THE SOCIETY OF NAVALARcHITECTS AND MARINE ENGINEERsOne W.rld Trade Center, Suite 1369, New York,N,Y,10048

P.P.,s to be $.,esmted .1 Extreme LoadsRe,Pon,eSYm$msiurnAd!n@on,W Oc+rabe,19-2019s1

Combining Extreme Environmental Loadsfor Reliability-Based DesignsAlaa E. Mansour, University of California, Berkeley, CA

ABST&T AT= OF PAPER

I?JTRODUCTION

The nature of the present-day toolsand procedures for calculating the res-~nse of a ship to environmental loadsis such that only separate individualcomponents of the response can becalculated successfully rather than theTore relevant combined response. Forexample, several computer programs existfor calculating the stillwater bendingmoment and shearing forces using theusual static balance procedure, once the

.,0

.,0

.,,

weight distribution and the ship geome-try are specified. Other programs andtools exist for computing the verticalbending moment due to the motion of theship in waves considering it as a rigidbody. Although the same program may beused to Calculate the horiz~~t~l ~n~torsional mcments, the results are usual-~Y 9+Ven aS separate entities and aSpecial procedure must be adopted torelate them to each other to obtain thecombined response. Similarly, secondaryresponse due to hydrostatic pressureacting on portions of the hull are com-puted using other tools and procedures.Even within the higher frequency, primarydynamic loads and responses such asslamminq and springing, con!putational

Fig. 1 Typical voyage variation of midship “ertical bending stress for a hulkcarrier. [1]

63

—

“0”,.,...7,,,.,!-

H ,,!,,,-,mm,*T”,*,

Fiq. 2 Decomposition of a stress time history of a Great Lakes vessel into low andhigh f>equency components.

methods are sufficiently different togive each a separate entity. The res-ponse due to thermal loads (e.g., air/water temperature variations ) is anotherexample where a drastically differenttool such as a finite-element programmay be necessary to obtain the response.

The reason for tbe separate cal-culations of the load/response compon-ents is not jwst due to computationaldifficulties. More important ly, thereason stems from differences in thebasic analytical approaches to eachload/response component and, whether ornot, such approaches fall within certainanalytical disciplines.

A ship designer is, therefore,faced with the important problem ofcomputinv the total response from theindividual components, which are cal-culated using separate programs. Forexample, he may calculate the stillwaterbending moment using the Ship HullCharacteristics Proqram, the wave “er-tical, horizontal and torsional momentsfrom a rigid-lmdy ship motion program,the higher frequency springing momentfrom a flexible-body vibration program,the slaving response from a slamminyprogram, the thermal response using afinite-element program, etc. If all ofthese load/response components were staticin nature, the problem would have beensimple, because only the magnitude anddirection of each load/response compon-ent would be necessary to obtain thetotal response. If the total responseinvolves dynamic cyclic individualresponses~blem becomes alittle more complicated since attentionmust be given to the - relationbetween the different response compon-ents. However, the actual problem thatfaces a ship designer, involves dynamic= individual responses that requirethe additional complications of deter-mining the degree of correlationbetween the different components and themethod of combining them which are thesubjects of this paper.

Undoubted y, there are certainsimilarities between decomposing responserecords of full-scale measurements intotheir basic components and cowk.in’incjana-lyticallycdculatexl components to obtain tbtotal response. Since decomposing

64

full scale measurements can be done witha certain degree of success, the approachtaken in this paper is to invert theprocedure in order to compute the com-bined response from the analyticallydetermined components.

In the next section of this paper,a brief discussion is given of the de-composition of full-scale records intotheir basic components. In the followingsection, a method is developed to combineanalytically-determined response com-ponents. In the following sections,application examples are given and SOmeconsiderations for reliability-baseddesigns are discussed together withsome concluding remarks.

DECOMPOSITIONS OF !KSASUP.ED RECORDS INTOTHEIR BASIC COMPONENTS

A typical measured stress timehistory of a bulk carrier is shown infigure 1 (from reference [1]). usually ,such a record consists of a rapidlyvarying time history of random amplitudeand frequency, oscillating about a meanvalue. The mean value itself is aweakly time-dependent function and mayshift from positi”e to negati”e (saqgingto hogging) . The two dominant factorswhich affect the mean value are (seereferences [81 and [91):

1. The stillwater loads which canbe accurately determined fromthe loading condition of.theship floating in stillwater.

2. The thermal loads which arisedue to variations in ambienttemperatures and differencesin water and air temperatures.

A closer look at the rapidly “aryinqpart shows that it also can be decomposedinto components. Figures 2 and 3 illus-trate records taken over shorter periodsof time (larger scale) . Two main cen-tral frequencies appear in these records.The smaller central frequency is associatedwith the loads resultinq from the motionof the ship as a rigid body (primarilyheave and pitch motions) . ‘This lowercentral frequency is, therefore, closein magnitude to the wave encounterfrequencies for wave lenqth nearly equal

-

VW*,Em 3,,1.3-

,m.L s,.,,,

Tmm. ?.s.)

i ,..,,,..,”.,m....,.,

?ig. 3 Decomposition of a stress timehistory of an ocean going bulkcarrier.

< to ship length. The higher centralfrequency is associated with loads re-sulting from the two-node mode responseof the ship when it flexes as a flexiblebody . This higher central frequency isthus close to the two-node mode naturalfrequency of the ship. Tbe high fre-quency response itself can be due to“springing” of the flexible ship whenexcited by the energy present in thehigh-frequency wave components as shownin figure 2. It can be due, also tothe impact of the ship bow cm the wateras the shin moves into the waves. i.e. .slamming (~ossibl,ytogethe,rwith” low-speed machinery -reduced vibrations) ,see figure 3. Though springing andslamming may occur simultaneously, it isunusual to see records which exhibitboth clearly. These two responses canbe distinguished from each other byinspecting the records V en”elope. Ingeneral, a decaying envelope (see figure3) indicates a slamming response whereasa continuous envelope 0f varying ampl i-tude, as shown in figure 2, indicatesa springing response.

The rigid body and the highfrequency responses do not always occursimultaneously in the same record. Quiteoften only the rigid body responseaPPears in a record; particularly, inthat of a smaller ship “hich has a hightwo-node mode frequency. Occasionally,only the high frequency springing res-ponse appears in a record when a shipis moving or resting in relativelycalm water. In particular, long flexi-ble ships with low natural frequenciesas those operating in the Great Lakesdo occasionally exhibit such recordswhen operating in calm water or in alow sea state composed mainly of shortwaves. Under these conditions, a longship will not respond as a riuid bodyto the short waves, but the two-nodersde frequency of the hull can besufficiently low to be excited by theenergy content of these short waves.Figure 4 (from reference [2]) showsa measured response spectrum of alarge Great Lakes vessel where theresponse is purely in the two-node

Ii=0 0.3120 0.72150.9165

FREQuENcy[HERTZI

Fig. 4 Stress response spectrum of alarge Great Lakes vessel. [2]

mode’ and higher frequencies with norigid body response appearing in thespectrum. The figure shows that responseat higher modes than the two-node modecan be measured, although smal1 andrelatively unimportant in most cases.

Slamming response on the otherhand never occurs separately withoutriuid hodv resuonse since. obviouslv,it-is a r=sult-of the riqid motion ~~the ship in the waves.

cOMBINING ANALYTICALLY DETESMINEDAESPONSE COMPONENTS

w inverse procedure, of that fordecomposing full scale measurementsinto the individual components, canbe adopted for combininu individuallycalculated response components. mmmain steps should be used in the pro-cedure for combining p~ responses.

Step 1. To combine the low frequencywave-induced responses(rigid body) with the high-frequency responses (spring-ing or slamming) .

Step 2. To add the mean value tothe response resulting fromStep 1. The mean value con-sists of the stillwater andthe thermal responses.

~:

Consider an input-output system inwhich the input is common to severalcomponents of the system; it is reguiredto determine the sun of the individualoutputs . Here, the input representsthe waves which can be in the form ofa time history if a time-domain analysisis sought, or in the form of a seaspectrum if a freauency-domain evaluationis preferred.

1 The two-node mode is labeled in thefigure as the first rode,

p_65

r -—-— -————-——- -— —_,1

)\

II

II

L_–--__-_-– _-– ~ii - i

Fig. 5 Schematic representation of amultiple system with cormnon input.

The components of the systemrepresent the components of the loadresponse of the ship to waves, e.g. ,the low-frequency wave-induced responseswhich consist of vertical, horizontaland torsional moments, the high frequencyresponses such as the springing loads,etc. It is required now to determinethe sum of these component responses,i.e. , to determine the output taking intoconsideration the proper relations orthe appropriate correlation of theresponse components (see references[81 and [9]).

Schematically, the procedure isrepresented by figure 5. IIIthis figure,‘“n”parallel linear components areconsidered which have common input’ ~ (t)and are summed up at the output to formy (t) The output of each system ismult~plied by a constant ai(i = 1, 2,..,n)before surnning up all the components ata common node to form ~(t) . These con-stants ai give additional flexibilityin the application of the model and canbe used to “weigh” the contribution ofeach linear system to the sum.

In a time domain, the output ~ (t)is given by the sum of tbe ~onvolutionintegral of each system.

jmhc(T)~(t -T)dT0

where

he(T) = ~ ah.(T)i=l 1 1

(1)

(2)

h. (T) is the impulse response function ofehch linear system, i.e. , the responseof each linear system to unit excitationmultiplied by time (response to theDlrac delta function) . h (T) is a com-posite impulse response ffinctionwhichsums the responses of the individualcomponents.

It should be noted that the impulseresponse functions of the individualcomponents ,,hi(T)‘<may or may not beeasy to obtain depending cm the complex-ity of the system. With suitableinstrumentation, it is sometimes pOs-sible to obtain a good approximationto h; (T) exDerimentallv. For the shiDsyst~rn, hi (~) can be d~termined for ‘most load components.

In a frequency domain analysis, asimilar procedure can be wsed. Infact , since the system function H. (w)is simply the Fourier transform oth, (t), i.e. ,. .

Hi(w) = / hi (t)e-jwtdt (3)o

therefore, we can define a compositesystem function Hc (w) as

.

HC(UJ)= ] hc(t)e-]wt dto

‘i~laiHi(w) (4)

It should be noted that for a singlesystem, the relation between the inputspectrum and the output spectrum isgiven by the usual relation:

[Syy(UJ)li= Sxx(UI)H*i(W)Hi(LU)

= Sxx(w) lHi(w)[2 (5)

where Sxx (w) is the sea spectrum whichrepresents .5common input, [Sis a response spectrum of an Yx&’:iAualload component, H*i (u) is the complexconjuqate of the system function ofa response component.

The mod.ulw of the indi”id”al systemfunction IHi (w)I is the response ampli-tude operator of the individual responsecomponents, i.e. ,

[R.A.O. ]i = [P(LU)]i= lHi(w)~ (6)

and, therefore, equation (5) representsthe familiar relation between the inputand the o“tp”t spectra of a singlelinear system.

For our composite system, an equa-tion similar to equation (5) can bedetermined for the n-response conrpenentsand the “weight” factors “ai” asfollows

Syy(lu) = Sxx (IJ)H*C(IJ)HC(UI)

n= sxx(ti)~ J aiajHi(m)H*j (M)

i=l 3=1

(7)2

A tilda over a variable indicates arandom variable.

66

—.—.

The double sum in equation (7) can beexpanded such that a final expres-sion for the total response spectrumSW (u), which combines the individualresponse spectra, may be written inthe form:

SYY(LO)= [; a.’/Hi(ti21Sxx(u)u)i=~~

+ [? t a.a.H. (W) H*j(IO)lSxx(ILI)i=~ j=~ ~ 1 1

(i#j)(8)

It should be noted that the firstterm in equation (8) represents simplythe algebraic sum of the individualresponse spectra, each modified by thefactor ai. Tbe second term, which canbe either positive or negative, repre-sents a corrective term which dependson the correlation between the loadccmmonents as can be seen from themul~iplication of Hi (u) by the complexconjugate of Hj (u).

If the system functions Hi (u) donot overlap on a frequency axis (i.e.,disjoint systems) , that is, if

Hi(LO)H*j(u) = O (9)

then the second term in equation (8)becomes zero and the load components areuncorrelated. In this case, the totalspectrum is simply the algebraic sum ofthe individual spectra of the load com-ponents, modified by the factors ai.Furthermore, if the wave input is con-sidered to be a normal random processwith zero mean, as usually is the case,then the respective output load responsesare jointly normal and are independent.Thus, the total response, in this specialcase, is a zero mean normal process witha mean square value given by:

co.

= ~ a.’ flHi(@)12Sxx(u)du (lo)i=l 1 0

In the more general (and morerealistic) case where some or all of theresponse components are correlated, themean square is given by

2 = ~-Syy(w)do‘Y

= ~ ai’ J-lHi(u) 12Sxx(~)dui=l Q

+ ~ ~ aiaj~i (IJ)H*j (u)sxx(u)dwi=l j=l(i#j)

(11)

It should be noted that ineWatiOns (10) and (11) the mean squareis taken equal to the variance of thecombined response since the mean valueof the wave responses is usually verysmall. As noted earlier, the otherresponses which consist mainly of thestillwater, and the thermal loads willbe added later in the second step of theanalysis to form a mean value for step1 combined responses.

In connection with equations (10)and (11) it should be mentioned thatthe usual Rayleigh multiplier used toestimate certain average quantities,such as average of the highest one third,one tenth response, etc. , are notgenerally applicable in the case of thecombined respnse, since these multi-pliers are associated with a narrow-band spectrum for which the amplitudescan be represented by a Rayleigh dis-tribution. It is only in the casewhere each of the load component res-ponses is narrow-band and each happen tobe closely concentrated around a conmmncentral frequency “u “, that it w~be reasonable to con~lude the combinedresponse SW (u) is, itself, a narrow-band process.

In the more general case, wherethe combined response spectrum is nota narrow-band spectrum, the variousstatistical quantities can be determinedfrom a more general distribution whichincludes the Rayleigh distribution asa special case. The general distribu-tion can be found in [3,41, and thefactors which, when multiplied by theroot mean square, givq the variousstatistical quantities, can be foundin [4,51.

Equation (11) can be written in adifferent form which is mre convenientto use in applications, and which makesit easier to define tbe correlationcoefficients between the differentresponse components.

where

Ui’= ~lHi(kJ)I‘Sxx(w)dw = variances or0mean squares of bhe responsespectra of the individual loadcomponents.

(13)

Pij = correlation coefficients ofindividual load components

67 $----

defined by,

.Pij = + f Hi(L@H*j (ti)Sxx(w)dw (14)

110

Equation (12) with definitions (13)and (14), which are derived fromequation (11), form the basis for com-bining the step 1 responses of a shiphull girder in a frequency domain analy-sis taking into cons ideration the cor-relation between the response components.If the response components are uncor-related, i.e., if P. . = O, the secondterm in equation (1m drops out andthe variance of the combined responses(output] is simply the algebraic sumof the individual variances modifiedby the factors a. , As discussed earlier,this occurs “henLthe system function ofthe various components do not overlapin frequency or overlap in a frequencyrange where the individual responsesare small. On the other hand, if theindividual components are perfectlycorrelated, pi may approach Plus orminus unity and the effect of thesecond term of eauation (12) on thecombined load va~iance SY2 can besubstantial.

The physical significance of thecorrelation coefficient can be furtherillustrated by considering only tworesponse components for simplicity. If

P12 is l?rge and positive (i. e.,approaching +1) , the values of the tworesponse components tend to be bothlarge or both small at the same time,whereas if PI

?iS large and negative (i.e.,

approaching - ), the value of oneresponse component tends to be large whenthe other is small, and vice versa.If P12 is small or”zero, there iS littleor no relationship between the two res-ponse components. Intermediate valuesof p12 between O and *1 depend on howstrongly the two responses are related.For example, the correlation coef-

ficient 0?3”Of the vertical and hori-

zontal ben xnq moments acting on a shipis expected to be higher than of thevertical and springing moments sincethe overlap of the system functions inthe latter case is smaller than in theformer case.

In a time domain analysis, theconvolution integral represented byequation (1) with the compositeimpulse response function as given byequation (2) form the basis for deter-mining the combined load response. Thequestion of whether the time or thefrequency domain analysis should beused depends primarily on what form therequired input data is available. Ingeneral, most wave data and practicalanalysis are done in a frequency domain,although in some cases where slanuningloads are a dominant factor, it may beadvisable to perform the analysis in timedomain.

6S

E!sP-2:In this step, the stillwater and

the thermal responses shcmld be.combinedto form the mean value for the rigidbody motion and higher frequency responses,The Stillwater and thermal responses areweakly time-dependent variables so thatin a given design extreme load conditionthey can be considered constants, sayover the duration of a design storm.Therefore, these two responses can betreated as static cases and can becombined for one or several postulateddesign conditions without difficulty.Alternately, if statistical data areavailable for each of these responses,the mean and variance of the combinedresponse can be easily determined.

The stillwater response can beaccurately determined for all loadingconditions using computer programs suchas the Ship Hull CharacteristicsProgram. Several postulated, extremebut realistic, weight distributionscan be assumed in the final stages ofdesign, and the corresponding Stillwaterresponse can be computed.

If a statistical description of thestillwater bending moment is adopted,data have shown that the qeneral trendassumes a normal distribution for con-ventional types of ships. A samplehistogram based on actual ship operationdata for a containership from reference[6] is shown i“ figure 6. A mean valueof the stillwater bending moment can beestimated based on this histogram forall voyages, or for a specific routesuch as inbound or outtound voyages.

.0. 0,. ..”.,,,!,,s

201

I10

I

0.,,0.....

-1

I

._.

.iL

STlm..m. m.”. [“w<,.., 100, r,.-,..s

D rmma... vox&c,s ,5, . . . . .

m meaumVWACES ,60“o”. 1

Fig. 6 Histogram of stillwater bendingmoment of a container ship.

Ih-l+5 1

&

b

~

Measured *WS----AI,-S,.?,.,,—

,7-

Fig. 7 Correlation between measured stillwater bending moment and air/seatemperature difference.

These mean values together with a setof standard deviations, wbicb can alsobe estimated from tbe histogram, canbe utilized to determine tbe extremetotal mmuent using a statisticalapproach.

Since the stillwater response, therigid body motion response, and thehigher frequency responses are allfunctions of the ‘ship weight and itsdistribution, it is preferred that thecombined response be calculated for agroup of selected loading conditions(and selected temperature profiles).

Primary thermal respomse is uswsllyinduced by differences in water/airtemperatures and by variations inambient temperatures. Figure 1 showsthat a consistent diurnal stress var-iation of magnitude of 3-5 ksi has beenrecorded onkoard a bulk carrier. Astudy of full-scale stress data measuredon a larger tanker indicates that thediurnal stress variations correlatewell, as shown in figure 7 withtemperature differentials beteen airand sea.

Taking the North Atlantic route asans%ample, the average diurnal changeof air temperature is about 10” F.The total diurnal change of deck platingtemperatures may vary from 10° to 50°F,depending upon the cloud cover condi-tions and the color of tbe deck plating.For estimating the thermal loads on aship bull, tbe sea temperature may beassumed as constant. Once the tem-perature differential along a shiphull is determined, the thermal stresses

can be calculated, using either ageneral purpose finite elenvent computerprogram or a simplified two-dimensionalapproach. The maximum thermal responsemay be then added to the stillwaterresponse for certain Pstulated desiqnconditions to form the mean value forthe low and high frequency dynamicresponses.

Although high thermal responses maynot happen in high seas, a heavy swellmay possibly occur under a clear sky.Therefore, several temperature condi-tions are to be taken intaconsid~ation indetermining the cotiined design response.

APPLICATION EXAMPLES

Since step 2 of combining theenvironmental loads acting on a shipimply a simple superposition of staticloads (either as constants or as randomvariables) only step 1 of conbiningthe dynamic random loads is consideredin the following application examples.

Generally, large tankers travelingin oblique seas may enccmnter horizontalbending moments of the same order ofmagnitude of the vertical bending moments,Therefore, the combined effect of thevertical and horizontal moments can becritical under certain conditions. Thedistribution of the primary stress inthe deck as a result of the combinedeffect becomes non-uniform and assumesa maximum value at one edge. The com-bined stress at the deck edge, gc isgiven by

_-

69

y , ~(t)$==v ‘h

(15)

where,

&c =

iv(t) =

~(t) =

Sv,sh=

combined edge stress

the vertical bending momentcomponent

the horizontal bending momentcomponent

the vertical or the horizontalsection modulus, respectively.

Defining the cotiined moment as thecombined edge stress multiplied by thevertical section modulus and “sinqequation (15), we can write,

tic(t)= %c(t)Sv = &v(t) + k!ih(t)

where,s (16)

K=:h“

APPlying equations (12), (13) and(14) and extending the results for thecase of a two-dimensional sea spectrum,tbe mean squ;re value of the combinedresponse u can be written as (seeequation (1%) :

2= OM”z + Kzc& + 20”hKc7‘Mc Mv”Mh (17)

where ‘JR 2 are the mean squarevalues o~’t~dv~%ical and horizontalbending moments, respectively, qi”en byequation (13) as’ :

Using equation (14) the correlationcoefficient is defined as :

- ~ ~WSxx(OJ,LdH”(w)H*h(@dwdP (20)

2

For simplicity of notation, the de-endence of the R.A.O. ‘S IH (u)I and

~Hh(IJJ)\ on the ship heading~ng withrespect to wave components is droppedfrom the notation.

lHv(.T%s~7:,n::t:%e~:::;;y,the system functions H“ (u) and Hh (w)can be determined from any typicalrigid-body ship motion computer program.

It should be noted that in eq”aticm(12) , the coefficient al was taken asunity and a2 was taken equal to

%a2

=K=r.,1

(see equation (16)) in determiningequation (17). It should be also notedthat the integrand in equation (2O)contains tbe information regardingthe phase between the horizontal andvertical moments and that the inte-gration of such information withrespect to frequency and the angle ubetween a wave component and the pre-vailing wave system leads to thedetermination of the correlation coef-ficient as given by equation (2o).Finally, the root-mean-square (r.m.s. )~of the coribined edge stress is qivenbv

(rms) = *c“(21)

where OMC is the r.m.s. of the combinedmoment given by equation (17).

The r.m.s. of the combined momentand the correlation coefficient asgiven by equations (17) and (20),respect ively, were computed in reference[5] forT;e3ar~~tanker of DWT 327,ooOtons. . . values of verticaland horizontal moments were computedusing a rigid body ship motion computerprogram and conbined using equation (17)to obtain the r .m.s. of tbe combinedmoment. The results of the calcula-tions are plotted in figures 8 and 9versus the ship heading for the casesof long-crested and short-crestedwaves, respectively. Significantwave heights of 7.4, 19.2 and 43.3 feetwere considered in order to examinethe general beha”ior of the responsesin low, moderate and high sea states.The discussion of these results inreference [5] indicated that thehorizontal bending moment is not smallcompared with the vertical bendinumoment (see figure 9). In severe seas,the maximum response of the combinedmoment (and the vertical moment ) isin head and following seas. Figure 10shows the variation of the combinedbending moment with the sea state asobtained by three different methods.The first is based on equation (17) .

and Pvh qiVen by~~~~t%~ ~!!Q), (19), and (20) res-pectively. The second method is basedon equations (17), (18),and (19) also,but with a correlation coefficient Pvhequal to O.32 obtained from the 1973lSSC Proceedings (determined empirically).The third method is based on P“h = O.53

I

70

Fig. 8 Variation of vertical andcombined wave bending moments withheading in long.cre$ted SeaS.

15

R

L,;:,,,s“ .,.FT.

10 M,. .

5 ‘ .,.-.vmlwR..,..’JLL*,.,, ‘,“,”,,CAL.0.,.,

0MODE,,,,SE,,7H’”.,,,2,,,

5(,0.,,,0,”,,, “c

3A “hM“

~ +~120 150 180

HEADING[DEGREES)

Fig. 9 Variaticm of “ertical , horizont~and combined moments with headingin short-crested seas.

as determined by averaging the responsesin short crested seas as determinedfrom equation (17) for all headingsand for the three representative seastates. The mean value of Pvh obtainedin this manner was O.53, significantly?.igherthan the IS&C “aIue. Iiowewr,the effect of Pvh cm the t-.m.s. com-iined moment is small, as Can be seenfrom figure 10.

As a second application example,=!e combined vertical and springingmment will be considered next. Using:requency domain analysis and using~ations (12), (13) and (14) with

al = a2 = I, we obtain the mean Square‘ralueof the combined response ~’o”=z”:x long-crested seas as

Z=g‘Vs

“’ + ~~z + 20“~ovo~ (22)

,o~S1GN!FtC&)4TW&VEHE,’HT,FT)

Fig. 10 Comparison of combined bendingmoment responses as computed bythree methods.

where,

2=‘v

2=as

PVs =

mean square of verticalbending moment

.mean square of the springingmoment.

f Sxx((u)IH5(UI) I’dw (24)0correlation coefficient

1[“’sxx(U)Hv(tJ)H*~(u)du (25)

‘Vus A

and H (u) and H (u) are the COr@eXsyste~ functions of the vertical wavemoment and springing moment respectively.The system response functiOns ~=n becomputed winq computer programs whichtake into consideration the ~ff~ctSof ship flexibility in the response,such as tbe spring ing-seakeeping program

~~e’~j2j5] . Application of, (23) and (24) to se”eIaI

Great Lakes Vessels where springing isimportant is gi”en in references [5]and [71 and will not be further discussedhere. These equations together “ithequations (1) and (2) for the time-domain analysis have a wide range ofappli~ability to any two or moredynam~c random responses including,combining primary and secondary’

4 Such as primary inplane loads ongrillages d“e to overall bending of thehul1 and secondary lateral pressurearising from the randomly varying watersurface.

!+71

responses IS), vertical, and torsionalmoments for shiDs where torsion isimportant, high” frequency loads withvertical and horizontal moments, etc.In all of the cases , the coefficients amust be appropriately determined. Whenldetermining the various statisticalaverages from the combined r .m.s.values, the more ‘general distributiongiven in [3], [4], and [5] for thepeaks should be used instead of theusual Rayleigb distrib”ticm.

DESIGN CONSIDERATIONS

For design purposes, operationalenvelopes must be postulated to projectthe loads<and responses of the hull.The operational en”elopes normallyinclude design variables such as seastate, ship speed and heading, shiploading condition, and thermal condition,A, good discussion of this isgiven ln references [8] and [9]. Fora reliability-based design, either ashort- or long-term procedure may beused. In the short-term procedure,only one or two se”ere design storms(sea states) are considered to~etherwith one or two ship loading conditionand thermal condition. IIIthe long-term procedure, the entire missionprofile of the ship must be postulatedincluding ship route, sea states eXPeC-ted to be encountered during operation,expected total years of service, etc.In either procedure, if a strictprobabilistic approach is used, thenthe extreme values of the combinedresponse must be estimated using aprobability measure and compared withthe ultimate strength of the hull todetermine the reliability level or therisk of failure. The basic methodologyof these two procedures was developedin reference [10] and is described inmore detail in that reference for loadsthat consider the static stillwater andthe dynamic (rigid body) vertical wawsbending moments only. The basicmethodology can be extended to includethe effects of the combined loads asdescribed briefly in this section.

In recent applications of theprobabilistic approach and structuralreliability to ships , it has beenfound that the shert-term procedure isuseful because of its simplicity andbecause it is less demanding in termsof the required input data and analysistime. In fact, in a postulated designsea storm, ship loading condition andthermal condition, the expected maximumcombined dynamic bending moment “BMmax,,in N maxima 5 during the storm, can

$ The negative maxima should be includedin N for “se in equation (26).

72

be calculated from [2]:

= ~{[ln(l _E2;AN,’/2E [B~axl 0

‘A - ‘/2+ 0.2886 [ln(l-E2) N] } (26)

Where m is the zero moment or thearea uncler “the combined responsespectr~ of the dynamic loads, i.e. ,m where o is given by equations(?lY ~~ (12) ; E )(s the bandwidth para-meter defined by

E2=1_~‘0m4

and

mm = ]-W’”Syy (u)duo

= f a,’ ~-UJmlHi(UI)IzSxx(u)dui=l 1 0

+ ~ ~ a.a, /mumHi(w)H*j (m)Sxx(w)dui=l j=l 1 10

(i#j)(27)

The static stillwater and thermalresponses should be then added to theexpected maximum combined dynamicresponse given by equation (26) beforeincluding it in any safety analysis(see for example reference [61) . Othermethods within the short-term analysiscan be used such as the distribution-free method given in reference [11] inconjunction with equations (8), (11)and (12).

On the other hand, the long-termanalysis is also useful and is essentialif the fatigue fracture modes are tobe included in the reliability analysissince the fatigue reliability partrequires the full loading and responsehistories. If a long-term full probabil-istic method is used, a general Weibulldistribution may be suitable torePre Sent the combined loads. TheWeibull distribution is given by

~ ~ L-1 -(x/k)L

fx(x) = E(E) ex >0

FX(X) =

where

x=

fx(x) =

~ - ~- (x/k)L x >0—(28)

random variable representingthe combined wave bendingmoment amplitude,

probability density fwmtionof x,

L

““l

FX (x) = distribution function of X,

L and k = Weibull distributionparameters.

The Weibull probability paper canbe used to fit the data and determinethe parameters L and k. Fox the com-monly used Weibull paper, the slope ofthe straight line which fits the datais “L” and the intercept with theaxis is “-L log k,’. Thus k and L canbe determined. Other methods forestimating L and k can be used suchas the method of moments or the maxi-mum 1ikelihood method.

Extrapolation of the availabledata to obtain the extreme values overthe entire ship life can be estimatedusinq several methods. One of thesemethods is based on order statistics.Using the Weibull distribution as aninitial distribution and including thestillwater and thermal bending nmnwnts.in the manner described earlier inthis paper, the probability law ofthe total extreme bending moment Zncan be written in the form:

z-ml-mm

L -1 - (~)IILZ-l!l~-~ ~

OZn (z) = ~(~)

.[l -

@zn(z) = [1 -

where

Zn =

‘1 =

‘m ‘

‘Zna“d Ozn =

z- ml-m L~i k ‘) ~n-l

z-ml-mm L,-(~) 1.

(29)

random variable repres-enting the total extremebending moment,

V.al”e of the stillwaterbending moment for aselected loading condition,

value of the thermalbending moment for aselected temperaturecondition,

the probability densityand distribution functionsof the random variable Zn.

Equations (28) or (29) must bechecked to see how well they fit theavailable data before they can be usedin design. In fact, at the presenttime, it is recommended here to usea semi-probabilistic approach such asstrength reduction and load magnific-ation factors [61 or a distribution-free method [111 until the validityof extrapolation procedures and the

fit of data in the distributions aredemonstrated when sufficient data arecollected.

CONCLUDING RE1.V@.SS

For reliability-based designs ,the load element must include thestillwater loads, the thermal loads,the low-frequency wave induced loads,and the hiqh- frequency loads. Theseloads can be combined as described inthis paper, and the “maximum’s combinedload should reflect an envelope of allpossible extreme values during the lifeof the vessel. The short- and long-term procedures of the reliabilityanalysis are briefly described in theprevious section for the case of thecombined load as developed in thispaper.

The second element in a reliability-based design is the strength or theultimate carrying capacity of the hull.This has not been discussed in thispaper, but reference [6] evaluatesbriefly the variables which introduceUncertainties in the hull strength.

At the present time, it is impor-tant that several levels be investi-gated in the strut’cu-al reliabilityanalysis of ships, whether tbe long-term or short-term procedure is used.The lack of lona records and sufficientlylarge amounts o? data necessitate such -diversified analysis.

Three distinct levels of structuralreliability may be wed at the presenttime [6]. In the first level, whichis closest to the procedures currentlyused in practice, the concept of aload magnification and a strengthreduction factor is used. Such factorsare determined frorcthe standard devi-ation or the coefficient of variationof the available data which providesa measure of the uncertainty orvariability in the desi~n parameter.The higher the degree of uncertainty,the higher these factors are, andtherefore, they are usually taken asmultiples of the standard deviationsof the “ariables. Reference [6] givesa more detailed discussion of thismethod.

The second level of str”ctualreliability (the distribution-freemethod] also relies on computing onlythe mean and standard deviation of thetotal combined bending moment and ofthe ult:mate collapse moment. TheUncertainty in estimating! the designparameters is thus reflected by thestandard deviations of the variables.On this basis, a safety index can bedetermined (see reference [11]). Thesafety index can be mathematicallyrelated to the probability of failureif the probability distributions of L--,.

73

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the combined bending moment and theul’cimate collcipsemoment are known.However, in *he absence of snch infor-mation, consistent designs can beachieved using the safety index on thebasis of some, though unspecified,reliability level.

The third level of structuralreliability is based on a strictlyprobabilistic concept. The combinedextreme bending moment and the ultimatecollapse moment are considered asrandom variables and are fitted intoaPPrOPria~e probability distributionssuch as given by equations (29) for theload. By this means, the reliabilityfunction @ conversely the risk offailure can be constructed and used asa measure of safety.

Combining the static and dynamicloads as described in this paper orby any other means is an essentialstep in any of the three reliabilitylevels briefly described above. It isrecommended, however, that only the firsttwo levels of the reliability analysisbe used at the present time. The useof the strict probabilistic reliabilityconcept is difficult at the presenttime because the available data aretoo limited to provide the exact formsof the probability distributions ofthe loads and strength or to provideadequate data to check the extrapolationprocedures to the extreme values. Thesample size required is of the order ofmultimillion pieces of records or data.In addition, the first two levels areconsistent with the simplified hydro-dynamic and structural analyses currentlyused to predict the sea loads and thestructural response. Finally, therules and procedures for designers nmstremain uncomplicated, thus simplifiedanalysis such as given by the firsttwo reliability levels is moreappropriate.

REFERENCES

1. R. S. Little, E. V. Lewis and F.C. Bailey, “A Statistical Studyof Wave Induced Bending Momentson Large Oceangoing Tankers andBulk Carriers, “ Trans a SNAMI,1971.

2. M. Critchfield, “Ev.al”ation of HullVibratory (Springing) Responseof Great Lakes Ore CarrierI@ Stewart J. Cort, ” DTMBReport 4225, Nov. , 1973.

3. S. O. Rice, “Mathematical Analysis~ f Ra~dOm Noise, ” Bell SystemTechnical Jounal, Vol. 23, 1944and Vol. 24, 1945.

4.

5.

6.

7.

8.

9.

10.

11.

D. Cartwriqht and ii.Longuet-Iliggins,“The Statistical Distributionof the Maxima of a Random Function, ”Proceedings of the Royal Societyof London, V=. 237, 1956..—

s. G. Stiansen and A. E. Mansour,“Ship Primary Strength Based onStatistical Data Analysis, “ Trans.SNAME, 1975.

S. G. Stiansen, A. E. Mansour, H.Y. Jan and A. Thavamballi, “Reli-ability Methods i; Ship Structures, ”The Naval Architect, Journal ofRINA, JUIY 1980.

S. G. Stiansen, A. E. MansOur andY. N. Chen, ‘SDynamic Response ofLarge Great Lakes Carriers toWave-Excited Loads, ” Trans. SNAMR,1978.

E. Lewis, D. Hoffman, W. Maclean,R. VanHoOff and R. Zubaly, ,,LoadCriteria for ship StructuralDesicp, “ Ship Structure ConnnitteeReport no. SSC-240, 1977.

J. Dalzell, N. Maniar and M. Hsu,“Load Criteria Application, ”Ship Structure Committee Reportno. sSC-287, 1979.

A. E. Mansour, “Probabilistic DesignConcepts in Ship Structural Safetyand Reliabilityr “ w. SNAME,1972.

A. E. Mansouz’, ‘“ApproximateProbabilistic Method of Calcu-latihq ShiD Longitudinal Strenath, “Journil of-~“Rese.arch, Vol. ’18,— ——No. 3, Sept. 1974.

ABSTRACT

The application of probabilisticand reliability methods to ship hullstructures requires the estimation ofone or more extreme loading conditions:Each consists of components such as stl31-water load, thermal load, low-frequencywave-induced load, and high-frequencySpringinq or slamming load. Combiningsuch a qro”p of time-dependent loadcomponents of various frequencies hasalways presented a problem to shipdesigners. The stillwater and thermalloads, though (weakly) time-dependentvariables, are of sufficiently differentcharacter from wave loads to necessitateseparate treatment. Tbe wave loadsthemselves consist of low-frequencywane-induced and high- frequency Slannninqor springing loads which require thesummation of random time series of dif-ferent frequency compositions. Thispaper examines and presents an approachsuitable for determining the combinedextreme response to the load componentsand the statistics of the sun.

74 ‘\