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design practice, especially if this refers to new designs.r 2007 Elsevier Ltd. All rights reserved.

Keywords: Tanker structures; Longitudinal strength; Structural reliability; Stochastic model of loads;

Uncertainties

1. Introduction

Oil tankers are being dimensioned on the basis of the rules of classication societies.Ship rules rely strongly on past experience and good engineering judgment, partiallysupported by theoretical analyses and model tests. These types of structural design rulesare deterministic rules, where all the uncertainties of the pertinent design parameters arecompensated by a single safety factor [1]. Experience with new ships in service has shownthat the structures designed in accordance with those deterministic ship rules are fairlysafe, i.e. structural damages in new ships are rare, especially ultimate failures of the hull-girder. It is also well known, however, that such rules are hardly applicable for the newtypes of structures or new methods for load or strength evaluation [2].

Nowadays, classication societies have been developing semi-probabilistic rules, basedon partial safety factors derived from the reliability formulations (PSFs) [3]. PSFs reectuncertainties in each of the pertinent variables instead of using only one global safetyfactor. Ideally, PSFs should be calibrated in such a way as to produce more uniform safetylevel when a large number of conventional ships is considered [1]. However, similarly to

their deterministic predecessors, rules based on PSFs are not readily applicable toindividual designs where there is not enough accumulated experience.Typical examples of such innovative design without feedback from past experience are

oil product tankers of new generation, which are presently under construction. The newgeneration oil tanker is a novel design, characterized by unusually low length-to-beamratio. One of the most interesting features of this ship design is that the design verticalwave bending moments are calculated by direct hydrodynamic and statistical analysis. Thisapproach is different from common practice, where design values of this load componentare determined by IACS UR S11 formulae [4]. Since new generation oil product tankerrepresents a new design and also the advanced method for load evaluation is employed,

obviously the present rules for the construction of ordinary merchant ships may notprovide good means to assess their safety. The main problem is the fact that the safetymargin between load and resistance in traditional ship rules is not explicitly quantied [2].

The reliability methods represent convenient means to quantify safety margin betweenload and resistance in novel ship designs. These methods are intended as a tool thatrationally takes into account uncertainties in loading, response and structural strength.The basic idea of reliability methods is to represent pertinent variables as randomvariables, each with an associated distribution type and parameters. After that, usingsuitable procedures, the probability of structural failure (or its counterpart, the safetyindex) may be calculated.

Hull-girder reliabilities of the new generation oil tanker and the conventional double-hull oil tanker are compared in the present paper. As the new generation oil tanker isdesigned based on vertical wave bending moments from direct hydrodynamic analysis, it isparticularly interesting to assess the consequences of such approach on ship safety. This

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question is nowadays very important, since there is a tendency to introduce direct designmethods in ship structural design [5]. The inuence of other structural features of newgeneration oil tankers, as unusually large ship breadth, that are not evident by puredeterministic analysis, may also be identied by reliability assessment.

In the present study, the reliability formulation and assumptions similar to thoseoriginally developed within the SHIPREL project, have been considered [6]. The basis of this reliability formulation is the ultimate collapse bending moment of the midship crosssection for one year of operation.

2. Description of analyzed ships

2.1. The new generation oil product tanker

The series of new generation oil tankers, ordered by one of the leading shippingcompanies, are presently under construction. The main particulars of the ship are reportedin Table 1 .

The specic characteristics of this tanker are lower draught and higher speed comparedto the traditional, medium-range oil tanker. The breadth of the new product tanker ismuch larger than of ordinary tanker of similar size (40 m compared to 32.2 m) resulting in30% increase of cargo capacity. Lower draught would enable easier approach to ports andreduced risk of grounding, and this should be a step forward in the protection of theenvironment. Another advantage of the new generation product tanker is that the ship isequipped with two independent propulsion systems in two separate engine rooms, twinpropeller shafts and two rudders. Since the machinery system failure is one of the majorcauses of tanker accidents, this tanker is obviously much safer than the conventionaldesign.

Since the hull form of the new generation product tanker deviates signicantly from thetraditional oil tanker, the rules for the construction of merchant steel ships may not bedirectly applicable. Rather, both the ship owner and the Classication Society requiresome additional analyses based on the rst principles. One of the most importantadditional analyses is the computation of design vertical wave bending moments by directhydrodynamic analysis.

The reason for this requirement is that the application of Rule formulae is limited toordinary hull forms with the ratio of ship length and beam higher than 5. From the mainparticulars of the new product tanker, it appears that the ratio Lpp /B reads 4.4 being muchless than 5. Therefore, IACS UR S11 for ship longitudinal strength is not applicable. Theperformed hydrodynamic and statistical analysis is described in detail by Parunov et al. [7].

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Table 1Main characteristics of new generation oil product tanker

Length between perpendiculars Lpp 175.5 mMoulded breadth B 40 mMoulded depth D 17.9mScantling draught T 13.0mDeadweight DWT 65200 dwt

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The main conclusion of that study is that the design vertical wave bending momentscalculated by direct analysis are 20% higher than the corresponding values from IACS URS11. The principal difference between conventional tanker and new generation tanker isthat wave bending moments from direct calculations are adopted as design values for thelatter while the former is designed using rule values. Consequently, the hull structure of thenew generation tanker is more robust and safer than the structure of other tankers. Theconcern of this study is to quantify this increase in robustness by means of reliabilityanalysis.

In the design check of the new generation tanker, conventional ship rules have beenemployed without considering the ultimate bending capacity as a design criterion. Initially,the midship section modulus at the strength deck is optimized to meet IACS UR S11 forlongitudinal strength. During the project development process, the section modulus wasincreased in order to meet IACS UR S11 with the rule wave bending moment replacedby the results of the direct hydrodynamic analysis. Therefore, the use of wave bendingmoments from direct calculations in ship design is the only reason why the longitudinalstrength of this vessel is above the minimum rule requirements.

2.2. The rule oil tanker

The rule ship analyzed in the present study is an existing large double-hull tanker fullysatisfying the contemporary rules for design and construction of steel ships including IACSUR S11. The particulars of the rule tanker are presented in Table 2 .

Direct hydrodynamic analysis for determination of extreme vertical wave bending

moments is performed for this ship using the same assumptions as for the new generationoil product tanker. Details of the analysis are presented in [7], where the overestimation of the rule wave bending moments by results of direct analysis is found to be even 30%. Theoverestimation of the rule wave bending moment is larger than in the case of newgeneration tanker, indicating that the rule wave bending moments may not ensureconsistent safety level through different ships. This inconsistency is another motivation toperform reliability analysis, as rational tool to deal with this problem.

3. Reliability formulation

When reliability methods are applied to assess structural safety, load effect componentsand structural strength components are considered as random variables. Demand on thestructure and structural capacity are related through a mathematical expression, known asthe limit state equation. That expression denes if the structures fulll their intended

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Table 2Main characteristics of rule tanker

Length between perpendiculars Lpp 270mMoulded breadth B 48.2mMoulded depth D 23.0mScantling draught T 17.1mDeadweight DWT 166300dwt

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purpose regarding a particular failure criterion or not. When considering longitudinalstrength of an oil tanker, it is usual practice to assume that ship hull behaves globally as abeam under transverse load subjected to the still water and wave-induced load effects.Among the load effects induced, the vertical bending moments at midship are of primaryimportance. The ultimate bending moment is considered nowadays as the most realisticlimit state condition concerning longitudinal strength of the ship hull [3]. The ultimatebending moment takes into account the loaddeection characteristics of the stiffenedpanels composing the ship hull, including their post-buckling behavior.

The limit-state equation with respect to hull girder ultimate failure under verticalbending moments, considered in the present study, reads:

wuM u M sw c wwwnl M wo 0, (1)where M u is the deterministic ultimate hull-girder bending moment; M sw the random

variable extreme still-water bending moment; M w the random variable extreme verticalwave bending moment; c the load combination factor between still water loads and waveloads; wu; ww; wnl the random variables representing modelling uncertainty of ultimatestrength, linear wave load and non-linearity of wave load.

The reliability analysis according to limit-state equation (1) is performed separately fortwo independent failure modessagging and hogging. The hull-girder reliability in each of the two failure modes is calculated for three elementary loading conditionsfull loadcondition (FL), ship in ballast (BL) and partial loading condition (PL). For rationalreliability assessment, the percentage of time that a ship spends in each of these loadingconditions has to be estimated. This estimate is made based on the statistical analysis of

load duration data for tankers performed by Guedes Soares [8], which is presented inTable 3 .

Annual safety indices bFL ; bPL ; bBL , and associated failure probabilities P f ;FL ; P f ;PL ;P f ;BL are calculated for each of the elementary loading conditions separately for failures insagging and hogging. The global annual safety index for each of two failure modes bt isthen determined as

b t F 1P f ;FL P f ;PL P f ;BL. (2)Furthermore, the reliability assessment is performed for as-built ships as well as for

corroded hulls, assuming 20-year corrosion according to ship rules [9]. The summing of failure probabilities in Eq. (2) is possible if the assumption is adopted that the operatingconditions are statistically independent. However, even if the correlation of failures indifferent loading conditions exists, Eq. (2) is still a good approximation if one of the threeloading cases dominates. It is well known that this is exactly what happens for oil tankers,where full load dominates for failure in sagging and ballast in hogging.

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Table 3Operational prole adopted for tankers

Load cond. Harbour Full load Ballast load Partial load

Percentage of spent time 15% 35% 35% 15%Voyage duration (days) 23.5 23.5 2.0


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The failure probabilities are calculated using the PROBAN program, which is part of the SESAM package for structural, hydrodynamic and reliability analysis [10]. In thisstudy, the probability of an event, given by limit state equation (1) is calculated usingFORM (rst order reliability method) implemented in PROBAN.

The sensitivity analysis is an important part of the reliability assessment since it enablesidentication of parameters that have the most important impact on safety indices. Thesensitivity factors used in this study are those dened as

a i

@G @ yi yi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi @G @ yi

2

yi s

, (3)

where G (.) is the failure hyper plane in the reduced space of the standard normal variables yi . Variables yi are the coordinates of the point in the reduced space closest to the origin.After transformation of yi in original space, the most probable failure point x i is obtained.The sensitivity factors calculated by PROBAN are further normalized to 100% sum.

4. Uncertainty model of ultimate vertical bending moment

The ultimate vertical bending moments of the example ships are calculated usingprogressive collapse procedure proposed by Bureau Veritas [9]. Calculations are performedfor two states of the hull:

as-built ship with gross thickness of structural elements, i.e. thickness of structuralelements as they are built-in in a new ship.

corroded state of hull, which is the anticipated state after 20 years of ship service. Forthe corroded state, the ultimate strength calculation is performed with netthickness, i.e. gross thickness reduced by the effect of corrosion. The thicknessreduction due to corrosion is taken according to BV rules [9], where corrosion reductionof plates and longitudinals is between 1 and 2.5 mm, depending on the location of thestructural element. For comparison, the new common structural rules (CSR) [3] for oiltankers propose corrosion reduction of plates and longitudinals for ultimate strength

calculation between 1.25 and 2 mm depending on the location of the structural element.Therefore, the differences between the two approaches for corrosion deduction aresmall with almost negligible inuence on the ultimate bending moment capacity, whichis also conrmed in practice. It is also worth noticing that applied CSR corrosiondeduction is not full corrosion for local strength assessment, but reduced corrosionproposed by the Rules specically for global strength assessment.

The results of the ultimate bending moment calculations of as-built and corrodedhulls are presented in Tables 4 and 5 , respectively.

The calculated moment is assumed to be the expected value of the ultimate bendingmoment. Therefore, the mean value of the random variable wu is taken equal to unity. Thisassumption is based on calibration by experiments of identical method for ultimatestrength calculation [11]. In most recent reliability studies, wu is assumed to follow a log-normal distribution with COV 0.15, which is adopted also in the present study. This


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COV takes into account both the uncertainty in the yield strength and the modeluncertainty of the method to assess the ultimate capacity of the mid-ship section, sinceboth variables contribute to the ultimate moment. The coefcient of variation of the yieldstrength of the steel normally range from 8% to 10%, while the additional modeluncertainty is assessed by engineering judgement, bringing the overall coefcient of variation to 0.15 [12].

5. Stochastic model of still-water bending moment

A Gaussian distribution is used as the stochastic model of still-water bending momentfor one voyage. The parameters of distribution were calculated from the loading manual,according to the method proposed by Guedes Soares and Dogliani [13]. For each of theelementary loading conditions (full load (FL), ballast (BL), and partial loading (PL)) themean values and standard deviations were calculated separately for departure and arrivalconditions. The mean values and standard deviations of the resulting normal distributionswere then obtained as average values for departure and arrival.

When the mean value msw and the standard deviation s sw of the normal distribution areknown, the extreme value distribution for a given time period T C may be approximatedusing Gumbel distribution with the following parameters [12]:

xe F 1sw 1 1nsw ; a 1 F sw f sw , (4)

where nsw is the number of occurrences of a particular load condition in the referenceperiod T C (1 year). F sw is the cumulative probability distribution, F 1sw its inverse while f swis the probability density function of normal distribution with parameters msw and s sw .According to the operating scenario presented in Table 3 , n

sw 5:4 for full load and

ballast, while nsw 27:4 for the partial loading condition. The mean value mse andstandard deviation s se of the Gumbel distribution are then given asme xe a 0:5772, (5)

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Table 5Ultimate bending moments of corroded ships, MNm

Sag Hog

New generation tanker 3911 6075

Rule tanker 11150 13118

Table 4Ultimate bending moments of as-built ships, MNm

Sag Hog

New generation tanker 4812 7238Rule tanker 12702 14832


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s e p

ffiffiffi6p a. (6)

Table 6 presents the parameters of the stochastic model of the still-water bendingmoment for one voyage and one year calculated according to the described procedure.

Surprisingly large differences in variability of still-water bending moments in ballastloading condition for the two ships may be noticed in Table 6 . Still-water bendingmoments for the new generation tanker change considerably during voyages. Thus, forballast loading condition the still-water bending moment at departure reads 1780 MNm,while at arrival its value is only 1300 MNm. The variability of still-water bending moment

for rule tanker is signicantly lower. For ballast condition, the SWBM at departure andarrival reads 3771 and 3609 MNm, respectively. Large variability of SWBM in ballast fordouble-hull tankers is documented also by Guedes Soares and Dogliani [13] where it isnoted that the SWBM may even change sign during the voyage. Obviously, the presenteddata of SWBM may not be considered as generally representative for double-hull tankers,since signicant differences between individual ships may be found. This conclusion justies the methodology of determination of parameters of the SWBM statisticaldistribution using individual ships loading manual rather than some general statisticalmodels.

Most of the previous reliability studies indicate that variability of SWBM is generallynot so important as the variability of wave loads and structural strength [14]. For thatreason, small model uncertainty of SWBM from loading manual compared to in-servicedata is neglected in the present study.

6. Stochastic model of vertical wave bending moment

6.1. Random variable extreme vertical wave bending moment M w

For both ships, the evaluation of wave loading was carried out by means of linear striptheory program WAVESHIP, a part of SESAM package [15]. The long-term distributionof vertical wave bending moment is calculated by POSTRESP program, which is also partof the SESAM package [16]. The POSTRESP uses standard long-term predictionmethod that considers the total lifetime response history as a series of short-term episodes.

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Table 6Parameters of still-water bending moment distributions, MNm (sagging SWBM is negative)

Ship L.C. One voyage One year

msw s sw mse s se

New generation tanker FL 615 310 1016 276PL 133 286 722 167BL 1177 512 1841 455

Rule tanker FL 1968 569 2706 506PL 384 957 2352 560BL 3829 87 3943 78


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Short-term responses are combined by a procedure that takes into account the relativeamount of exposure to the various levels of sea severity.

The computed long-term distribution is highly dependant on the assumptions used in theanalysis. The choice of wave scatter diagram containing the probabilities of occurrence of various short-term sea states is the most important parameter in the long-term predictionof extreme wave loads. The characteristic values of vertical wave bending moments fromthe long-term distributions may differ up to 50% when different wave climate descriptionsare used [17]. Assumptions of heavy weather manoeuvering, such as ship speed reductionand course changes, could also have large impact on tails of long-term distributions [18].To standardize the procedure for computation of extreme wave loads, IACS has issuedRecommendation Note No. 34 as guidance for statistical analysis [19]. The basicassumptions proposed by IACS for calculation of long-term distribution of wave bendingmoments are:

The IACS North Atlantic scatter diagram should be used. This scatter diagram coversareas 8, 9, 15 and 16, as dened in Global Wave Statistics (GWS) [20]. The data fromthe GWS are further modied by IACS in order to take into account the limited wavesteepness more properly [21].

Only ship speed equal to zero is to be taken into account. The well-known two-parameter PiersonMoskowitz spectrum (ITTC spectrum) isrecommended.

Short-crested waves with the wave energy spreading function proportional to cos 2Whave to be used. All heading angles should have equal probability of occurrence and maximally 30

1

spacing between headings should be applied. Proper corrections for non-linear effects are to be applied.

A Weibull 2-parameter model is usually used to approximate the long-term probabilitydistribution of vertical wave bending moment computed by the above described procedure:

F x p 1 e x p

y l

, (7)where y and l are the Weibull scale parameter and the shape parameter, respectively. It is

well known that the Weibull distribution is an excellent approximation of the amplitude of various ship responses in waves. F x p in Eq. (7) represents the probability that theamplitude of the response variable is less than x p in one randomly chosen cycle. Theprobability that the response amplitude remains less than a given value xe over a longertime period, e.g. 1 voyage, 1 year or 20 years, is given by the Gumbel law

F xe e e

xe xea (8)

whose parameters xe and a are derived from the parameters of the Weibull distribution (7)by the following relationships:

a yl ln n1 l =l , (9)

xe yln n1=l . (10)

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Symbol n in (9) and (10) represents the number of response cycles in a given long-termperiod, while xe calculated by (10) represents the most probable extreme value in n cycles.The mean value and the standard deviation of the Gumbel distribution are given byEqs. (5) and (6), respectively. The Gumbel distribution obtained by this procedure isactually the inherent uncertainty of extreme vertical wave bending moment, as representedby random variable M w in Eq. (1). The coefcient of variation of this uncertainty is usuallyin the range of 610%.

The long-term distributions of vertical wave bending moments calculated for threeloading conditions are shown in Figs. 1 and 2 for the new generation oil tanker and therule tanker, respectively. The linear IACS UR S11 vertical wave bending moments arealso included in Figs. 1 and 2 as horizontal lines. Table 7 describes the stochastic models of the vertical wave-induced bending moment for application in hull-girder reliabilityassessment.

It is interesting to notice from Figs. 1 and 2 that the relative magnitudes of VWBM aredifferent for FL, BL and PL as a consequence of differences in hull form and massdistribution. It is also obvious that for both ships the partial loading condition leads to thelargest VWBM comparing to other loading conditions. This phenomena may be explainedby the fact that partial loading conditions considered in the present study are thoseresulting in the largest still-water bending moments and shear forces among all loadingconditions. Consequently, such loading conditions result in the largest wave-induced loads,as well. It should be kept in mind, however, that other non-homogenous partial loadingconditions could result in different values of VWBM. Another aspect of the problem is thesensitivity of VWBM to the draught-to-length ratio ( T /L) [22]. Due to the Smith correction

factor, namely, the loading conditions with lower draught, as ballast conditions, couldhave larger wave pressure exerted on the ship bottom that would tend to increase VWBM.

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0

500

1000

1500

2000

2500

1 2 4 5 6 83 7

-log 10 (Prob)

V e r

t i c a

l W a v e

B e n

d i n g

M o m e n

t [ M N m

]

FL

BL

PL

IACS

Fig. 1. Long-term distributions of VWBM of new generation oil product tanker.


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calibrated specically for the strip theory, while different uncertainty models would be

more suitable for the 3D method. In this way, the results of reliability assessment areactually independent on the choice of seakeeping method for wave load assessment.

6.2. Uncertainty of wave load evaluated under linear assumptions ww

Simplications, assumptions and inaccuracies of the linear engineering models used topredict extreme wave loads on ship hulls are taken into account by the modellinguncertainty ww, that appears in Eq. (1).

Direct consequences of the modelling uncertainty are large discrepancies in long-termpredictions of extreme wave bending moments performed by various researchers or

institutions for the same ship, i.e. the same physical reality. The obvious example of thesediscrepancies is the comparative study of long-term distributions of vertical wave bendingmoments performed by leading classication societies for the containership with noforward speed [4]. The maximum and minimum predicted lifetime vertical wave bendingmoments differed by as much as 80%. These differences could have a signicant impact onship design practice, since direct calculation methods are obviously not sufcientlystandardized to assure a consistent level of ship structural integrity.

For the present study, ww is assumed to be a normally distributed random variable withthe mean value equal to 0.9 and coefcient of variation equal to 0.15. This statistical modelis mainly based on the benchmarking study for the program WAVESHIP [25]. In thatstudy, the modelling error parameter f , dened as the ratio of the lifetime bendingmoment based on the measured transfer functions and the lifetime bending moment basedon the calculated transfer functions, is determined for a number of ships and shown inFig. 3 . The modelling error may be simulated by a normal distribution with the mean value

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0

1

2

3

4

5

6

7

1.50.90.80.70.60.5 1.41.31.21 1.1

modelling error

n u m

b e r o

f o

b s e r v a t

i o n s Experimental data

Normal distribution

Fig. 3. Variability of the modelling error parameter.


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of 0.9 and the coefcient of variation of 10%. The coefcient of variation of ww is nallyincreased to 0.15, accounting for other uncertainties of linear engineering models.

6.3. Uncertainty in non-linear effects wnl

The most important source of non-linear behavior in wave loads are the non-verticalship sides. The main consequence of this non-linearity is the difference between saggingand hogging vertical wave bending moments. Using linear theory, it is not possible todistinguish the sagging and hogging bending moments since only a single linear bendingmoment is obtained. Usually, the sagging bending moment is higher, while the hoggingbending moment is lower than the linear prediction. The non-linear hydrodynamicmethods, such as quadratic strip theory, are capable of predicting differences in saggingand hogging bending moments [26]. Despite the existence of such methods, the linear strip

theory is still the most popular tool for seakeeping computations. Non-linearity, especiallythe difference in sagging and hogging wave bending moments is approximately taken intoaccount by introducing non-linear correction factors F S and F H for sagging and hogging,respectively. In that way, the non-linear bending moments in sagging M S and hogging M H are obtained as M S F S M L and M H F H M L , where M L is the vertical wave bendingmoment computed by linear analysis. In this work, the non-linear correction is assumed tobe the same as in the IACS UR S11, which means that the ratio between the wave bendingmoments in sagging and hogging depends only on the block coefcient C b. It is furtherassumed that the linear value of vertical wave bending moment is the mean value betweensagging and hogging wave bending moments. By applying these assumptions, one obtains

the correction factors of wave bending moments in sagging and hogging as

F S M S M L

2R1 R

; F H M H M L

21 R

, (11)

where R represents the ratio of the vertical wave bending moments in sagging and hoggingfrom IACS UR S11:

M S M H R

C B 0:71:73 C B

. (12)

For the reliability assessment in the present study, the uncertainty of non-linear effects

wnl is assumed to be a normally distributed variable with mean value equal to non-linearcorrection factors calculated by Eqs. (11) and (12). The coefcient of variation of thisuncertainty is assumed to be 0.15 [12,14].

7. Combination between still water and wave bending moment

The load combination factor c that appears in Eq. (1) is used to account for the fact thatthe extreme values of wave loads and still-water loads do not occur at the same time.Accordingly, the total extreme vertical moment must be somewhat reduced. The loadcombination factors are usually calculated using the Ferry-Borges and Castanheta loadcombination method based on an assumed operating scenario [27]. In the present analysis,the load combination factors used for the reliability assessment of the oil tankers werethose calculated by Guedes Soares and Teixeira [12]. These load combination factors aregiven in Table 8 .


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It should be noted that these values do not change signicantly for different amplitudesof the load variables within the same operational prole and outcomes of various loadcombination studies for oil tankers lead to similar results [28].

8. Results of reliability analyses

The annual safety indices are calculated by the PROBAN program using the FORMmethod [10]. Safety indices b are calculated for two different ship states (as-built shipand corroded ship according to 20-year corrosion from ship rules), for three differentloading conditions (full load, ballast and partial load) and for two separate failure modes(hogging and sagging). The summary of the stochastic model adopted is shown in Table 9where the notation adopted in Eq. (1) is used.

8.1. Results of reliability analyses of as-built ships

Results of the reliability calculations for as-built state of hulls are presented in Table 10together with the associated global annual failure probabilities P ft calculated by Eq. (2).

As may be seen from Table 10 , the full load condition dominates for the sagging failuremode for both ships. For the hogging failure mode, ballast is the most important loadingcondition. Partial loading condition is of less importance, although it may generate someextreme wave and still-water load effects, as elaborated in 6.1. The reason is that tankersgenerally spend less time in partial loading conditions compared to ballast and full load.

Consequently, the number of cycles of wave loads and the load combination factors arereduced for partial loading condition. Such considerations are not used by conventionaldeterministic approach, which points out the advantage of using the reliability methods.

From Table 10 , one may clearly see the benets of using direct hydrodynamic methodsin design of an oil tanker. The hull girder reliability of a new generation tanker issignicantly higher than the reliability of the rule tanker for both sagging and hoggingfailure modes.

Although the calculated reliability index of 2.25 for as-built rule tanker in sagginglooks rather low, other researchers obtained similar results. For example, the averageannual reliability index of three as-built large tankers in sagging reads 2.21 according tothe analysis presented by Teixeira and Guedes Soares [28]. The same authors calculated theaverage value of reliability indices for as-built bulk-carriers in sagging as 2.13 [12]. Thisleads to the conclusion that the reliability indices calculated in the present study arecomparable to other similar analyses.

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Table 8Load combination factors

Load condition c

FL 0.92BL 0.91PL 0.80


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8.2. Results of reliability analyses of corroded ships

The results of the reliability calculations for corroded hulls are presented in Table 11together with the associated global annual failure probabilities P ft calculated by Eq. (2).

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Table 9Summary of stochastic model adopted (sagging SWBM is negative)

Variable Distribution Mean COV

M sw (MNm) Gumbel New generation tanker FL 1016 0.27PL 722 0.23BL 1841 0.25

Rule tanker FL 2706 0.19PL 2352 0.24BL 3943 0.02

M w (MNm) Gumbel New generation tanker FL 1734 0.09PL 1730 0.10BL 1717 0.09

Rule tanker FL 6019 0.09PL 5790 0.10BL 5659 0.09

M u (MNm) Deterministic New generation tanker As-built Sag 4812Hog 7238

Corroded Sag 3911Hog 6075

Rule tanker As-built Sag 12702Hog 14832

Corroded Sag 11150Hog 13118

ww Gaussian 0.9 0.15

wnl Gaussian Sag 1.03 0.15

Hog 0.97 0.15wu Log-normal 1 0.15

c Deterministic FL 0.92PL 0.80BL 0.91

Table 10Safety indices in sagging and hogging for as-built ships

Ship Cnd. bFL bBL bPL P ft bt

New generation tanker Sag 2.97 6.01 5.63 1.49E03 2.97Hog 6.89 3.50 5.90 2.33E04 3.50

Rule tanker Sag 2.25 5.11 4.86 1.22E02 2.25Hog 5.10 2.88 3.81 2.06E03 2.87


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Some interesting conclusions may be drawn from the results for corroded hulls thatare presented in Table 11 . The annual failure probability of corroded rule tanker insagging is about 2.5 times higher than the corresponding value of the new generationtanker. It may be noted that the ratio of failure probabilities of rule tanker and newgeneration tanker is much reduced compared to as-built ships, where this ratio readsmore than 8. The relatively large breadth of the new generation tanker may explain thisphenomenon, since the main deck has signicant impact on the strength characteristics of ahull-girder. When such large deck is corroded, the hull-girder properties are rapidlyreduced as well as the reliability index with respect to the ultimate bending moment. Theseresults emphasize the importance of the corrosion-protection system and its inuence onthe safety of new generation tanker.

Despite the effect described above, it may be noted by comparing Tables 10 and 11 , that

the reliability of corroded new generation tanker in sagging b t 2:06 is only slightlylower than the reliability of as-built rule double-hull tanker b t 2:25. This is alsoone illustrative example how, within the framework of reliability-based methods, relativeincrease of safety of new generation oil tanker may be described compared to thetraditional design.

9. Sensitivity analysis and parametric study

9.1. Sensitivity analysis

Sensitivity analyses are performed for cases that resulted in the lowest safety indices, i.e.full load in sagging and ballast load in hogging for corroded hulls. The sensitivity factorsa i are calculated by PROBAN according to Eq. (3) and further normalized to 100% sum.These normalized sensitivity factors are presented in Tables 12 and 13 . In the same tables,the coordinates of the design point xi , representing the most probable combination of random variables in the case of failure are also included.

By analyzing the results in Tables 12 and 13 , it is obvious that the uncertainty of ultimate longitudinal strength is generally the most important random variable. However,for the new generation oil tanker in ballast condition, the importance of the vertical still-water bending moment overcomes that of the ultimate longitudinal strength. It is alsoobvious that for new generation oil tanker the importance of the wave-induced loadvariables ( M w, ww and wnl ) decreases from full load to ballast. Similar conclusions were alsoobtained by Guedes Soares and Teixeira [12]. For the rule oil tanker, however, the sameconclusions are not valid as the importance of the still-water bending moment is quite

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Table 11Safety indices in sagging and hogging for corroded ships (20-year corrosion from ship rules)

Ship Cnd. bFL bBL bPL P ft bt

New generation tanker Sag 2.06 5.46 4.99 1.97E02 2.06Hog 6.34 2.81 5.17 2.48E03 2.81

Rule tanker Sag 1.65 4.76 4.46 4.95E02 1.65Hog 4.72 2.26 3.27 1.25E02 2.24


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marginal for both failure modes. Also, the importance of wave-induced load variablesdecreases only slightly from full load to ballast compared to the new generation tanker

case.This example is another evidence of the advantage of reliability methods compared todeterministic or even semi-probabilistic methods. Partial safety factors in semi-probabilistic rules should reect the relative importance of pertinent variables. However,it is obvious from the presented analysis that the relative importance of wave-induced andstill-water loads may vary considerably between individual ships of the same type.Therefore, only direct reliability analysis performed for each individual design may providesufciently accurate information about the importance of the uncertainty of the pertinentvariables.

9.2. Parametric study

To estimate the mean values and COVs of modelling uncertainties in a rational manner,large amount of collected full-scale measurement data from actual ships would berequired. Since this would be quite an expensive and time-consuming request, theresearchers are faced with only a limited quantity of available data. Consequently, theparameters of modelling uncertainties ww, wnl and wu specied in Table 9 represent the bestestimates that might be changed if more data for comparison of full-scale measurementsand numerical computations were collected. To explore the implications of these potentialvariations of parameters on the calculated safety indices, a parametric study is performed.This is done in such a way that COVs and the mean values of modelling uncertainties arevaried within reasonable range of their values. As in the sensitivity analysis, the parametricstudy is performed for ships in corroded states, for the sagging failure mode in full loadcondition and for the hogging failure mode in ballast condition. Only one of the

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Table 13Sensitivity factors and coordinates of design points for ballast load in hogging failure mode (corroded hulls)

Ship wu ww wnl M w M sw

New generation tanker ai ; % 35.1 3.6 3.6 1.6 56.0x i 0.77 0.97 1.05 1745 3066

Rule tanker ai ; % 57.0 16.5 16.5 9.9 0.1x i 0.77 1.02 1.10 5954 3936

Table 12Sensitivity factors and coordinates of design points for full load in sagging failure mode (corroded hulls)

Ship wu ww wnl M w M sw

New generation tanker ai ; % 43.2 12.5 12.5 6.1 25.7x i 0.81 1.00 1.14 1786 1285

Rule tanker ai ; % 46.6 18.6 18.6 9.3 6.8x i 0.84 0.99 1.14 6202 2835


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parameters is varied in each reliability analysis, while all the others retain their values asspecied in Table 9 .

9.2.1. Variation of COVsThe best estimate of COVs of ww, wnl and wu reads 0.15 according to Table 9 . In the

parametric study, the safety indices are calculated also for COVs of 0.1 and 0.2. The resultsof the parametric study are presented in Tables 14 and 15 for the new generation tankerand the rule tanker, respectively.

As may be seen from Tables 14 and 15 , the COV of wu has the most important inuenceon the safety indices in all cases. The inuence of COV of wave load uncertainties ww andwnl is much less important. Furthermore, it may be noted that the same value of safetyindex is obtained if COV of ww or wnl are varied for the same amount. These results are fullyconsistent with the sensitivity factors specied in Tables 11 and 12 : the sensitivity factor for

wu is the largest in all cases, while the sensitivity factors of ww and wnl are exactly the samefor both ships in all loading conditions and failure modes.

9.2.2. Variation of mean valuesThe parametric study of the mean values is performed using the following mean values

of modelling uncertainties:

mean value of wu between 0.95 and 1.05 (the best estimate in Table 9 is 1.0); mean value of ww between 0.85 and 0.95 (the best estimate in Table 9 is 0.9); mean value of wnl equal to 1.14 in sagging and 1.12 in hogging. (the best estimates inTable 9 are 1.03 and 0.97 in sagging and hogging, respectively).

While the means of wu and ww are small variations of their best estimates, the meanvalues of non-linear correction factors wnl are signicantly larger for both sagging and

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Table 14Safety indices for various coefcients of variation of modelling uncertainties (New generation tanker, corrodedhull)

COV wu COV ww COV wnl0.1 0.2 0.1 0.2 0.1 0.2

bFL (Sag) 2.39 1.75 2.14 1.97 2.14 1.97bBL (Hog) 3.15 2.46 2.84 2.77 2.84 2.77

Table 15Safety indices for various coefcients of variation of modelling uncertainties (Rule tanker, corroded hull)

COV wu COV ww COV wnl0.1 0.2 0.1 0.2 0.1 0.2

bFL (Sag) 1.95 1.39 1.75 1.55 1.75 1.55bBL (Hog) 2.75 1.85 2.38 2.13 2.38 2.13


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hogging failure modes. This large increase of non-linear correction factors is based onsome recent studies showing that even full form tankers of medium size may havesignicant non-linear contributions to the total vertical wave bending moments [5]. This isexplained by the simultaneous effect of ship length and large bulbous bow. In such case,both hogging and sagging vertical wave bending moments may be signicantly higher thanthe linear prediction. This is contrary to the IACS UR S11, where difference betweensagging and hogging is almost negligible for full-form ships with large block coefcients, asoil tankers. For this reason, large variation of mean value of non-linear effects is includedin the parametric analysis. The results of the analysis are presented in Tables 16 and 17 fornew generation tanker and rule tanker, respectively.

From Tables 16 and 17 one may conclude that the variation of mean of non-lineareffects mwnl has the largest impact on the calculated safety indices. This is expected as therange of variation of this parameter is the largest. However, it is interesting to notice thatin most cases relatively similar results are obtained with variation of the mean value of theultimate strength mwu, although its range of variability is much lower. This is anotherevidence of the large sensitivity of the calculated reliability indices to the modellinguncertainty of ultimate strength.

The results of the parametric study for COVs and mean values are summarized in Fig. 4 .It may be concluded from Fig. 4 that the largest variability has the safety index of therule tanker in hogging, that varies +0.49/ 0.45 around the value calculated with thebest estimates of parameters. The variability of other safety indices is somewhat lower.Thus, the variability in sagging reads +0.33/ 0.31 for the new generation tanker and+0.3/ 0.32 for the rule tanker. Obviously, the variability of safety indices is not

negligible and they emphasize the importance of proper uncertainty assessment of pertinent variables for a credibility of a reliability study.

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Table 16Safety indices for various mean values of modelling uncertainties (New generation tanker, corroded hull)

mwu mww mwnl0.95 1.05 0.85 0.95 1.12 1.14

bFL (Sag) 1.83 2.28 2.21 1.92 1.79bBL (Hog) 2.61 3.01 2.89 2.73 2.60

Table 17Safety indices for various mean values of modelling uncertainties (Rule tanker, corroded hull)

mwu mww mwnl0.95 1.05 0.85 0.95 1.12 1.14

bFL (Sag) 1.42 1.88 1.83 1.48 1.33bBL (Hog) 2.00 2.51 2.44 2.09 1.81


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The results of the presented parametric study might be useful when comparing thereliability analyses based on differing assumptions. The ranges of variability of the safetyindices calculated in the present work may be used to approximately estimate the

implications of different parameters of modelling uncertainties on the results of reliabilitystudies.

10. Conclusion

The new generation oil tanker, which is characterized by shallow draught and lowlength-to-beam ratio, is designed based on the vertical wave bending moments obtained bydirect hydrodynamic and statistical analysis instead of using rule values. The aim of thispaper is to quantify the changes in notional hull-girder reliability as a result of suchinnovative approach in ship design. For that purpose, a comparative hull-girder reliability

study of a new generation oil tanker and an existing conventional rule tanker is carriedout. The reliability assessment is performed for as-built state of ships and forcorroded ships, assuming 20-year corrosion, as given by ship rules.

Results of the analysis of as-built ships show that the structural reliability of the newgeneration oil tanker is much higher than the reliability of the rule oil tanker. Forcorroded hulls, the ratio of failure probabilities between the rule tanker and thenew tanker design is reduced, as a consequence of the unusually large breadth of the newgeneration oil tanker. Nevertheless, the reliability of the corroded tanker of newgeneration is almost as high as the reliability of the as-built rule tanker.

The sensitivity analysis shows that for various ships, even of the same type, the relativeimpact on structural safety of still-water and wave-induced bending moments may be quitedifferent. While for the new generation tanker the calculated reliability index is verysensitive to the still-water bending moment in ballast loading condition, the reliability of the rule tanker is almost completely insensitive to this variable. The sensitivity analysis

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0

0.5

1

1.5

2

2.5

3

3.5

R e l

i a b i l i t y i n d e x

New generation tanker -sag

"Rule" tanker - sag

New generation tanker - hog

"Rule" tanker - hog

Fig. 4. Ranges of reliability indices calculated by parametric variation.


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shows that the uncertainty of the ultimate bending moment capacity is generally the mostimportant variable in reliability assessment.

The parametric variation is performed for plausible ranges of COVs and mean values of the modelling uncertainties. The results of the parametric study conrm the results of thesensitivity analysis regarding the huge importance of the uncertainty of ultimate strength.It is also shown that different assumptions about non-linear effects may lead to signicantchanges in the calculated safety indices. The results of the presented parametric studymight be useful for comparison of reliability analyses that are based on differentparameters of modelling uncertainties.

References

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[3] ABS, DNV, Lloyds Register. Common Structural Rules for Double Hull Oil Tankers, 2006.[4] Nitta A, et al. Basis of IACS unied longitudinal strength standard. Mar Struct 1992;5:121.[5] Guedes Soares C. On the uncertainty in long-term predictions of wave induced loads on ships. Mar Struct

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[25] Parunov J, Senjanovic I. Incorporating model uncertainty in ship reliability analysis. Trans SNAME2003;111:377408.

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