” that a safe,aPPrOxlmate treatment of thermal effects is toshift the base line by one-half the amount ofthe average total change in effective thermal“bending moment.”

Fig. 16 Lonq-Ter-mDistribution of Combj”edBending Mo~ents: Wave Bending (VerticalLateral) and Sti11 Water Bending [7]

Finally, another low-frequency effect--that of local loads--is usually calculated interIiISof stresses. From the viewpoint ofstructural design, however, more importanceattaches to the effect of such loads on ultimatestrength. For example, the hydrostatic pressureon the under side of the double bottom mayaffect the longitudinal buckling load of thedouble bottom structuve. Hence, informatim o“such secondary loadings should be supplied tothe structural designer.

Thus, insofar as the possibility of damageor U1timate failure by buckling or permne”tset is concerned, the complete low-frequencyhul1 loading picture can often be presentedin the form of two sets of long-term curves, onefor outbound a“d the other for-inbound--as inFig. 16--adjusted for thermal effects.

The combining of dynamic or vibratiomlloads with those previously discussed poses anumber of more difficult problems. Not onlydo these loads act at different frequencies,but some are transient (slamming and whipping)and others are steady state but intermittent(springing,which may occur in head and beamseas, for example, and not in following seas).

SAG

HOG

andFig. 17 Typical Long-Term Distribution of WaveBending Moment, Sag and Hog, with ThermalEffect Superimposed [7]

47

—

;alzell, et al [9] made an extensive analysisof stress records showing vibrational effectsfor a typical ocean-going bulk carrier and ahigh-speed container ship.

Considering springing first, it can beassumed, as previously discussed, to be a sta-tionary stochastic process--when it occurs.Hence, a long-term cumulative distribution canbe calculated in a manner similar to that usedto determine such a distribution for low-frequencywave bending moment. But serious problemsarise in attempting to combine these twodistributions directly, since account must betaken of the manner in which the low and high-frequency bending moments combine. Furthermore,these distributions must be compared with greattaution because the meaning of probabi1itychanges when mixed frequencies are involved.For example, when both low and high-frequency(springing) bending are present, the springingoscillations will be superimposed on the low-frequency bending (Fig. 3), and there will bemany more bending moment maxima. The responsespectrum for a combined record wil1 have twodistinct peaks, as show” in Fig. 18, a“d willtherefore be a type of broad spectrum. Hence,a Rayleigh distribution will not apply to thepeaks. The customary practical solution has beento assume that use can be made of a ge”er.alizedRayleigh distribution with broadness parametercalculated from spectral moments. Such adistribution defines all maxima [or minima), notjust the principal peaks between zero crossings.To obtain the statistics of the principal peaks,a different distribution applies, as discussedby Battjes [56], who refers to the principalpeaks as crests. See Fig. 19. So far asis known, this distribution has not been appliedto the problem.

Another alternative, that also apparentlyhas had only limited application, is to makeuse of the probabilities of the highest maximum(or minimum) in a period of say 20 -30 minutesinstead of the distribution of al1 maxima. Thehighest expected maximum in 20 minutes of combinedlow and high-frequency bending can be derivedfrom the generalized Rayleigh distribution.Van Hooff 157] has shown how the resulting long-term distribution, based on the highest maximumin 20 minutes, can be interpreted in comparisonwith one based on rms, o.

Probably the best available approach forcombining low-frequency wave bending andspringing, for ships having a significant amountof sprinaina. is to calc,,latea long-term dis-tribut lg moment. To dothis requires that the RAOS for the combined

.. . . . . . . .:ionof combined bendir

response be used, in conjunction with ageneralized Rayleigh distribution that takes

T-A----,-----,%

MIDsHIP SITWSS STSD.

1 1,

1.00 2.00

Frequency ~

Fig. 18 Typi’al Stress Response Spectrum forGreat Lakes 8ulk Carrier, Showing Two OistinctEnergy Peaks [4]

account of the spectrum broadness parameter forthe double-peaked response spectra [58].

As previously noted, the number of springingmaxima in a given period of time will be muchgreater than the number of low-frequency bendingmaxima. Hence, the resulting long-term distri-bution of cmnbined bending will representprobabilities governed by the high-frequency, orsPrin9~n9, response. Fig. 20 [34] shows acomparison of long-term distributions for wavebending, springing, and combined loads, andindicates how the latter two must be shifted onthe probability scale so that they will all becomparable in time. However, if the 8attjes [56]formulation could be used. which counts onlvthe highest m.iximumbetween zero crossings,”noadjustment of the probability scale for the

0 zero crossings of ~(t)

L d k 1 \ Aflf, t

~t Q upcrossings of ~(t) through h

X crest height above h

Fig. 19 Sketch Showing Relations Among Stress Maxima, “Crests,” and Level Crossings [56]

4s

-.

—-J--J.-

.—-..

Q(- ..%,,,,

1

Fig. 20 Theoretical Long-Term Distributions of Sti-essfor wave (LOW-FreqUeIICy),Springing (High-Frequency),and Wave and Springing Combined,

Showing Shifts to Bring into a Comnon Time Scale [34]

combined curve wil1 be required.

As a practical matter, springing loads aremoderate in most ships and reach high levelsonly in very full, flexible ships such as GreatLakes bulk carriers.

Slanmningalso occurs under certain conditionsonly--depending on sea severity and on shipspeed, heading and draft forward. But when itdoes occur it is transient and occurs atirregular intervals. Furthermore, it is followedby a number of cycles of whipping that die outas a result of damping. It does not combinewith wave-induced bending moment in a randomway, for there is a fairly 1imited range ofphase angles between a slam and the precedingwave bending moment peak [7]. Hence, not onlynwst the occurrence of slamming be predicted,but the phasing must be allowed for when it doesoccur. And account must be taken of the factthat speed and course will be adjusted by themaster to prevent S1ams exceeding some subjectivelevel of severity. For the present our onlyrecourse seems to be, as previously noted, tomake use of full-scale ship measurements todetermine an allowance for the amount by whichmak bending moments are permitted to increasebecause of slamming and whipping on differentship types. See Aertssen.[59], Lewis et al [1][7], and Dalzell, et al [9].

A further problem within the domain of thestructural designer must be considered inrelation to the effect of high-frequency loadson U1timate strength. Even if one knows themagnitude of a peak of combined low and high-frequency bending moment, is this a realisticload to be considered in relation to possibleultimate failure? Such failure involvesabsorption of a large quantity of energy bythe structure, but the short duration of thehigh-frequency load may mean that its effectis significamtly t-educed. Hence, fo~ theTonwnt the problem is to furnish as completeinformation as possible on the low and high-frequency loads, leaving it to the structuralresearcher to determine their combined effect.

Finally there is the problem of combiningCYC1ic load spectra in relation to fatigue.Since the stil1 water bending moment staysreactively constant for long periods of time--between cargo changes or major ballast shifts(often an entire one-way voyage)--it has theeffect of periodically changing the baselineabout which the other bending moments vary.This constitutes, as previously noted, “irregularCYC1ic load reversal with fluctuating meanload.” A crude overall picture of the totalCYC1ic 1oad spectrum can be obtained by directsuperposition of low and high-frequency loadspectra, provided they are all calculated forthe same 1ifetime period. The problem ofobtaining more realistic spectra for fatigue.testing of critical structural details is givenby Nibbering and Scholte [60], as discussedsub$equently.

UNUSUAL BENDING MOMENTS

So far, consideration has been given onlyto ship behavior in the open sea assuming1inearity of waves and response. Hence, pre-dictions of long-term wave bending moments arevalid only for such conditions. Therefore, inorder to establish the most extreme loads fardesign, it is important to give attention topossible wave conditions that would cause evenmore severe bending moments. Some such condi-tions may be mentioned:

Steep, breaking (non-linear)waves(mult~-directional).

Shoaling water (de ths of the same7order as ship-length waves .

- Opposing currents.- Wave reflection from steep, rocky shores,

or jetties.- Effects of earthquakes.

The mathematical model of the ocean waveenvironment discussed earlier in this paperassumes simple storm-generated waves uncontam-inated by seas or swel1s from other storms,and without influence of shallow water, currentsor nearby shores. The combination of point wave

%-

419

i

spectra of different shapes and areas, with aspreading function to account for shortcrested-ness, seems to provide a reasonably good repre.sentation of this simplified situation. Onlywhen conditions become very severe does theassumed 1inearity come into question. St.Oenis [61] has warned that “when the sea attaimsuch a severity that its surface takes on the*ppearallceOf mOuntains in turmoi1, no represen-tation based on 1inear or quasi-1inear theorycan possibly suffice to describe it and onewould need have recourse to a fundamentally non-linear theory.....S, However, it is fortunatethat the non-1inear trend as waves becomehigher and steeper is in the direction ofsharper crests and flatter hollows. (Note thefavorable “Smith effect” on calculated bendingmoment in going from a sine wave to a “trochoidal”or Stokes wave form). Work of Dalzell has shownin fact that when experimental bending mmsntsobtained in regular waves in a model basin areplotted against wave height, the values usuallytend to fal1 below the 1inear trend when waveheights are high [62]. Fig. 21. Hence, theassumption of 1inearity is on the safe side.Dalzell also found that much steeper waves--and therefore, higher bending nmments--could begenerated in the model tank than are evermeasured on full-scale ships at sea. Althoughnon-1inear wave theories are available forregular waves, a non-1inear model of irregularwaves becomes highly complicated, as showfibySt. Oenis [63].

One problem with very steep waves is thebreaking crests, which can be dangerous for smal1craft and can cause damage to deck structure andoutfit, as noted by Buckley [64], even thoughlongitudinal bending may not be as severe.

Assuming that a 1inear storm sea model isgenerallY acceptable, consideration must begiven next to the fact that simple storms seldomdevelop and decay in an otherwise undisturbedocean. Usually the sea’s surface is alreadydisturbed by the waves remaining from a previousstorm or by a swel1 from a distant storm orstorms. Lewis [40] attempted to account forthis effect by making use of samples of actuallymeasured spectra which showed irregularities anddouble peaks that could be accounted for only bythe superposition of different storms. Ochi [43]carried this further by developing a mathematicalformulation of families of point spectra withdouble peaks. However, neither of theseapproaches allOw for the fact that UW1lY thesuperimposed storms have different dominantdirections. Fig. 22. As noted in the reportof Committee 1, Environmental Conditions, tothe ISSC in 1973 [65] “Sea waves and swell bothfrom different directions are the rule, ratherthan the exception, on the ocean surface...However, the climatology of such situations ofsea and swell combinations is inadequate...The most desirable solution is the measurementof directional spectra. Experience has shownthis to be very difficult and very expensive--justifiable only for a limited number of reseai-chsituations... Wave prediction techniques offerthe best potential for an adequate accounting ofthe mixture of sea and swell which normallyexists on the ocean surface...’,

...!../K+

~ -.0014 ...1.

‘?? PO%16LE,MP*.Z7 >EFFEC75 , ‘.50$ ‘0””

a ,00!0.%“ ...o*- *

.. ..$ -

...0.

.. . . .

.

.0..,

,...4 -

-ml....8

O..*< t i

Fig. 21 Model Test Results for Heave, Pitchand Bending Moment Coefficient,- Modelin Regular Head Seas (Wave length, a = 1.OL;FroudeNo. = 0.12 - 0.14) (Oavidson Lab. ReportR-926, Part II)

Since that time (1973), as a matter of fact,considerable progress has been made in theaPPliCat~On Of forecasting techniques to theship designer’s problem. In particular, theocean wave spectrum mdel developed at theFleet Numerical Oceanographic Center (FNOC),Monterey, Calif. (Lazanoff and Stevenson, 1975)[66] has been expanded to cover the NorthAtlantic and North Pacific Oceans, as well asthe Mediterranean Sea, and will be extended tothe southern hemisphere. Not only are theresulting routine operational forecasts ofwave spectra of great value to ship operators,but the accumulation of data over many years isof potentially inestimable value to ship designers.

i.o

.-sPECIRALPERIOD,1(SEC)

Fig. 22 O$rectional Sea Spectrum, ShowingSuperposition of Wind Sea and Swell [61]

I

1—

50

I

Table II Typical Hindcast Directional Spectrum [68]

LCCA,40N

5s292+4,,.2WW

?4.....28..6w,..s.2,.6.“!,,cm, “s,,.@

\[y ; .m .ss.tll .$,,a? ,C92m? ,072,m,.J<wmu .U3a D,.(,,0.,

m M .0,,m m m m ,., ,m m ,m ,m

I.00 .m m ,1 a.n

w .* .16.16.17.07.m .@ .@ .m .m .m .Ca.s4.03 ., ,%.69,*,,*F+CE- .1s.27.25.30.22.0,.Z4m .m ,s0.a m ,W ~ ,,, ~,=

.!4,,.” .W .2!,38.38.s6.42.,,.02.Ja.W .m .m m .m m 2.3 nmew .,,.s ,50.,,,@ .,, ,, ,m ~ m ~ ,@ m ,W ,,, #,m

m .!2.,s.,,.!4.Co.0,.m .@ m ,m ~ ,m ~ ,m , *,8,=

%.uMI‘g “ ‘~ ‘“‘“ “ .“ .23 .~ .~ m m w .M OY . .=“10 ,0.3,,,

S.”,F,CAN,w.”,“s,0”,

To nmke the data available in useable form, acwrehensive project was undertaken some timeago to develop a worldwide climatology as ajoint DTNSRDC/FNOC project (Bales and Cumins[67]), using the computer bank of meteorologicaldata to “hindcast’!the wave spectra over aperiod of many years. The principal advantagesof such hindcast spectra (Table 11) for designuse are:

- Wide coverage of the world‘s oceans.- Availability of input wind data for many

years in the past.. Inclusion of directional properties of

seas and cross seas or swells.

There are certain 1imitations to be considered:

- Oata are calculated rather than measured,a disadvantage that can be overcome by systematicand extensive comparison with actual measurements.

- Accuracy is of necessity reduced inareas where wind observations are scarce.

- Range of wave frequencies covered maynot be adequate for all purposes.

Recent papers by Cunnninsand Bales [68] [69]give some preliminary results and suggest thepotential value of this work. Of particularimportance is the need to select suitable para-meters to describe the directional spectra andthen to give statistical data on these para-meters, in order to reduce the vast number ofhindcast wave spectra to manageable form.

Special consideration also needs to be givento the waves generated by hurricanes and typhoons.These severe storms that move at relativelyhigh speeds along erratic paths cannot beaccounted for in the normal collection of routinewave observations, forecasting and hindcasting.Not only are the winds unusually strong, butthey change direction continual1y, so that theresulting high, confused seas do not conformto the ideal model. Some studies have been madeof hurricane seas [70], but much more work needsto be done. Buckley has called attention tothe strong correlation between ship damages andthe occurrence of unusual or freak sea conditions[64]

D, .,CT*—Tm.Lm,,.”

Other conditions that may lead to moresevere, i.e. steeper, waves are shallow waterand opposing currents. The report of ISSCConnittee 1.1, Environmental Conditions, 1976[71], discusses these effects in Chapter 9 onAbnormal Waves (based on a contribution byR. Wahab]. It is stated that “much attentionhas been paid recently to abnormal wavesoccurring off the east coast of South Africa..Other areas where unusual1y high waves have beenobserved include the Sea of Casquets (N.W. France)and the vicinity of the Oogger Bank in the NorthSea.” The South African example is explainedon the basis of storm seas or moderate swells0PPosin9 the Agulhas Current that runs along thecoast. The North Sea example is explained asbeing caused by wave refraction by shelving andshoaling water (Pierson [72]). Other locationsin which shallow water has apparently been afactor in ship losses or damages are the Contin-ental Shelf approaching the English Channel, theGrand 8anks, the Continental Shelf in thevicinity of Cape Cod, and the bar at the mouthof the Columbia River in Oregon.

The report of Committee 1.1, EnvironmentalConditions, to the lSSC, 1979 [73], states,“Conditions which are somewhat similar to thoseoff the African coast, as far as the possibleoccurrences of freak waves are concerned,occasionally prevail along the north wall ofthe Gulf Stream northeast from Cape Hatteras,when north-easterly winds generate a sea runningcounter to the Gulf Stream (James [74]). Anadditional disturbing factor is the instabilityof the initially cold polar air masses, heatedby the relatively high and gusty winds near sealevel, resulting in a rougher sea.” The lossesof the tanker Texaco Oklahoma and bulk carriersAnita and Norse~l and 1973 mayfibeen related to such abnormal waves.

Another condition of unusually steep wavesis the standing waves caused by the reflectionof an oncoming sea by vertical cliffs andbreakwaters. Such steep waves have been reportedin the 8ay of Biscay off the coast of Portugal.

Earthquakes are known to cause severe wavescalled tsunamis or “tidal waves.” Little isknown about the magnitudes and farms of suchwaves, but on the other had, it may not be k-

)/’51

necessary or desirable to design for such rareand unusual occurrences.

Although the incidence of abnormal1y highor steep waves in certain areas of the world’soceans has been known for a long time, compara-tively 1ittle has been done by way of directmeasurement of such waves that would be usefulto the ship structural designer. In recentyears, large moored buoys have collected meteoro-logical and wave spectral data for NOAA, a fewof them giving directional spectra. Unfortunatelythe buoys have been located in coastal watersand generally not on main shipping routes [75].The potentiality exists for making more extensiveuse of such moored buoys to provide systematicwave data at critical locations. A data buoywas deployed for “at least a year” at the edgeof the Continental Shelf at the entrance tothe English Channel [76], but results of dataCO1lection are not yet known to be available.No matter how good our open sea wave data andour methods of predicting long-term trends ofwave bending moments in the open sea may become,we can never be sure that the resulting designloads are acceptable until we know more aboutthe possibly more severe loads that may beencountered under unusual circumstances.

APPLICATION TO OESIGN

Ultimate Strength

Having reviewed the loads affecting theU1timate strength of the hul1 girder and con-sidered methods of predicting and combining themin probabilistic form, it will be well to consi-der how these loads can be applied in structuraldesign. Of course, this involves considerationof strength or capability, which is the subjectof other papers at the Symposium. But there isa need to try to link demand with capabilityat some time early in the meeting.

Considering classification society methodsfirst, there has been a definite trend towardnmre rational approaches in recent years. TheRules of all Societies have adopted proceduresthat embody the fol1owing features:

- Separate determination and addition ofstill water and wave bending moments.

- Determination of wave bending moment interms of a coefficient (or “effective waveeight”) arrived at by comparative calculationsMI the basis of wave spectra and probability-.

- Establishment of allowable stress values-t ●re essentially comtant (a smal1 variationmm ship length in some Rules is explained by~~ -nts of absolute values of corrosion●n-) a“d include an empirical fa~tOp Of-.

~ section nmdulus can then be determinedfi~ly ~ the Rules, taking into account the$alk~ factors: ship type, length and breadth,* Mictznt and material of construction.

[t is si~if icant that al1 classification~ - also made provision for a more-G ~ to design, taking account of~m ~, exlu?ctedsea conditions in

the intended service, more advanced techniques,etC. Attention is focussed on a Droabilisticapproa~hto design bending moment”,while retainingan emp]rlcal value of allowable stress. Asexplained in the ABS Rules (1981) [77] (Section6), “Considerationwill be given to the wave-induced bendina nwment calculated bv means ofa statistical ~probabilistic] analy~is based onthe ship motion calculation in realistic seastates.” Lloyds’ Register Rules (1978) [78](ChaDt8P 4) amDlifv this: “In direct calculationprocedures”capable-of deriving the wave inducedloads on the ship, and hence the required modulus,account is to be taken of the ship’s actual formand weight distribution.... The Society’s directcalculation method involves derivation of responseto regular waves by strip theory, short-termreponse to irregular waves using the sea spectrumconcept, and long-term response predictions usingstatistical distributions of sea states.” Thisapp:oach has become quite general among classifi-cation socletles (Appendix 1).

Thus classification societies have advancedto the point of accepting a probabilistic approachto demand (loads) while retainin a deterministic

7approach to c?pability.(strength . Furthermore,strength IS viewed basically in terms of allow-able stress rather than ultimate strength(although separate consideration may be given toavoidance of local bucklins). A number of writershave pointed out that the &pabil ity of thestructure to carry loads, or what Vasta called“load-carrying ability”, is a very different thingfrom the load corresponding to material yield orultimate stress.

It is significant that the classificationsocieties, as well as ISSC, are interested indeveloping even more advanced, rational, directmethod. Quoting Aldwinckle [79], “The statistical[probabilistic] analysis of the demand/capabilitymethod continues to show great promise in movingtowards a completely rational approach tostructural safety..“. It is apparent that furtherresearch and development is needed to study theeffects of variation in constructional standards... With the improved shipbuilding technologyin the world... it should be possible withinthe next decade to introduce the reliabilityapproach. . to benefit both the shipbuilder andthe shipowner.”

For present purposes the reliability approachrefers to the concept of designing for anacceptable level of failure probability. It wasdeveloped in the field of Civil Engineering overmany years, under the leadership of A. M.Freudenthal [80] and has since been applied inAeronautical Engineering [81] and many otherengineering fields.

When suitable probabilistic expressionsfor hull structural capability become available,the probabi1ity of failure can be explicitlyderived. See Fig. 23. Abrahamsen et al. [82]have shown that a double integration of demandand capability is involved. Since our normalprocedure is to integrate to obtain the lifetimeprobability of demand exceedance, only one moreintegration, involving an assumed normal densityfunction of capability remains to obtain theprobability of failure. I

52

Fig. 23 Probability of Failure Determined fromDensity Functions of Demand and Capability

If y is demand and z is capability, the prob.ability of failure may be defined as the pyoba.bility that the ratio z/y is less than unity.Since Z[Y is a sort of ideal safety factor, wecan also say that the failure probabi1ity is theprobability that the safety factor is less thanunity. It can be shown that the cumulative dis-tribution of the safety factor i$

( )/.p;?;=Q(Y) P(Z) dz

00

(22)

where, Q(y) is the 1ifetime probability of demandexceedance,

P(z) is the probability density functionof capabjllty.

The integral can be evaluated numerically andthen its value at zo/yo = 1 determined.

In a discussion by the present authors ofCorrmitteeV.I. Report (ISSC, 1979) [83], anexample was given of a tentative application ofthis approach to a typical tanker, assuming aconstant (deterministic)value of sti11 waterbending moment and neglecting high-frequencyresponse. Use was made of a density functionof ‘apabi1ity based on ValUeS of avet-ageStt-enqt.hand Coefficients of variation (standard deviationsas percent of mean) for critical panels inbuckling, from Mansour and Faulkner [84~. Thelifetime curve for demand was obtained fromdata in the same paper on stresses vs. waveheight, cm the basis of the binomial or Poissonmodel, as described by Karst [54], as given byEquation (2o). See Fig. 24.

For the tanker under consideration thelow-probabilityportion of the demand curvecalculated on the above basis is shown in Fig.25. Using the demand curve in conjunction with

P@ j I-Q(y). Lifetime,robab,lit,ofMCeedmce

1-PW1 = Rmk.aJ3iLityof EX.eedance P..cycle

Fig. 24 Determination of Lifetime Probabilityfrom Probability Per Cycle [83]

53

the capability curves shown leads to lifetimefailure probabilities for grillage failure of0.017 and strut-panel failure of 0.047. In asubsequent section on choice of probabilitylevels, figures given will show that these aresomewhat high to be acceptable (i.e., 2 to 5ships in 100).

However, the assumed standard deviations ofstructural capability appear to be too severe.If the coefficients of variation were reducedfrom the calculated value of 11.2%, the resultsfor grillage failure would be:

Coeff. of var. 11.2% 8% 5%Failure Prob. 0.017 0.0026 0.0002

Assuming that a failure probability off0.0026 is acceptable (2.6 ships in 1000 , a

coefficient of variation of about 8% would makethe design satisfactory. The figures show howsensitive the failure probability is to thecoefficient of variation of capability, andtherefore the importance of further work todefine the latter--as well as the mean value--more accurately.

It was noted [83] that available data onprobabilistic aspects of capability, in general,seem to apply to local panel failure rather thancomplete failure of the entire girder flange incompression or tension. For example, for atanker such as that under consideration here,the local compression buckling of a deck panelin the center tank would shift the load to thetop of the side tank structure--deck,side shelland longitudinal bulkheads. This structurewould probably carry considerable higher loadbefore there would be further buckling, suchthat ultimate failure or collapse could be saidto have occurred. As stated by Faulkner indiscussion of the lSSC Committee 11.2 report(1979) [85], “Abetter understanding of theultimate moment, Mc, which allows for the spreadof elasto-plastic instability, etc., through thecross section is arguably the most importantoutstanding item to complete the capabilitymodeling.”

An outstanding contribution to the deter-mination of a realistic value for averagecapability (ultimate strength) of the hull gir-der is that of Smith [86], whose procedureconsiders the simultaneous stress-strainbehavior of the various structural elements,including “hard corners,” as compressive loadis incrementally increased. It takes accountof the shift of load from buckled componentsto the more rigid members in a reasonable way,and allows for the shift of elastic neutralaxis. 8illingsley [87] has made a valuablecontribution to this approach and focusses onhow the buckling and post-buckling behavior ofindividual elements is related to “the responseof the overall section.” The distinction hedraws between the aims and methods of the shipstructural designer in contrast to the “marinesafety structural analyst” should, we believe,be gradually eliminated as emphasis in designis shifted from stresses to load-carryingability.

0,5

L

0,2

0, I

o

,VEMAND PER CYCLE

DEMAND

I-R(y)PER CYCLE /

k -

WAVESANDdATEl? 5TILLWATER;TRE5S

ST RUT-F,4NeL

1

WAVES

STl

i,

\

5 10 15 20

5TRE55 , TON5/lN.2

Fig. 25 Plot of Probability Data for Calculation ofProbability of Failure for Typical Tanker [83]

The recent paper by Mansour and Thayambilli[88] gives a great deal of valuable informationon ultimate strength. But the above writers donot deal with the question of variability, andthis remains perhaps the area of greatest needfor research. Billingsley does suggest certainvariable factors indirectly when discussing hisanalytical methods:

- Conditions of edge fixity, and- “Quality of construction in terms of

plating fairness, residual stress, and align-merit.”

To these can be added other objective uncertain-ties such as dimensions, scantlings, materialproperties, plus subjective uncertainties assoc-iated with the reliability of theories andexperimental data utilized. Some data are alreadyavailable (Stiansen, et al) [B9] and a new SSCproject on Major Sources of Uncertainties shouldfurther clarify what is known and what remainsto be investig~ted.

Attention must also be given to humanerror. Ractliffe mentions errors “at the levelof design, or construction, or maintenance” ascauses of most major structural failures, andYamamoto mentions ship handling (discussion ofJoint Session 2, ISSC 1979) [90]. Anotherconsideration is the increasing use of shiDboardload or stress monitoring equi~ment, as discussedin the report of Connnittee1.3, lSSC 1979 [91],which can potentially reduce the probability ofstructural failure if properly used.

m

In general, it appears that fatigue is notspecifically considered in all classificationsociety design procedures. But presumablyfatigue is considered indirectly in establishing

allowable stresses (as in the case of ABS Rulesfor aluminum ships, for example) and in eval-uating specific structural details during planapproval.

One aDDrOach to Structural desian relativeto the fatigue loading is simply to ~ake surethat the probability of exceeding the yieldpoint at critical areas of stress concentrationis at an acceptable level, considering the costof repairing nuisance cracks. Presumably, low-cycle fatigue WOU1d thereby be avoided. However,although this can be readily done in relation tomajor points of concentration, such as hatchcorners, it is difficult to accomplish in respectto all of the many welded structural details inlongitudinal structural members. In any case,this approach can lead to excessive scantlings,and therefore it appear$ that a complete pictureof CYCIic 1oadings should be obtained for theuse of the structural analyst (and researcher).

For the fatigue viewpoint, the type ofloading (discussed earlier) is one of cyclicload reversal, usually with fluctuating mean(still water) load. Lifetime spectra forCYC1ic loadings of different frequencies can beprovided in useable form on the basis of avail-able techniques, as explained in a previous sec-tion.

Again it appears that the more difficultproblem today is the determination of the capa-bility of the ship’s structure. In general,the materials used in shiDbuildina are not sub.iectto fatigue failure when uhflawed ~amples are -subjected to cyclic loadings of the numbercorresponding to low and high frequency bendingin a ship’s lifetime. But the longitudinalstructure of a ship contains many points ofstress concentration, arisiilgeither from grossdesign discontinuities, such as hatch corners,

S4

or from local discontinuities occurring instructural detai1s and at welding f1aws. Ofparticular concern is a concentration factor of2 or 3 at a location such as a hatch corner whichresults in mean stresses wel1 below yield pointbeing magnified beyond yield. A comparatiWl ysmal1 number of stress reversals (several hun-dred) can then lead to so-called “low cvcle,’fatigue cracking. The effect may be e~agg& atedby effects of corrosion. See Fig. 26 fromNibbering [55].

Ideally design for fatigue needs full-scalefatigue tests of all of the critical structuraldetails in the ship, using an irregular loadingspectrum and varying mean value correspondingto the ship’s expected lifetime experience [60].This is obviously not feasible, but much can belearned from such tests of representative detailscorresponding to situations of local stressconcentrations. Here the problem is primarilyone of controlling crack propagation, since--as pointed out by Nibbering and Scholte [60]--tiny cracks are almost always present at criticalpoints and can seldom be detected at an earlystage. Fortunately, such cracks propagate slowlyand can be detected and reDaired before damaaebecomes serious. (Guidance to ship operator; isneeded on this point.)

A current SSC project by Professor W. H.Funse is assembling data from structural engi-neering sources on details applicable to ships.Results should go a long way toward defining thecapability of local structure to withstand cylicloading. (See his paper at the present Symposium.)

The report of the ISSC Committee V.1, Oesign(1976) [92] develops a theoretical probabilisticapprOach that is intended to be a substitute forfatigue tests for checking fatigue strength of“primary structural members, such as deck, sideshell plating, .... in terms of crack initiation.”

u 1

u

‘t bFATIGuE LIFE

~ 6000 ‘PLAINMS SPECIMENS’3~= 5000

Fig. 26 Example of Application of CyclicLoading Curves to Study of Fatigue [55]

Attention is focussed on welded butt joints, anduse is made of experimental SN-curves (takinginto account initial imperfections and corrosiveatmosphere), Goodman’s correction for meanstress, and Miner’s law for assessing cumulativedamage to arrive at a “capability function”(defining fatigue strength).

The analysis then proceeds to predictfatigue failure (crack initiation) probabilityon the basis of cyclic demand consisting of anassunwd normal distribution of still waterbending moment (stress] and a comparativelysimple exponential distribution as an approxi-mation of wave-induced moment (stress). Resultsfor this ideal case are presented for a particu-lar ship in the form of a graph of probabilityof fatigue crack initiation vs. characteristicextreme value of wave stress in 108 cycles. Atable shows the effect of variations in certainrandom variables on the predicted probability.

The above procedure involves a number ofunce~taiqt~es, such as doubt regarding theappllc?blllty of Miner’s law, focussing oncrack Initiation rather than propagation, lack i.of consideration of high-frequency loads inthe demand spectrum, etc. But it represents apromising approach for defining fatigue capabilitythat presumably could be extended to the case ofgross design discontinuities such as hatch corners.Seeding [49] has also dealt with fatigue on aprobabilistic basis, considering cumulativedamage with respect to both crack initiationand propagation, including hatch corner effects.

Note that if the general application of morerational design standards should in time resultin reduced hull scantlings, then the incidenceof fatigue cracking might increase to an unaccept-able level. In this case some modification instrength standards by classification societiesWO.U1d be expected. The problem is to balante

~

initial cost against costs of crack repair, asdiscussed in the next section.

Choice of Probabi1ity Levels

The final step in establishing a completelyrational probability approach to design for bothultimate strength and fatigue will be thedetermination of an acceptable level of failure(or damage) probability that will give asatisfactory balance between demand and capa-bility.

One approach is simply to ascertain whatlevels of failure have occurred in the past andthen to select a suitable design value that isas good or better. This was discussed at lengthin report SSC-240 [7]. On the basis of aninformal study by J. F. Oalzell of some Lloyds’statistics on merchant ship losses (presumed tobe due to structural failure) it was concludedthat a figure somewhere between 0.003 and 0.006would be an estimate for the probability offailure that has been tacitly accepted over thepast twenty years for large ocean going ships.In proposing a specific figure for a new designcriterion, Oalzel1 suggests 0.001. This impliesthat merchant ships should be designed with aprobability of ultimate--or catastrophic--failureof no greater than 0.001, i.e., that a new ship

55

.

u:.-z nave a chance of not over one in a thousand:Z ‘ailure during a normal life span.

A similar approach can be taken to damage:0 longitudinal structure requiring repair (fromall causes, including fatigue). Hence, there isobviously a need for collecting more data onship structural damages and failures in relia-bi1ity format. This requires the compilation ofstatistics on ships suffering hull failures--orships lost in which structural failure is sus-pected--and relating these to ship-years ofservice. Such damage data must be analyzed to:

a) Separate damages affecting longitudinalstrength from al1 others.

b) Classify these as to type--fatigue,brittle fracture, or buckling.

c) Relate the data for a specific timeperiod to number of ships in service during thattime.

d) Collect data on estimated costs ofrepair of damages.

Another more basic approach discussed in[7] to determining the probability level to beused in a design criterion i5 that of “expectedcost”, which has been summarized in a convenientform by Freudenthal and Gaither for a plication

!to maritime structures in general [80 . It isbased on the principle that the best design isone that minimizes the expected total cost,where the latter consists of the sum of initialcost, possible cost of failure, and cost ofrepairs of less serious damages--such as fatiguecracks. Al1 damages not involving the main hullgirder are excluded, since presumably they wouldnot be affected by a“y change in the main hullstructural design.

Expressed as an equation, the total expectedcost to be minimized is,

L= I+ PIF+(l-P1)ZCP2 (23)

whet

c=

initial cost of the ship (or structure),probability of failure (in a lifetime),anticipated total cost of failwe(replacement cost + cargo 10ss + tem-porary charter of replacement ship +loss of business from customer reactions+ cost of pollution or other environ.mental effects, etc.),anticipated cost of damage or “failureof function of survivino structure”(“the success cost”), i~e., cost ofrepairs and other associated costs,

P2= the expectation or expected number ofsuch damages.

The probability, pl, of a failure that canlead to the complete loss of the ship is verylow. The expectation of other damage, pz, thatwould require more or less extensive time out ofservice for repair depends on any one of themodes of failure previously discussed--particularly,fatigue cracking and local buckling. In fact, aship might experience a number of such damagesin several modes during its lifetime.

After the probability of failure in alifetime is estimated for each fail”pe ~de, and

56

for each damage mode the expectation of damage,they should be multiplied by their respectivecosts in the above equation. In prin’iple,the total expected cost, L, can be evaluatedfor several alternative hull designs and theoptimum design determined graphically.

The above analysis assumes that safety ofhuman life is not a problem. But if it cannotbe assumed that lifesaving equipment willadequately protect human life, then statisticson loss of life in different industries andmodes of transportation can be used to determinerisk values to be used in design [93] [94].

CONCLUSIONS

It is concluded that great progress hasbeen made in recent years toward the goal ofrational design of the ship hull girder.Methods have been developed to predict theprobability per cycle of extreme bending momentsor stresses, and refinements of these methodscontinue to be made. One such refinement, ad-justing short-term responses for variations inthe number of cycles per unit time for differentship headings, has been shown here to be ofnegligible importance in the case of wavebending moment response. However, there are anumber of serious gaps in the procedure thatmust be filled before such an approach can beused in any other than a partial or comparativefashion. One of the objectives of this paperhas been to describe the ideal overall approach,and in so doing it has called attention to theprincipal gaps. These will be listed below asa guide to further research and study:

1. More complete ocean wave data on roughweather routes, particularly clarifying direc-tional properties with two or more seas combined.

2. More data on sea spectra for abnormalconditions of shoaling water, opposing currents,etc.

3. More statistical data on still waterand dynamic (vibrational)loads in service,both as to increases in extreme loads affectingultimate failure and dynamic cyclic loads affect-ing fatigue.

4. Improved techniques for combining loadsof differing frequencies that do not alwaysoccur simultaneously.

5. Further development of the probabilisticapproach.to,structural capability--including,(a) predict,on of ultimate strength and itsvariability; (b) prediction of fatigue strengthand its variability in relation to criticaldetails.

6. Collection bv classification societiesandlor regulatory bodies of ship damage andfailure data in reliability format.

Successful completion of the above wouldpermit the calculation with some confidence ofprobability of ultimate failure and of fatiguecrack propagation (beyond critical length) plusacceptable levels of these probabilities fordifferent ship types in the’most sever-eservicesfor which they are intended to be certified.

It should be emphasized that the completeideal procedure for evaluating the longitudinalstrength in reliability format envisioned here

~.

I

is not intended for routine application toindividual ship designs--except in the case ofunusual ships, or other marine structures, forfiich no empirical data are available. Itsrain value should be in continuing the develop-ment of more rational techniques for routinedesign--in particular, the refinement andincreased rationalization of the rules ofclassificationsocieties. This involves apply-ing the reliability principles to many differ-ent ship types, over a wide range of sizes, andcorrelating predictions of failure probabilityagainst actual experience in terms of hullfailure per ship-year “at r-isk.j’ Similarly,the frequency of fatigue crack propagation toa size requiring repair should be correlated,ith predictions.

ACKNOWLEDGMENTS

The authors wish to acknowledge theassistance of many members of the staff, pastand present, of the Center for Maritime Studiesat Mebb Institute of Naval Architecture fortheir many contributions to the work discussedhere, as well as the research support providedzy the Ship Structure Committee and American3ureau of Shipping over many years. Althoughraterial in the paper came from many sources,as indicated by the References, we wish tothank particularly Or. M. K. Ochi.of U. of Fla.,W. H. H. Chen of ABS and Mr. N. Maniar of?. Rosenblatt k Son for many valuable discussionssf topics presented hepe. The triennial (ISSC)internationalShip Structural Congresses haveCrovided a great deal of valuable data anduseful discussions, particularly in the meetingsof Comnittee 1.3 Oesign Loads and the newCzmtnitteeV.2 Oesign Proposals, chaired byK, V. Taylor of Lloyds’ Register of Shipping.Cvelyn Palumbo and Michael Rabinowitz of theCM3 staff provided valuable assistance in thecomputer calculations to determine the effectof including cycles per unit time in long-termcalculations.

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——

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59

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APPENOIX1

AOS- “ebb R.I.&. Lloyd. M.*. Veribs

swro.$ ofInfmnmtion Lwi*treml .1967 R.1.M.ReFortNo.1Z3, R.Gc.admn,SOUthhamQ-Jbtnhamw,‘RecentCa-m Prmedum by&.Gmmara,Onc. *C7Is~.ium,7970 v.lqmmtrinM. Prac-

19n fim-m~,Tin@. N.E.C.1. *ical %il.so*y of4%5 Sip Sirwti,-alLb.iwl,

w, Swing,!547;or( N.V.Publ.No.T/,7971

u.”,O.h

scum.: Pie.sm-Maskwitzfor Maldmdatuf..North H.gb.n& Lumb; C.+JIW %11,.6 U.ld.”,.A*NorthAtlmti. Atl.nti.[N.V.) datahen available. f.. NwthAtlentio

Fo.Fat: Obswv.dswdra P1ot- T.b.lwdistributionof T.bul..di.t?ib.+inn Cmti.xp....edinfomtedin WJup. of .- f+ri.ds and high% of pwicda and heights. of W.ibull distribution..stint.iy.ht.

v...Spectru u,.- Sa.c@i,gof~.~ulaiim

N.v. N.v. HodiFed Pie.s.m-Wa.k&..*1 apactlm f.. witzN9rth Atbntic

p~V,-c.n: Wdd ti.bw dcula. St.iptheorycalcula- Pr.fer =udal *..* (*w- Strip +Jmw o.lculati-

tiaIs. Calculmiic+M pm- tin. w ~d.1 t..ti ten’ti. . . randc.mwiti . . . . til bat.,fw..d. (wn.mllyml.ul.tim.) ?ey.s.icm-1.v.i.)

Striptheorycal..l.ticm.AllIwadirq.? Y+* n. [7+) Y*=,UPb !4*..Nd diatr.ofhemdin~Ewalp’cbnbility

P.EqualF.rc.hability Variablep~bili+y

w. of-“. lmlgth.Nomlly eqalPrc+.ab

13+ “p*O51 ~7[norr-llY)

Sort-t...Predict.R...lta i. t.m$of ..s Standwd (subject of v..Wsf.m for, .i. tt.m):STip .p..d. 4$7iP headings

1+ 47 7

2 (m.. 3)14 5 (mx.71

tin..f~cy$$ *..“mm10 8

Sig.=.. hei~ta 5}

~Y .U.b...16

shipler@.h.z!ml”..of

& 5 7 T~

ma-t.<.Pr.die+. 1.*.9..*.pb=abiliti.. I.teg..t. p.cb.biliti.. ln~.grat. probability.e Intq..t. prc+abiliti.sw+er ship h..ding .t W.. wave p..icd., WV. 4w. w... pcricd., WV. ave. w.. writ-d., *v.CLW.tantWW. height; ht.., +ip headirgs. heights, & ship headings; hts.; fit to H.ibull dieintagrat. .v.r -.. ht. and fair result w.F4li- t.. 6 Mt.g..t. over shiv

..1 ly . headings.

Probability lw.1 (W N - 108 f.. .~Par.ti.e Mg.p;ny ~ . ,Oz=d O.pmds .pw r.spc.m. un. N . 10° f.. .Orpe..tiveUu.be, of CWle, ) fw P..P... WIY. ultimb 10 de.studyD●*iw.

P.,P . . . . . Ult..al..de-valuede~end,w facto. wndam f..+..ofsa-.afaf.ty,etc. fety,.*C.

60

93,

94.

95.

Safet at Sea, ?roceedings of the Z“d~ean Conference on Mat-i”eTechnology, London 1977.

Report on ConnnitteeV.1 Design, lSSC 1979.

D. Hoffman, “Wave Data Application forShip Response Predictions,” Webb Instituteof Naval Architecture Report to GHRProgram, October 1975.

PROCEDURESFOR PREDICTINGLONG-TERMWAVE LOAOS [34]

H.%diw, %hiffwd Hafen, Wipbuildi.gResearchAs*c-J.M.Planeix,t+.Huth.r, A.J.Paxicadji,Pr-aoe.dinwCc+.1971,H.-S.$chultz, .i.timof.hpan,R.p.ar+ R.Cv+ois,‘Salicitati.msof*. R.m...hInstituteofJSW 1969. NO.69,s+t. 39P. .xtemwseti.tw.asde. Saafleet.

navi.e.. lam.,,ATMA, 1971,N lx and1972,N 1479P. V.V.Kezljakw.Sud..tr.ay..iy.

1*6,N 8

F(o116 WaldenforN.aAhA* Umld.nforNorthAtlanticR-11;Hcqben& Lu,Ib- N.A.Ianti.

Roll,s ** f..Nort$AtlanticHqbm & Lutob;Ymancuchi& @a..- Pscifi. I

di$trb.ofsig.hts.& p+?i- Tabulardist.ibutimof F..M*w Offi..dstaTat- Integral..w.ofthew-ababi-~. f..N.A.,P.....t.d9..- P.,iodsandheights. 1..ofw.iod.& hghti.Tat- lityof.xc..di.gofthem..FMc.llY.swml.tivedist.. 1..ofspwimlwds. stat...s. functimofH 34.

I. S. S.C.,with I.s.s.c.\ - ‘1/3

‘o’ -3% ’.’’(:$) ::j-’;c”,b“’““d m Obse’_-HI,31*aiqn.ht. (HA=733 H1,3)T itmans.PP.rm.icd

Strip +h6ery talc.bticm strip theo.y calculati.ms C6w.rally .*rie theory cd- Strip +hOrj cal..hticm.mhti.n.

57

77 5

22 8<5

usually1 s1

aL!m4ally6 8

5 6

Integ.ahpm+.abilitim m%~tiPrtibi Ii ties Caimminaahavt-ta. P.* Integmt.probabilities overwe. wave pri.ads,wave over WV. perids, WW. babilityf..eachu@,. w.. p..icd.,WV. heights,h% am shipheadings. hti.andshiphemdi.gs. *W~l ..w.,oultiply d!ipheadi.gs.

bypm+..of*w*um &tiding.xnt.gmti(%t.ms)

N -106 ?.. -rative tit N .701P*T-=...MIY

61

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APPENDIX 2

Ful1-Scale Midship Stress Measurement Programs

This 1ist includes ships on which the most comprehensive stress measurement and analysis programs havebeen done. It is not an exhaustive 1ist of all such efforts. An unpublished Webb literature searchin 1977 identified 124 papers describing measurements taken on about 150 ships, most of which containmore 1imited data. A sunanaryof some of the other available data was given in [34].

)

USA-ABS

uSA-SSC

USA-SNAMEBelgium

United KingdomJapan

Abbreviatims:

(1)Heading Angle

O Following306090 Beam Seas120150180 Head Seas

~

Universe IrelandIdemitsu MaruEsso MalaysiaR. G. FollisFotini LWolverine State and

Hoosier StateCalifornia BearMormacscanSealand McleanBostonEdward L. RyersonJordaensRoi BaudoinMineral Serai”gDart EuroDeEncounter”Bay ClassJapan AceWakahata MaruChidoruan k!aruJapan Adler

ECUS-East Coast U.S.5A- South America

Seas

~

TankerTankerTankerTankerBulkcargo

,,

CargoCargoCont.Cent.OrecargoCar FerryBulkCont.Cont.Cont.oreoreBulk

mPG/NEPG/JaP.PG/NEPG/ECUSPacif.N. Atl.N. Pac.N. Pac.N. Atl.N. Atl.N. Atl.G. LakesN. Atl.N. SeaNE/SAN. Atl.NE/AusPacif.Pacif.Pacif.Pacif.

NE- North EuropePG-Persian Gulf

y

1076.01069.31000.0754.6800.0496.0

!,

528.5458.0880.5496.D711.B479.5362.9715.3715.3745.5574.2787.4912.06B9.0

W

174.9163.3154.8104.5106.071.5,,

76.068.01D5.571.575.065.949.9105.0100.0100.0B2.7120.B146.0104.9

Aus.-Australia

APPENDIx 3

Short-term B.M. Response, SL-7 at 25 Knots, H 1/3=24.5 ft.

Mean(2)

B.M. Response,FT-TONS

105X1.76641.6B451.4B271.36161.49471.74121.B611

Instrumentedq Sea Time

0.860 11 Voy.0.830 14 Voy.0.830 13 Vov.0.820 18 VO~.0.840

0“:54) :5:

0.625’0.6300.590

0.6920.5420.8000.600

0.5660.B20.820.80

5 Voy

3 seasons2 seasons4 seasons

10 months14 ‘my

2 years

13891 records7352 records10517 records

Avg. Response for equal no. cycles each heading = 1.5964 x 105 FT-TONSAvg. Response for equal time each heading = 1.6029 x 105 FT-TONS

Ibte: P(c)*, column (5), is p(.) xl2. g

62

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