Neutral Axis

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Statistical properties of bulk carrier residual strength

A. Campanile a, V. Piscopo b, A. Scamardella b

a The University of Naples Federico II, Department of Industrial Engineering, Via Claudio 21, 80125 Naples, Italyb The University of Naples Parthenope, Department of Science and Technology, Centro Direzionale Isola C4, 80143 Naples, Italy

a r t i c l e i n f o

Article history:

Received 24 December 2014Accepted 29 June 2015

Available online 17 July 2015

Keywords:

Residual strengthDamage scenariosBulk carrierCorrosionMonte Carlo simulation

a b s t r a c t

Time-variant residual strength of bulk carriers under corrosion wastage is investigated by Monte Carlosimulation. A new formulation of the incrementaliterative approach is presented to account forinstantaneous neutral axis rotation, in case of asymmetrical damage conditions. Relevant incidence on

hull girder residual strength is preliminarily studied. Hence, two main aspects are investigated: theformer regards the correlation among input variables, commonly assumed as uncorrelated in classicalprocedures. In this respect, three correlation models are considered: no correlation, full correlation, andfull correlation among variables belonging to the same group of compartments. Subsequently, a z-test isperformed and the hull girder residual strength probability density functions are determined for bothsagging and hogging conditions, as when correlation exists, LindebergFeller Central Limit Theoremcannot be applied. Three damage scenarios are analysed, according to the last requirements ofHarmonized Common Structural Rules for Bulk Carriers and Oil Tankers, assuming as reference casethe bulk carrier section scheme, proposed in the last ISSC Report. Finally, the residual strength factordistribution vs. time is determined, together with relevant lower/upper bound values, with certain non-exceedance probability level, that in turn depends on correlation among corrosion wastages of allstructural members.

&2015 Elsevier Ltd. All rights reserved.

1. Introduction

During the last decades, due to increasing environmental andsafety concerns, several attempts have been undertaken by Inter-national Organizations, Classication Societies and researchers, toincrease ship performances after collisions or groundings, onceaccidents occur. Even if ship collisions and groundings have beentraditionally related to damage stability or cargo spills fromdamaged hulls (Hussein and Guedes Soares, 2009), attention hasbeen focused on hull girder residual strength performances, too. Inthis respect, within the research programme Tanker Safety, theGermanischer Lloyd developed the rst guidelines to evaluate the

absorbed plastic deformation energy in a ship-ship collision, basedon minimum hull-girder ultimate load capacity and damagepenetration depth (Egge and Bckenhauer, 1991). Ships wereclassied, with regard to resistance against collision, by means ofthe additional class notation COLL, followed by an index rangingfrom 1 to 6 and indicating that a critical situation, such as therupture of cargo tanks with subsequent spillage of cargo or water

ingress into dry cargo holds, is likely to occur only if deformationenergy due to collision exceeds the one of a non-strengthenedreference ship by more than a specied value. Subsequently, theAmerican Bureau of Shipping introduced the additional classnotation RES and developed some guidelines for assessing thehull girder residual strength for oil tankers (ABS, 1995a) and bulkcarriers (ABS, 1995b). They provide assumptions about types ofaccidents to be considered, damage location and extension, as wellas residual strength check criteria, in terms of minimum residualhull-girder section modulus and maximum allowable shear stressin side shell platings. In the same years, the International MaritimeOrganization (IMO) developed some guidelines to evaluate the

amount of oil spills from oil tankers (IMO, 1995, 2003), based ondamage locations and extensions, due to collision or groundingscenarios. Subsequently, the Det Norske Veritas introduced theadditional class notation CSA-2, to be assigned to ship designscomplying with enhanced fatigue and ultimate limit state criteria,including residual strength due to collision or grounding occur-rences (DNV, 2009). Even if the CSA-2 notation was applicable toall ships, guidelines were furnished only for tankers, LNG/LPGcarriers and containerships. Besides, in 2010 the InternationalMaritime Organization adopted the International goal-based shipconstruction standards for bulk carrier and oil tankers (IMO,2010a), mainly based on a ve-tier system, as proposed by

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Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2015.06.0600029-8018/&2015 Elsevier Ltd. All rights reserved.

E-mail addresses: [email protected](A. Campanile),[email protected] (V. Piscopo),[email protected] (A. Scamardella).

Ocean Engineering 106 (2015) 4767
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Bahamas, Greece and the International Association of Classica-tion Societies (IACS), namely: (I) Goals; (II) Functional require-ments; (III) Verication of conformity; (IV) Rules and regulationsfor ship design and construction; (V) Industry practices andstandards. According to IMO resolution, residual strength checkwill be mandatorily required for oil tankers and bulk carriers of150 m in length and above, (i) for which the building contract isplaced on or after 1st July 2016, (ii) the keels of which are laid onor after 1st July 2017, (iii) the delivery of which is on or after 1st

July 2020 (IMO, 2010b). Furthermore, in the relevant guidelines forverication of conformity with goal-based ship constructionstandards (IMO, 2010c), the IMO conrmed that rules have toprovide hull-girder residual strength reasonable levels afterdamage, taking into account: (i) applied methodologies; (ii) ood-ing scenarios due to collision or grounding events; (iii) environ-mental conditions and periods of exposure, representative of seastates in case of damage occurrences; (iv) acceptance criteria;(v) validation of residual strength assessment procedure, throughthe analysis of a range of representative ship designs and loadingconditions.

Finally, in 2014 IACS published the Harmonized CommonStructural Rules for Bulk Carriers and Oil Tankers (CSR-H), inaccordance with IMO Tier (IV) requirements (IMO, 2010a), making

residual strength check mandatory for bulk carriers and oiltankers, equal to or greater than 150 m in length (IACS, 2014a).Hull girder residual strength criteria have been dened by struc-tural reliability analysis, assuming that the annual probability offailure, due to collision or grounding scenarios, shall be less thanthe one of an intact ship in open sea, subjected to 25-year extremeloads (IACS, 2014b). Strength check criteria have been determinedon the basis of the following key elements: (i) annual probabilityof damage; (ii) probability of damage size; (iii) reduction incapacity as a function of damage size; (iv) increase in still waterbending moment due to damage; (v) environmental conditionswhen damage occurs; (vi) exposure time to environment, beforereturn to port. Residual strength check, prescribed by CSR-H, ismainly based on the net scantlingapproach, which implies that

one-half corrosion additions have to be subtracted from the gross

offered thickness of any structural element. Besides, residualstrength calculation is performed according to the incrementaliterative method, commonly applied for ultimate strength analysis,deleting from the structural model all platings, stiffeners andprimary supporting members lying in the damage area, withoutaccounting for instantaneous neutral axis rotational shift, in caseof asymmetrical damage conditions. In this respect, two mainproblems arise: the former is related to the incrementaliterativemethod, not accounting for neutral axis rotation; the latter,instead, concerns the corrosion wastage model and the hull-girder residual strength statistical properties, mainly related tocorrelation among input variables.

As concerns the rst aspect, in the recent years several studieson the assessment of hull girder residual strength in damagedconditions have been carried out. Paik et al. (1998a)studied theresidual strength of hull structures, based on section modulus andultimate bending strength, developing a rapid procedure toidentify the possibility of hull girder failure after collision orgrounding damages, based on some closed-form formulas.Wanget al. (2002) reviewed the state of art research on collision andgrounding, focusing on issues that relevant design standards needto address: denition of accidents, scenarios, evaluationapproaches and acceptance criteria.Fang and Das (2005)applied

structural reliability analysis to damaged ship structures, present-ing some limit state design functions and determining the hullgirder probability of failure. Hussein and Guedes Soares (2009)studied the residual strength of three double hull oil tankers,designed according to IACS Common Structural Rules (IACS, 2006a,2006b). They analysed several damage scenarios at side andbottom with different size, to derive the hull girder residualstrength lower bound values, which may be accounted for futuredesign. The progressive collapse method was applied togetherwith rule failure modes.Dec et al. (2012)proposed a frameworkfor the assessment of structural safety of ships under differentoperational conditions, based on reliability and redundancy indi-cators applied to hull girder ultimate capacity, accounting forageing affects due to corrosion wastage, deriving several polar

plots of both reliability and redundancy indexes for different sea

Nomenclature

A hull girder sectional areaAi area ofith structural element sectionb damage penetration breadthC1 annual corrosion rateCOV[MH] coefcient of variation of hull girder hogging capacity

COV[MS] coefcient of variation of hull girder sagging capacityCOVC1 annual corrosion rate coefcient of variationd damage penetration depthE maximum percentage error of Monte Carlo simulationE[MH] expected value of hull girder saggingE[MS] expected value of hull girder saggingh damage penetration heightIy0 hull girder vertical moment of inertiaIy0z0 hull girder product of inertiaIz0 hull girder horizontal moment of inertiaIQR inter-quartile rangeMH hull girder hogging capacityMS hull girder sagging capacityn iteration number of Monte Carlo simulation

nreq required iteration number of Monte Carlo simulationp condence level of Monte Carlo simulationRSF hull girder residual strength factor

T timeTc coating protection lifetg as built thickness ofith structural elementtr corrosion wastageyi horizontal coordinate ofith structural element centre

of massYG horizontal coordinate of hull girder section centre

of masszC condence coefcient for normal distributionzCL hull girder neutral axis vertical position at centrelinezG vertical coordinate of hull girder section centre

of masszi vertical coordinate of ith structural element centre

of massZD hull girder elastic section modulus at deckZK hull girder elastic section modulus at bottom neutral axis rotation about the horizontali strain in theith elementC1 annual corrosion rate expected valuex time-variant thickness expected value correlation coefcienti stress in theith elementx time-variant thickness standard deviation hull girder curvature

A. Campanile et al. / Ocean Engineering 106 (2015) 476748


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states, ship speeds and heading angles.Alie (2012)and Alie et al.(2012) investigated the incidence of neutral axis rotation onresidual hull girder strength for asymmetrical damaged shipsunder longitudinal bending, applying the progressive collapsemethod and nding an explicit expression of neutral axis rotation,as a function of hull girder cross section biaxial curvature. Afterperforming a series of progressive collapse analyses on severalbulk carriers and oil tankers, it was found that residual hull girder

strength, considering the instantaneous neutral axis rotation, isgenerally lower than the relevant one, obtained by constrainingthe rotation, as it is commonly assumed in classical procedures.Saydam and Frangopol (2013)provided a probabilistic frameworkfor performance assessment of hull girders, under suddendamages, due to collision or grounding accidents. They combinedthe effects of damage scenarios with progressive deterioration dueto corrosion by an optimization-based version of the incrementaliterative method applied to an oil tanker. Relevant reliability indexwas presented in polar plots for different operational conditions.Finally,Choung et al. (2014)investigated the ultimate strength of avery large crude carrier, based on a probabilistic assessment ofdamage extent, due to collision or grounding accidents. Theyexpressed the damage extent probability distribution as a functionof several non-dimensional parameters and furnished a polyno-mial expression for estimating the hull girder residual strength, atdifferent cumulative probability levels.

As concerns the latter aspect to be investigated and mainlyrelated to both applied corrosion wastage models and time-varianthull girder ultimate strength statistical properties, some advancescan be cited in the last years. Akpan et al. (2002) developed anapproach for the risk assessment of ultimate strength of agingships, degraded by corrosion and fatigue, based on time-dependent random function models for corrosion growth andcorrosion-enhanced fatigue cracks. Second-order reliabilitymethod was applied to evaluate the hull girder reliability index.Paik et al. (2003) developed some mathematical models forpredicting corrosion as a function of ship age, developing twosets of time-dependent corrosion wastage models for 23 (34)longitudinal member groups of bulk carriers (oil tankers), basedon statistical analysis of corrosion measurement data.Guo et al.(2008) applied a semi-probabilistic approach to assess the time-variant ultimate strength of ageing tanker deck platings, by a non-linear corrosion wastage model based on statistical analysis of theAmerican Bureau of Shipping corrosion wastage database.Ivanov(2007)applied Taylor series expansion method to evaluate the hullgirder geometric properties in a probabilistic term. Wang et al.(2008) presented a database of as-gauged thickness measure-ments and performed a statistical study of time-variant hull girderstrength for gradually degraded hull structures. Ivanov (2009,2012)developed a procedure for calculating hull girder reliability,based on a probabilistic format of cross-section geometric proper-ties, presented as annual distributions for any given lifespan.Kim

et al. (2012)studied the ultimate strength ofve containerships toinvestigate the impact of corrosion wastage throughout the entireship lifetime, based on double hull oil tanker corrosion require-ments, applied as initial guidelines. Saad-Eldeen et al. (2013)analysed the hull girder ultimate strength of corroded shipstructures, based on both experimental and numerical assess-ments, carrying out several tests on a box girder, under verticalbending moment. Kim et al. (2015) applied various types ofindustry corrosion additions to investigate the ultimate strengthperformances of four bulk carriers, to understand relevant effects,in terms of ultimate bending capacity. Finally, Campanile et al.(2014) investigated the time-variant ultimate strength of bulkcarriers under corrosion wastage in intact conditions, applyingboth Taylor series expansion method and Monte Carlo simulation,

mainly focusing on two main aspects, the former regarding the

correlation among input variables, the latter instead the applic-ability of LindebergFeller Central Limit Theorem, in case of partialor full correlation among input variables.

In the present paper statistical properties of bulk carrierresidual strength are investigated by Monte Carlo simulation,focusing on both incrementaliterative method and correlationamong input variables. A new formulation of the incrementaliterative method is preliminarily proposed, to account for instan-

taneous neutral axis rotation, in case of asymmetrical damageconditions. Equilibrium of axial forces and horizontal bendingmoment is imposed to derive both neutral axis vertical positionand rotation above the horizontal. The incidence of neutral axisrotation on hull girder sagging and hogging capacities, as well asthe minimum iteration number required to perform Monte Carlosimulation, are preliminarily studied. The incidence of correlationamong corrosion wastages of different structural members is fullyinvestigated, despite of commonly applied procedures, whereinput variables are assumed uncorrelated (ISSC, 2012). In thisrespect, three correlation models among input variables areassumed: (i) no correlation; (ii) full correlation; (iii) full correlationamong corrosion wastages of structural members belonging to thesame group of compartments. Time-variant residual strengthstatistical properties are subsequently determined by the corro-sion wastage model proposed byPaik et al. (2003). According toCSR-H requirements, three damage conditions are analysed:(i) side damage; (ii) asymmetrical bottom damage; (iii) symme-trical bottom damage. Monte Carlo simulation is applied to thewell-known ISSC bulk carrier (ISSC, 2012); sagging and hoggingultimate capacity mean values and coefcient of variations (COVs)are determined up to 25-year ship lifetime, with 2.5-year step. Az-test is performed to verify that hull girder residual strengthfollows the normal distribution, independently from correlationamong input variables, as far as LindebergFeller Central LimitTheorem cannot be applied when correlation among input vari-ables exists. Moment vs. curvature diagrams are reported, togetherwith residual strength frequency histograms, derived by Freed-manDiaconis rule. The residual strength factor (RSF) distributionvs. time is also investigated, with reference to the analyseddamage scenarios. Correlation among corrosion wastages of dif-ferent structural elements is further discussed, with reference toRSF lower and upper bound values, derived under a certain non-exceedance probability level. Finally, time-variant RSF plots arereported for the analysed damage scenarios for both sagging andhogging conditions.

2. CSR-H requirements for residual strength check

Harmonized Common Structural Rules for Bulk Carriers and OilTankers (CSR-H), in force from 1st January 2014, prescribe thatship structural strength has to be checked against yielding and

buckling criteria (IACS, 2014a), including ultimate strength assess-ment of hull girder capacity. Furthermore, all ships having a lengthgreater than 150 m have to withstand still water and vertical wavebending moments in damaged conditions, due to collision orgrounding scenarios. Residual strength calculations have to beperformed, to check the hull girder ultimate bending capacity,accounting for structure plastic deformation and post-bucklingbehaviour. For collision assessment, damage is assumed to belocated on ship side, from the freeboard deck downward, for adistance of 0.75D and 0.60D for single and double side shellarrangements and transversally inboard, for a distance equal toB/16, having denoted by D and B the ship moulded depth andbreadth respectively, as shown inFig. 1. For grounding assessment,instead, damage is assumed to be located on the bottom, in the

most unfavourable transversal position, extending upward for a

A. Campanile et al. / Ocean Engineering 106 (2015) 4767 49


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distance equal to min (B/20; 2) and min (B/15; 2) for bulk carriersand oil tankers and transversally for a distance of 0.60B. In anycase, damage penetration should not exceed the minimum ruledouble side width and double bottom height, so that plates andstiffeners of inner bottom and hull longitudinal bulkhead have tobe considered intact.

According to CSR-H requirements, residual strength check canbe performed by the incrementaliterative approach, mainlybased on the Smith method (Smith, 1977), under the followingassumptions (Hussein and Guedes Soares, 2009): (i) the ultimatestrength is calculated at a hull girder transverse section betweentwo adjacent webs; (ii) the hull girder transverse section remainsplane during each curvature increment; (iii) steel material beha-viour is assumed to be elastic-perfectly plastic; (iv) the hull girdertransverse section is divided into a set of elements, actingindependently each one from other, namely longitudinal stiffenerswith attached platings, transversely stiffened plate panels andhard corners. Furthermore, for residual strength calculations, thefollowing additional assumptions are undertaken in modelling thedamage area (IACS, 2014b): (i) stiffeners have to be consideredintact, unless the connection with the attached plating is included

in the damage extent; (ii) primary supporting members aredeleted from the model; (iii) neutral axis rotation is not takeninto account, even in case of asymmetrical damage condition. Thelast assumption is slightly non-conservative, as it doesnt accountfor hull girder capacity reduction due to neutral axis rotation, evenif it is partially balanced by a damage size larger than the mostprobable one at failure, as obtained by structural reliabilityanalysis, carried out on a comparative basis between intact anddamaged conditions (IACS, 2014b). In this respect, as collisionevents are more likely to occur in coastal areas, having more densetrafc but less severe environmental conditions than open sea(Rusaas, 2003), the wave bending moment is likely to be signi-cantly lower than for unrestricted service in North Atlanticconditions with 25-year return period, due to short exposure

time and lower waves, where most collisions and groundingstake place.

3. Theoretical background

3.1. Time-variant residual strength statistical properties

Damaged ship structures reliability analysis is generally carriedout by a time-independent rst order formulation, correspondingto one-year operational conditions, accounting for uncertaintiesdue to hull girder ultimate capacity, as well as still-water andvertical wave bending moments. In any case, mean value of hullgirder residual strength is determined by the incrementalitera-tive approach, on the basis of gross offered thickness, reduced byone half corrosion addition, as a function of structural elementtype and location. As concerns relevant statistical properties,Mansour and Howen (1994) suggest that hull girder capacityfollows the normal distribution with COV0.15; Fang and Das(2005) suggest it is normally distributed too, with lower COV,assumed equal to 0.10 (Teixeira, 1997);Hussein and Guedes Soares

(2009),instead, assume it follows the lognormal distribution, withCOV of 0.08 (Guedes Soares et al., 1996). Finally, a similar value,ranging from 0.10 to 0.15, is suggested in the SSC-459 Report (SSC,2011).

One of the rst attempts to evaluate the time-variant reliabilityindex has been performed only recently bySaydam and Frangopol(2013), who applied a linear time-variant corrosion model, toestimate the hull girder residual strength by the Latin-Hypercubetechnique. Annual corrosion rates, in terms of mean value andcoefcient of variation, have been determined for several classes ofstructural elements (Akpan et al., 2002): side shell platings andstiffener webs; deck platings, deck and bottom stiffener webs;bottom shell platings. Furthermore, as shown byCampanile et al.(2014)for the time-variant ultimate strength statistical properties

h

d

b

h

B

D

Damage penetration

Height, h

Depth, d

Side shell arrangement

Single side

0.75D

B/16

Double side

0.60D

B/16

Collision damage extent

Damage penetration

Height, h

Breadth, b

Ship type

Bulk carriers

min(B/20;2)

0.60B

Oil tankers

min(B/15;2)

0.60B

Grounding damage extent

Fig. 1. Collision and grounding damage extents (IACS, 2014a).



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of a bulk carrier in intact condition, coefcient of variations notonly depend on time, by means of the chosen corrosion model, butalso on the assumed correlation among corrosion wastages of allstructural elements. Particularly, they proposed three differentcorrelation models (all random variables independent; randomvariables fully correlated; full correlation between groups ofvariables belonging to the same category of compartments),founding that ultimate hull girder capacity COV may vary up to

one order of magnitude.At design stage, several scenarios in CSR-H Rules (IACS, 2014a)are envisaged for both side and bottom damages, for whichstrength calculations have to be performed. Anyway, during shipsoperation, it is impossible to predict the time when damage willoccur. Therefore, in actual analysis the strength of damaged ship isdetermined as function of that time; statistical properties of hullgirder residual capacity are determined accounting for time-variant corrosion wastage of structural elements. For the correctevaluation of damaged ship strength, two main aspects have to befurther investigated: (i) neutral axis rotation has to be taken intodue consideration in the incrementaliterative approach; (ii)corrosion wastages statistical properties of all structural elementshave to be properly accounted, to correctly estimate the time-variant residual strength COV, choosing the most appropriatecorrelation model among input variables.

3.2. Modied incrementaliterative method

In the so-called incrementaliterative method (IACS, 2014a) themoment-curvature curve is derived by determining the bendingmomentMacting on the hull girder cross-section, due to the imposedcurvature , corresponding to the hull girder section rotation aroundits instantaneous neutral axis, the vertical position of which isdetermined by equilibrium of axial forces. Denoting by the strainin any structural element, due to the imposed curvature , for a givenneutral axis position, relevant stressis derived from element stressstrain () curve, which takes into account the structural behaviour inthe non-linear elasticplastic domain. The neutral axis vertical positionhas to be adjusted iteratively, till sums of forces above and below it areequal, so that equilibrium is achieved. Sagging (hogging) ultimatebending moment is the minimum (maximum) of the obtainedmoment-curvature curve. Besides, according to CSR-H requirements(IACS, 2014a), the load-end shortening curves for stiffened elementsrefer to beam column buckling, torsional buckling and web localbuckling of anged proles/at bars, while plate buckling is

considered for transversely stiffened plate elements. Finally, theelasto-plastic failure mode is applied for both lengthened stiffenedplate elements and shortened/lengthened hard corners.

The previous procedure can be easily applied when hull-girdercross-section is symmetric and a pure vertical bending moment isapplied, as neutral axis remains horizontal and moves onlyvertically during the progressive collapse behaviour. On the con-trary, when the cross-section is damaged at asymmetric positions,

or horizontal and vertical bending moments are applied simulta-neously, both rotation and translation of neutral axis need to betaken into account. In this respect, several efforts have beenundertaken in the past to underline a suitable procedure forultimate strength analysis of asymmetric cross-sections. Smithand Pegg (2003) modied the incrementaliterative approach,accounting for a given ratio of biaxial curvature or moment, interms of secant modulus corresponding to actual bendingmoment-curvature slope. zg and Barltrop (2008) combinedvertical and horizontal bending moments in the incrementaliterative approach, deriving several interaction curves for differentloading combinations.Choung et al. (2012)provided two conver-gence criteria, to nd translational and rotational locations ofneutral axis plane, for both intact and damaged ship conditions,accounting for asymmetries due to material properties, appliedloads and hull girder cross-section geometry, based on both forceand moment equilibrium criteria, to be simultaneously veried.Alie et al. (2012)proposed an incremental formulation of biaxialbending collapse behaviour, furnishing explicit expressions ofneutral axis position and rotation, as a function of cross-sectionrigidities that, in turn, depend on elements tangential axialstiffness, by means of relevant stressstrain curves. It is noticedthat in any case torsional (Campanile et al., 2010) and shear effects(Campanile et al., 2009a, 2009b) are commonly neglected.

All the above mentioned procedures furnish quite similar results,in terms of hull-girder capacity. Anyway, it is possible to develop anincrementaliterative method, taking explicitly into account thecross-section geometrical asymmetries, due to collision or ground-ing scenarios. In this respect, let us consider an asymmetric hull-girder cross-section, as shown in Fig. 2, referred to a globalcoordinate system Oyz, with z-axis lying in the ship symmetryplane, previous to damage event.

Denoting byG(yG;zG) the structural centre of mass after damageoccurrence (represented by a grey area in Fig. 2), let us dene alocal reference systems Gy0z0 with axes parallel to the globalreference ones. Denoting by Iy0 (Iz0) the vertical (horizontal)

y

z

O

G y0

z0

n

a

zCL

Fig. 2. Global and local reference systems.



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moment of inertia about y0 (z0) axis and by Iy0z0 the product ofinertia, the angle (counter-clockwise positive) between theneutral axis and the horizontal can be easily determined as follows(Hughes, 1988):

tanIy0z0Iz0

1

Anyway, when buckling or post-buckling occurs, instantaneousneutral axis moves vertically and rotates, depending on theapplied curvature , according to the following equation:

zna zCL y tan 2

having denoted byzCL() and () the neutral axis vertical positionat centreline and rotation about the horizontal, respectively. Dueto the applied curvature , the neutrairder transverse sectionrotates about its instantaneous neutral axis, so that the axial strainiin the ith structural element, having centre of massGi(yi;zi) and

transverse sectionA i, can be determined according to the follow-ing formula:

i zizCL yi tan

cos 3

Once the stress i() is derived by the element stressstraincurve, neutral axis vertical position and rotation about the hor-izontal can be determined, satisfying both axial force and hor-

izontal bending moment equations:

XNi 1

i Ai 0

XNi 1

i yiAi 0

8>>>>>>>>>:

4

having denoted by N the number of elements which the cross-section is subdivided into. Eq. (4) can be iteratively solved,adjusting for any curvature , both neutral axis vertical positionat centrelinezCL() and rotation about the horizontal (), recalcu-lating element strains, stresses as well as total sectional force andmoment, until both equations are satised. It is noticed that

Eqs.(2)

(4)are coincident with the classical incremental

iterativemethod (IACS, 2014a), in case of symmetric hull girder cross-sections (0). Besides, it has been veried that deformationreversal doesnt occur in all performed calculations.

Table 1

ISSC Bulk carrier main dimensions.

Length between perpendiculars 285.0 mRule length 281.3 mMoulded breadth 50.0 mMoulded depth 26.7 mDesign draft 19.8 m

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

5

5

7

7

7

7

7

7

7

7

34

48

83

3

5

5

4

4

3

3

3

2

2

2

2

1

1111111111111

141

1313

1211

1110

99

2 22

22

3

1

18.5*

18.0

**

24.5***

18.5***

17.5**

16.5*

14.5*

13.5*

4400

1800

3100

9700

2400

4400

1100

700

Space:870

Space:800

14.5

*

18.5

*

12.5

*

12.5

*

12.5

*

12.5

*

12.5

*

2790

R1800

Space: 880

2640 2640 2640 2640 3520 4400 4400 2120

Space: 880

20.5*

18.5*

17.5**

17.0**

991

4000

StraightCamber

Space:880

Yield Stress*: 313.6 MPa

**: 352.8 MPa***: 392.0 MPa

Frame spacing: 870

(in mm)

Spac

e:880

No. Dimensions Type Yield Stress (MPa)

1 390x27 Flat-bar 392.0

2 333x9+100x16 Tee-bar 352.8

3 283x9+100x14 Tee-bar 352.8

4 283x9+100x18 Tee-bar 352.8

5 333x9+100x17 Tee-bar 352.8

6 283x9+100x16 Tee-bar 352.8

7 183x32.5x9.5 Bulb-bar 235.2

8 283x9+100x17 Tee-bar 352.8

9 333x9+100x18 Tee-bar 352.8

10 333x9+100x19 Tee-bar 352.8

11 383x9+100x17 Tee-bar 352.8

12 383x10+100x18 Tee-bar 352.8

13 283x10+100x21 Tee-bar 352.8

14 300x27 Flat-bar 392.0

12320Space: 880

R650

Space:880

Web frame spacing (double bottom and hopper side tank): 2610Top-side tank web frame spacing: 5220CVK transverse stiffener spacing: 1305

Fig. 3. ISSC Bulk carrier section scheme.



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3.3. Rule and time-variant corrosion wastage models

Hull-girder ultimate bending capacity decreases in time, asthickness of platings and stiffeners reduces due to corrosionwastage. According to CSR-H requirements (IACS, 2014a), onlyone-half corrosion addition must be considered for hull girderultimate and residual strength check. One-side exposure wastageallowances, determined on the basis of approximately 600,000

thickness measurement data, sampled from single hull tankersand single side skin bulk carriers, with age ranging from 5 to 27years, are generally considered appropriate for 25-year ship life-time, if suitable hull structure maintenance programmes areundertaken (IACS, 2014b).

Anyway, if a more reliable corrosion wastage estimate isrequired, the easiest alternative to the deterministic net-scantlingapproachis to base corrosion prediction on statistical analysis ofpast data for comparable situations (Paik et al., 2003). In thisrespect, corrosion models can be divided into two phases, theformer without corrosion occurrence due to coating protection,the latter instead represented by an uncertain but constantcorrosion rate, which results in a linear decrease of plate thicknesswith time (Yamamoto and Ikegami, 1998; Hu et al., 2004). Even ifnon-linear corrosion models seem to be more appropriate thanlinear ones, due to the nature of corrosion rate, decreasing withtime after coating degradation (Qin and Cui, 2003), few data areavailable in literature, so that in the following the linear corrosionmodel developed byPaik et al. (1998b)will be applied:

tr T C1 TTc 5

having denoted byTcthe coating protection life, equal to 7.5 years,and C1 the annual corrosion rate, the expected value C1 andcoefcient of variation COVC1 of which have been determined byPaik et al. (2003), for 23 different structural elements of single skinbulk carriers, in the hypothesis of severe corrosion rate, corre-sponding to the 95th percentile of relevant probability function.Hence, time-variant statistical properties at time T4Tc can beeasily derived for any structural element (Campanile et al., 2014):

X tgC1 TTc

X C1 T Tc COVC1

( 6

having denoted by tg the as built thickness. Finally, the best-tprobability function of annual corrosion rate follows the twoparameter Weibull distribution. Scale and shape parameters havebeen derived as a function of mean values and coefcient ofvariations given by Paik et al. (2003) for different structuralelements. It is noticed that Eq. (6) is in general approximated,because the as-rolled steel product thickness, assumed as adeterministic variable in actual analysis, follows a normal or log-normal distribution, which implies that the resulting probabilisticdistribution will follow neither the normal nor the Weibulldistribution and will vary with the ship life.

3.4. Monte Carlo simulation

Several techniques can be applied to problems involving ran-dom variables, such as Taylor series expansion method, LatinHypercube sampling or Monte Carlo simulation, depending onsystem complexity and computational effort. Monte Carlo methodhas been extensive applied in the eld of operational research andnuclear physics, where there are a variety of problems beyond the

available resources of theoretical mathematics (Hammersley andHandscomb, 1975). Nowadays it is also widely applied in the eldof structural mechanics, including ultimate strength of stiffenedpanels (Garbatov et al., 2011), buckling of platings subjected tonon-uniform corrosion wastage (Silva et al., 2011) and hull girderultimate bending capacity (Vhanmane and Bhattacharya, 2011;Campanile et al., 2014). The method is mainly based on thegeneration of random values assumed as input variables, eachone with a given distribution. In case of one-dimensional

Table 2

Comparison with the methods for ultimate strength analysis reported by ISSC (2012).

Symbol Method Working organization MH(GN m) MS(GN m)

ANSYS (PNU) FE elasto-plastic analysisANSYS Pusan National University 17.500 15.800ANSYS (ISR) FE elasto-plastic analysisANSYS Indian Register of Shipping 18.326 17.726ABAQUS (CR) FE elasto-plastic analysisABAQUS China Corporation Register of Shipping 18.396 16.855ALPS/HULL (PNU) FE progressive collapse analysis Pusan National University 16.602 15.380CSR (BV) Incrementaliterative approach Bureau Veritas 14.822 11.521CSR (CR) Incrementaliterative approach China Corporation Register of Shipping 18.338 14.921CSR (PNU) Incrementaliterative approach Pusan National University 18.360 14.500RINA Rules (UoG) Incrementaliterative approach University of Genova 17.482 13.952ISSC 2000 (Rigo) Incrementaliterative approach Rigo P. c/o University of Liege 18.714 14.340Modied P-M (PNU) Modied Paik Mansour formula Pusan National University 16.576 14.798CSR-H Incrementaliterative approach Actual calculation 18.364 14.708

Table 3

Comparative analysis between classical and modied incrementaliterativeapproach.

T

(years)

Classical

incremental

iterative approach

Modied

incremental

iterative

approach

Sagging

difference

Hogging

difference

E(MS)

(GN m)

E(MH)

(GN m)

E(MS)

(GN m)

E(MH)

GN m

(%) (%)

Up to7.5

12.427 15.927 11.911 15.140 4.150 4.940

10.0 11.902 15.402 11.405 14.599 4.176 5.21712.5 11.379 14.874 10.910 14.069 4.121 5.41415.0 10.861 14.339 10.417 13.529 4.089 5.64417.5 10.341 13.802 9.911 12.975 4.163 5.98920.0 9.827 13.264 9.422 12.425 4.120 6.33022.5 9.319 12.725 8.940 11.870 4.064 6.72225.0 8.812 12.186 8.457 11.312 4.022 7.167

Table 4

Minimum iteration number nminas a function of sample size n.

n Case (I) Case (II) Case (III)

Sagging Hogging Sagging Hogging Sagging Hogging

100 2 1 230 183 189 67200 2 1 225 180 245 72300 2 1 238 188 187 62400 2 1 236 187 199 66500 2 1 234 185 224 73750 3 1 236 187 211 70

1000 2 1 249 197 204 661500 2 1 226 180 205 68



8/21

problems, several techniques are available, such as the inverse-transform one, among others (Kroese, 2011). In case of multi-variate problems, instead, correlation among input variables has tobe investigated. In fact, while in case of uncorrelated variables,each one can be easily generated by its own distribution, whencorrelation exists, the random generation procedure has to beproperly varied, by means of several approximate techniques(Kroese, 2011). Time-variant residual hull girder capacity is cer-

tainly a multivariate problem, as corrosion wastage of eachstructural element can be regarded as a random variable. Hence,correlation among input variables becomes a basic issue to beinvestigated, as standard deviation of both sagging and hoggingcapacities strictly depends on it (Devroye, 1986). In this respect,Monte Carlo simulation will be applied, considering three differentcorrelation models among input variables:

Case (I): input variables are uncorrelated. Annual corrosionrates are generated for each structural element based onrelevant Weibull distribution.

Case (II): input variables are fully correlated. A uniform randomnumber q is preliminarily generated in the interval [0,1];annual corrosion rate of each structural element is subse-quently determined by relevant inverse cumulative distributionfunction, evaluated at probability level q .

Case (III): input variables are partially correlated. Structuralelements are preliminarily subdivided into several compart-ment categories, namely double-bottom, hopper tank, wingtank and transversely stiffened side shell in the examined case.Full correlation ( 1) is assumed among elements belonging tothe same category of compartments, while no correlation

( 0) exists for elements belonging to different categories. Auniform random number qi is generated for each category ofcompartments in the interval [0,1]; hence annual corrosion ratefor each structural element belonging to the ith group isdetermined by relevant inverse cumulative distribution func-tion, evaluated at probability level q i.

It is noticed that Lindeberg

Feller Central Limit Theorem (amore general form of well-known LindebergLvy CLT) can beapplied in Case (I) only, as input variables are uncorrelated, but notfollowing the same distribution law (Hunter, 2011; Campanileet al., 2014). On the contrary, when correlation exists, as for Cases(II) or (III), a z-test (Sprinthall, 2011) needs to be performed, toverify that hull girder residual strength follows the normaldistribution. In this respect, FreedmanDiaconis rule (Freedmanand Diaconis, 1981) is applied to estimate the optimum bin sizelength of hull girder capacity frequency histograms:

Bin size 2IQRxn 1=3 7

having denoted by n the Monte Carlo iteration number and IQR(x)

the data inter-quartile range (Graham and Cook, 1996). Finally, theminimum iteration number nminrequired to perform Monte Carlosimulation can be determined according toDriels and Shin (2004):

nmin min nA :nZnreq n

8

having denoted by nreq(n) the required iteration number, as afunction of hull girder capacity mean value condence levelp andmaximum percentage error E:

nreq n 100zc p COVn

E

29

In Eq.(9) COV(n) is the hull girder sagging/hogging coefcientof variation, while zc(p) is a coefcient derived by the normaldistribution and depending on the assumed condence level p . IfEq. (8) is satised and Monte Carlo simulation is run for nminiterations, the target function mean value will not differ more thanE from the true one, with probability level p. In the present studyMonte Carlo simulation has been performed by a dedicated codedeveloped in Matlab Mathworks (MathWorks, 2014), mainly con-sisting of two parts: the former is devoted to residual strengthanalysis by the modied-incremental iterative method, accountingfor rule and time-variant corrosion wastage models; the latter,instead, performs the statistical analysis of Monte Carlo results.

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

0 100 200 300 400 500 600 700 800 900 100011001200130014001500

n

min

n

Limit value

Case (I) -Sagging

Case (I) -Hogging

Case (II) -Sagging

Case (II) -Hogging

Case (III) -Sagging

Case (III) -Hogging

Fig. 4. Minimum iteration number vs. sample size collision scenario.

y

z, z

G y0

0

n a

Sectional area A 5.693 m2

Centre of mass vertical position zG 11.224 m

Centre of mass transversal position yG 0.000 m

Vertical moment of inertia Iy0 699.540 m4

Horizontal moment of inertia Iz0 1811.702 m4

Product of inertia Iy0z0 0.000 m4

Neutral axis rotation 0.000 deg

Neutral axis vertical position at CL zCL 11.224 m

Elastic section modulus at deck ZD 44.625 m3

Elastic section modulus at bottom ZK 62.325 m3

Ultimate hogging capacity MH 18.364 GNm

Ultimate sagging capacity MS 14.711 GNm

Fig. 5. Section scheme and gross sectional propertiesintact condition.



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4. The ISSC bulk carrier

4.1. Main data

The vessel selected as test case for time-variant residualstrength analysis by Monte Carlo simulation is a Capesize singleside bulk-carrier, the main dimensions of which are listed inTable 1andFig. 3, where material properties, scantlings and frame

spacing are also detailed. The ship is longitudinally stiffened,except for side shell platings comprised between the hopper andwing tanks.

The hull has been extensively studied by Amlashi and Moan(2008), who carried out a series of nonlinear FE analyses, undervarious loading conditions. It has also been benchmarked in thelast ISSC Ultimate Strength Committee report (ISSC, 2012), whereultimate hogging and sagging capacities have been determined byseveral Working Organizations, by different techniques, namely FEelasto-plastic analysis, incrementaliterative approach and mod-ied Paik-Mansour formula. Actual values of gross scantling hullgirder capacity in intact condition are well in accordance withnonlinear FE calculations. In fact, actual hogging and saggingultimate bending capacities are equal to 18.364 GN m and14.708 GN m, respectively, while relevant FE values, derived byfull elasto-plastic analyses, range from 17.500 to 18.396 GN m forhogging and from 15.800 to 17.726 GN m for sagging conditionrespectively, as reported inTable 2.

It is noticed that, according toFig. 3,the length of elements is870 mm for side shell, 5220 mm for longitudinal stiffeners withintop side tank, 2610 mm within double bottom and hopper side

tank, excepting the center vertical keel where the length is1305 mm, because of brackets between oors.

4.2. Incidence of neutral axis rotation

Before performing Monte Carlo simulation, the incidence ofneutral axis rotation is investigated. With reference to the collisionscenario detailed in the next paragraph, residual bending capa-

cities are determined by classical (IACS, 2014a) and actual incre-mentaliterative approach. Table 3 reports time-variant residualstrength expected values, for sagging E(MS) and hogging E(MH)conditions, up to 25-year ship lifetime, with 2.5-year step. Meanvalues are determined replacing the gross thickness of anystructural element with the one at time T4Tc, as obtained bythe rst of Eq.(6).

Sagging and hogging residual strength capacities decrease intime, as predictable. Furthermore, if compared with classicalapproach, the actual incrementaliterative method always leadsto slightly less hull girder capacities, with percentage differencesaround 4% and ranging from 5% to 7% for sagging and hoggingconditions, respectively. In any case, obtained results are inaccordance with those ones derived by Alie et al. (2012), who

found that maximum residual strength reduction, due to neutralaxis rotation, is about 8% for a similarly sized single side bulkcarrier.

4.3. Minimum iteration number

The minimum iteration number required for Monte Carlosimulation is preliminarily investigated with reference to thecollision scenario detailed in the next paragraph, assuming acondence level p 95% and a percentage error E1.0%. In allcases, sagging and hogging capacities follow the normal distribu-tion, according to the performed z-test, which implies that for theassumed condence level zc is equal to 1.96. Table 4reports theminimum iteration number nmin required for Cases (I)(III), for

both sagging and hogging conditions, as a function of sample sizen, ranging from 100 to 1500.

The minimum iteration number required in Case (I), i.e. whenall variables are uncorrelated, is very low. On the contrary, whenfull correlation exists, as in Case (II), the minimum iterationnumber is about 300 for both sagging and hogging conditions.Finally, Case (III) shows intermediate values, as partial correlationamong input variables is assumed. Similar results are also reportedinFig. 4, where the minimum iteration number nminis plotted. vs.the sample size n. All points below the magenta curve satisfyEq. (9), while continuous and dashed lines refer to sagging and

y

z

O

G y0

z0

n

a



Centre of mass transversal position yG -3.183 m



Product of inertia Iy0z0 -129.892 m4

Neutral axis rotation -5.420 deg





Ultimatesagging capacity MS 11.911 GNm

Fig. 6. Section scheme and gross sectional propertiesside damage (collision).

Table 5

Mean values and COVs of hull girder ultimate bending moment capacity - Intactcondition fromCampanile et al. (2014).

T

(years)

Mean value Case (I) Case (II) Case (III)

E[MS]

(GN m)

E[MH]

(GN m)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

Up to7.5

14.708 18.364

10.0 14.100 17.680 0.073 0.044 0.805 0.803 0.777 0.51712.5 13.490 16.960 0.153 0.094 1.738 1.567 1.595 1.09915.0 12.890 16.260 0.262 0.157 2.589 2.687 2.559 1.62517.5 12.290 15.550 0.338 0.202 3.769 3.743 3.760 2.39620.0 11.700 14.810 0.440 0.295 4.940 4.959 4.939 3.15922.5 11.110 14.100 0.562 0.345 6.228 6.326 6.000 4.08625.0 10.510 13.360 0.679 0.461 7.330 7.879 7.280 5.619



10/21

hogging capacities, respectively. Hence, in the following analysesMonte Carlo simulation will be carried out, performing in all cases1000 iterations, with 95% condence level and 1% maximumpercentage error.

5. Time-variant residual strength analysis

Hull girder ultimate strength capacities in intact conditions arereported fromCampanile et al. (2014), while time-variant residualstrength analysis is performed for the following damage scenarios,according to CSR-H requirements (IACS, 2014a):

1. Side damage (collision);2. Bottom damage (1st grounding scenario);3. Bottom damage (2nd grounding scenario).

For each damage scenario the reference section scheme, as wellas gross scantling sectional properties are reported. Time-variantmean values and COVs of sagging and hogging hull girdercapacities are subsequently determined, up to 25-year ship life-time, with 2.5-year step. Cases (I)(III) ofSection 3.4are consid-ered to investigate the incidence of correlation among inputvariables on hull girder statistical properties. Finally, after per-forming a z-test, frequency histograms obtained by Monte Carlosimulation and best-t normal distributions are reported for both

sagging and hogging capacities after 25-year ship lifetime.

5.1. Intact condition

As previously said, ultimate strength analysis in intact condi-tions is necessary to carry out a comparative analysis with time-variant hull girder residual strength capacity, due to collision orgrounding scenarios. Statistical properties of ISSC bulk-carrierlongitudinal strength in intact conditions have been alreadyinvestigated byCampanile et al. (2014).Fig. 5shows the analysedsection scheme, with relevant gross sectional properties, whileTable 5 reports detailed statistical analysis results.

5.2. Side damage (collision scenario)

The damage scenario for collision assessment is determinedaccording to CSR-H requirements for single side bulk carriers,assuming a penetration height h20.175 m and a depthd 3.125 m (seeFig. 1).Fig. 6shows the analysed section scheme,with relevant gross scantling properties. Damaged structures havebeen deleted from the model (IACS, 2014a).

Sagging and hogging capacities are determined up to 25-yearship lifetime, with 2.5-year step.Fig. 7shows the hull girder cross-section behaviour as function of the curvature without corrosionwastage (TrTc): black curve refers to the hull girder bendingcapacity, while blue and red lines are relative to neutral axisvertical zzCL() zCL(0) and rotational ()(0) shifts, asregards the elastic neutral axis values, when any structuralelement composing the hull girder cross-section does not undergoinelastic effects, such as yielding or buckling (Choung et al., 2012).

Mean values and COVs of hull girder sagging and hoggingcapacities are reported inTable 6. Also in this case COVs increasewith time and substantially depend on the chosen correlationmodel. While for Case (I) they reach a maximum value of 0.790%,when full correlation among variables or groups of them exists, forCases (II) and (III), COVs reach a maximum value of 8.046% and7.283%, respectively.

Finally,Fig. 8shows sagging and hogging capacity histogramsat T25 years, for Cases (I)(III). Frequency histogram bin size isderived according to Eq. (9). In all cases data follow the normaldistribution (red curve), according to z-test null hypothesis thatthey are a random sample from a normal distribution. In the samegraphs mean, standard deviation, skewness and excess kurtosisare also reported. The last two parameters are equal to zero for the

normal distribution.

-2.5-2.25 -2 -1.75-1.5-1.25 -1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

x 10-4

-20

-18

-16-14-12

-10

-8-6-4

-20

2

468

10

121416

1820

Curvature (1/mm)

Bendingca

pacity(GNm)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 10-4

-10-9

-8

-7-6

-5-4-3

-2-1

01

2

345

67

8

910

Neutral axis vertical shift

Bending moment

Neutral axis rotational shift

z(m),

(deg)

Fig. 7. Moment-curvature diagram at time T0 yearsside damage (collision).

Table 6

Mean values and COVs of hull girder ultimate bending moment capacity side

damage (collision).

T

(years)


E[MS]

(GN m)

E[MH]

(GN m)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

Up to7.5

11.911 15.139

10.0 11.405 14.599 0.074 0.065 0.833 0.741 0.801 0.49712.5 10.910 14.069 0.191 0.110 1.757 1.519 1.672 0.99915.0 10.417 13.529 0.279 0.171 2.834 2.474 2.631 1.53917.5 9.911 12.975 0.377 0.235 3.682 3.215 3.589 2.19420.0 9.422 12.425 0.541 0.320 4.983 4.402 4.80 0 2.80922.5 8.940 11.870 0.634 0.387 6.032 5.385 5.859 3.4 4025.0 8.457 11.312 0.790 0.490 8.046 7.159 7.283 4.154



11/21

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 110

1

2

3

4

5

6

7

8

Bending capacity (GNm)

pdf

Mean=8.457 GNm

Std=0.067 GNm

Skewness=0.062

Kurtosis=0.205

9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 140

1

2

3

4

5

6

7

8


pdf

Mean=11.312 GNm

Std=0.055 GNm

Skewness=-0.013

Kurtosis=0.104

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 110

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=8.457 GNm

Std=0.680 GNm

Skewness=0.438

Kurtosis=0.078

9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=11.312 GNmStd=0.810 GNm

Skewness=0.383

Kurtosis=-0.069

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 110

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=8.457 GNm

Std=0.616 GNm

Skewness=0.357

Kurtosis=-0.219

9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=11.312 GNm

Std=0.470 GNm

Skewness=0.305

Kurtosis=-0.151

Fig. 8. Frequency histograms for Cases (I)(III) at time T25 yearsside damage (collision).



12/21

5.3. Bottom damage (1st grounding scenario)

Bottom damage scenario is assessed considering the damagepenetration height h and breadth b of 2.500 m and 30.000 m,respectively (seeFig. 1). According toIACS (2014a)requirements,damage area has to be located on the bottom, in the mostunfavourable transverse position. In the actual grounding scenario,damage area involves the hopper tank and only partly the doublebottom, as shown inFig. 9, where the analysed section scheme,

gross scantling sectional properties and damage section are

shown. Sagging and hogging capacities are determined up to 25-year ship lifetime. In this respect, Fig. 10 shows relevant grosssection moment-curvature diagram. As well as in the previouscase, neutral axis rotational shift is appreciable when bucklinginception occurs.

Mean values and COVs of hull girder sagging and hoggingcapacities are reported inTable 7. While for Case (I) COV reaches amaximum value of 0.794%, maximum values for Cases (II) and (III)are 8.882% and 6.680%, respectively. Finally,Fig. 11shows saggingand hogging capacity histograms at timeT25 years. Also in thiscase data sampled from Monte Carlo simulation follow the normaldistribution (red curve), according to the performed z-test.

5.4. Bottom damage (2nd grounding scenario)

The second analysed grounding scenario is assessed consider-ing the previous damage penetration height and breadth, locatingthe damage area symmetrically respect to the ship symmetryplane. Obviously, in this case there is no neutral axis rotation, asshown inFig. 12, where the analysed section scheme, as well asthe relevant gross scantling sectional properties are reported.

By a comparative analysis of gross scantling sectional proper-ties listed inFigs. 9 and 12for the asymmetrical and symmetricaldamage scenario, ultimate hogging and sagging capacities arequite similar between them, which implies that the incidence ofdamage area transverse location on hull girder residual strength isalmost negligible. Finally, Fig. 13 reports the gross-scantling

moment-curvature diagram for the analysed section scheme. It is

y

z

G

y0

z0

n

a



Centre of mass transversal position yG -1.980 m


Horizontal moment of inert ia Iz0 1633.214 m4








Fig. 9. Section scheme and gross sectional propertiesbottom damage (1st grounding scenario).

-2.5-2.25 -2 -1.75-1.5-1.25 -1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

x 10-4

-20

-18-16-14-12

-10-8-6

-4-20

2468

101214

161820

Curvature (1/mm)

Bendingcapacity(GNm

)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-10-9-8-7-6-5-4-3-2-101234

5678910


Bending moment


z(m),

(deg)

Fig. 10. Moment-curvature diagram at time T 0 yearsbottom damage (1st grounding scenario).

Table 7

Mean values and COVs of hull girder ultimate bending moment capacity bottomdamage (1st grounding scenario).

T

(years)


E[MS]

(GN m)

E[MH]

(GN m)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

Up to7.5

13.267 13.453

10.0 12.681 12.900 0.112 0.114 0.838 0.969 0.708 0.70312.5 12.137 12.195 0.176 0.300 1.682 2.030 1.493 1.47815.0 11.584 11.606 0.270 0.381 2.752 3.277 2.244 2.28817.5 11.034 10.979 0.355 0.416 3.819 4.615 3.238 3.21720.0 10.497 10.329 0.462 0.517 4.786 5.780 4.138 4.25722.5 9.966 9.723 0.564 0.658 6.216 7.557 5.0 68 5.60225.0 9.238 9.212 0.681 0.794 7.40 0 8.882 6.6 80 6.545



13/21

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120

1

2

3

4

5

6

7

8


pdf

Mean=9.238 GNm

Std=0.063 GNm

Skewness=-0.019

Kurtosis=-0.188

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

1

2

3

4

5

6

7

8


pdf

Mean=9.212 GNm

Std=0.073 GNm

Skewness=0.072

Kurtosis=-0.003

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf


Skewness=0.249

Kurtosis=-0.057

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=9.212 GNm

Std=0.818 GNm

Skewness=0.302

Kurtosis=-0.136

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=9.238 GNm

Std=0.617 GNm

Skewness=0.213

Kurtosis=-0.194

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=9.212 GNm

Std=0.603 GNm

Skewness=0.274

Kurtosis=-0.164

Fig. 11. Frequency histograms for Cases (I)(III) at time T25 yearsBottom damage (1st grounding scenario).



14/21

noticed that neutral axis rotational shift is always zero, indepen-dently from the considered curvature, as it could be predictable forthe analysed case.

Sagging and hogging mean values and COVs are reported inTable 8for Cases (I)(III): relevant maximum values are equal to0.750%, 9.173% and 6.777%, respectively. It is noticed that max-imum COV value is obtained for Case (II), the minimum value forCase (I) and this relationship occurs for other intact and damageconditions, as well. Finally, Fig. 14 shows sagging and hogging

capacity histograms at time T25 years. Also in this case data

sampled from Monte Carlo simulation follow the normal distribu-tion (red curve), according to the performedz-test.

6. Residual strength factor statistical properties

Ship residual strength, due to collision or grounding scenarioscan be assessed in terms of the residual strength factor RSF(T),dened as the ratio of time-variant hull girder capacity indamaged conditions to the intact gross scantling one (Akpanet al., 2002):

RSF T

Mdam T

Mint 10

Relevant statistical properties, in terms of mean value and COV,can be determined as follows:

ERSF T E Mdam T

Mint11

COV RSF T COVMdam T 12

having denoted by E[Mdam(T)] and COV[Mdam(T)] the expectedvalue and coefcient of variation of hull girder sagging/hoggingcapacity in damaged conditions, as determined in previous para-graphs (seeTables 68). As it was veried that the residual hullgirder capacity is normally distributed, independently from corre-

lation among input variables, residual strength factor will be in

y

z,

G y

0

z0

an



Centre of mass transversal position yG 0.000 m










Fig. 12. Section scheme and gross sectional propertiesbottom damage (2nd grounding scenario).

-2.5-2.25 -2 -1.75-1.5-1.25 -1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

x 10-4

-20

-18-16

-14-12

-10

-8-6

-4

-202

4

68

101214

16

1820

Curvature (1/mm)

Bendingcapacity(GNm)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 10-4

-10

-9

-8-7-6

-5

-4-3-2

-10

12

34

5

67

8

910


Bending moment


z(m),

(deg)

Fig. 13. Moment-curvature diagram at time T0 yearsbottom damage (2nd grounding scenario).

Table 8

Mean values and COVs of hull girder ultimate bending moment capacity bottomdamage (2nd grounding scenario).

T

(years)


E[MS]

(GN m)

E[MH]

(GN m)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

COV

[MS]

(%)

COV

[MH]

(%)

Up to7.5

13.812 13.122

10.0 13.230 12.442 0.087 0.121 0.856 1.080 0.707 0.71812.5 12.677 11.926 0.175 0.241 1.751 2.090 1.509 1.41915.0 12.116 11.276 0.245 0.316 2.609 3.175 2.480 2.21917.5 11.555 10.691 0.322 0.426 3.751 4.657 3.400 3.16920.0 10.996 10.057 0.424 0.569 4.722 5.742 4.339 4.25222.5 10.439 9.453 0.546 0.675 6.149 7.4 42 5.716 5.07925.0 9.885 8.951 0.658 0.750 7.623 9.173 6.777 6.290



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7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

1

2

3

4

5

6

7

8


pdf

Mean=9.885 GNm

Std=0.065 GNm

Skewness=-0.001

Kurtosis=-0.027

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120

1

2

3

4

5

6

7

8


pdf

Mean=8.951 GNm

Std=0.067 GNm

Skewness=-0.071

Kurtosis=-0.272

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf


Skewness=0.382

Kurtosis=-0.070

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf


Skewness=0.401

Kurtosis=-0.103

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=9.885 GNm

Std=0.670 GNm

Skewness=0.284

Kurtosis=-0.257

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


pdf

Mean=8.951 GNm

Std=0.563 GNm

Skewness=0.274

Kurtosis=-0.257

Fig. 14. Frequency histograms for Cases (I)(III) at time T25 yearsbottom damage (2nd grounding scenario).



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0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

Fig. 15. Residual strength factor distribution vs. timeside damage (collision scenario).



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turn normally distributed, too. Hence, relevant lower bound value,corresponding to 5% non-exceedance probability level, can bedetermined as follows:

RSFT 5% ERSF T 1 1:645COV RSF T 13

while the upper bound one, corresponding to 95% non-exceedanceprobability, becomes:

RSFT 95% ERSF T 1 1:645COV RSF T 14

Fig. 15shows the residual strength factor distribution vs. timefor the side damage scenario. Sagging and hogging residualstrength factors are plotted for Cases (I)and (III): continuouslines refer to RSF mean value, as determined by Eq.(11), while greyareas represent the range between RSF lower and upper boundvalues, as determined by Eqs.(13) and (14). Sagging and hoggingRSF values for both collision and grounding scenarios after 25-yearship lifetime, are reported in Tables 9 and 10, together withrelevant lower and upper bound values, as determined for Case(I)(III), respectively.

If input variables are uncorrelated as for Case (I), the rangerepresented by the grey area is very small, with RSF lower boundvalues equal to 0.568 and 0.611, after 25-year ship lifetime, forsagging and hogging conditions, respectively. On the contrary,when full correlation exists, as for Case (II), grey areas becomewider, with RSF lower bound values equal to 0.499 and 0.543 forsagging and hogging conditions. Finally, when full correlationexists among variables belonging to the same group of compart-ments, as for Case (III), 25-year RSF lower bound values are 0.506and 0.574, respectively. Similar results are reported inFigs. 16 and17for the two analysed grounding scenarios, with asymmetrical

and symmetrical damage penetrations, respectively. For the asym-metrical one (1st grounding scenario) represented inFig. 16, RSFlower bound values for sagging and hogging conditions, after 25-year ship lifetime, are: 0.633 and 0.495 for Case (I), 0.563 and0.428 for Case (II) and 0.570 and 0.448 for Case (III). For thesymmetrical damage one (2nd grounding scenario) represented inFig. 17, RSF lower bound values are: 0.665 and 0.478 for Case (I),0.588 and 0.411 for Case (II) and 0.597 and 0.434 for Case (III),which implies that the incidence of damage transverse position islow, at least for the analysed section scheme.

Obtained results show that correlation among input variablesstrongly inuences lower and upper bound values of time-variantresidual strength factor. In this respect, if the hypothesis ofuncorrelated variables is on the safe side, the full correlation

model is too much conservative. Hence, the assumption of partial

correlation among variables belonging to the same group ofcompartments, as assumed in Case (III), seems to be the mostsuitable one. Besides, minimum sagging/hogging capacity residualstrength factors are recognized after 25-year ship lifetime for sidedamage and symmetrical bottom grounding, respectively. In fact, ifpartial correlation exists among input variables, RSF minimumlower bound values reduce up to 0.499 and 0.434. The incidence ofdamage transverse position, in the hypothesis of groundingscenario, is very low.

7. Conclusions

Statistical properties of time-variant hull girder residual strengthhave been investigated by Monte Carlo simulation, applying thecorrosion wastage model proposed by Paik et al. (2003) and amodied-incremental iterative method, to account for instantaneousneutral axis rotation, in case of asymmetrical damage conditions.Relevant incidence on hull girder residual strength has been prelimi-narily investigated, while minimum iteration number, required toperform Monte Carlo simulations, has been determined on the basis ofcondence level of sagging/hogging capacity mean value and max-imum allowable error. Three different damage scenarios have beenanalysed up to 25-year ship lifetime, according to CSR-H (IACS, 2014a)requirements: side damage, asymmetrical and symmetrical bottomgrounding conditions. Monte Carlo simulation has been applied to thebulk carrier section analysed in the last ISSC benchmark study,considering three correlation models among input variables: nocorrelation, full correlation, and full correlation among variablesbelonging to the same group of compartments. After an extensivestudy on hull girder residual strength statistical properties, four main

results have been achieved:

(i) In case of asymmetrical damage conditions, the modiedincrementaliterative approach leads to hull girder capacitypercentage reductions between 4% and 7% for sagging andhogging conditions, respect to classical Smith method (seeTable 3), in accordance with results obtained by Alie et al.(2012)for a similarly sized bulk carrier.

(ii) Hull girder residual strength is normally distributed, inde-pendently from correlation among input variables and effec-tiveness of LindebergFeller Central Limit Theorem thatcannot be applied when correlation exists.

(iii) Sagging/hogging residual strength COVs not only increasewith time, as it was predictable by the applied corrosion

wastage model, but they mainly depend on damage scenarios.

Table 9

RSF mean, lower and upper bound values after 25-year ship lifetime sagging condition.

Damage scenario Mean value Case (I) Case (II) Case (III)

RSF RSF5% RSF95% RSF5% RSF95% RSF5% RSF95%

Side damage (collision scenario) 0.575 0.568 0.582 0.499 0.651 0.506 0.644Bottom damage (1st grounding scenario) 0.641 0.633 0.648 0.563 0.719 0.570 0.711Bottom damage (2nd grounding scenario) 0.672 0.665 0.679 0.588 0.756 0.597 0.747

Table 10

RSF mean, lower and upper bound values after 25-year ship lifetime hogging condition.

Damage scenario Mean value Case (I) Case (II) Case (III)

RSF RSF5% RSF95% RSF5% RSF95% RSF5% RSF95%

Side damage (collision scenario) 0.616 0.611 0.621 0.543 0.689 0.574 0.658Bottom damage (1st grounding scenario) 0.502 0.495 0.508 0.428 0.575 0.448 0.556Bottom damage (2nd grounding scenario) 0.484 0.478 0.490 0.411 0.557 0.434 0.534



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0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

f

Fig. 16. Residual strength factor distribution vs. timebottom damage (1st grounding scenario).



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0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (years)

RSF

Fig. 17. Residual strength factor distribution vs. timebottom damage (2nd grounding scenario).



20/21

In this respect, reference values, generally lying between 8%and 10% and commonly applied in ship structural reliabilityanalysis in damaged conditions, may be avoided and fullstatistical analyses need to be performed.

(iv) Residual strength factor is normally distributed too, withlower and upper bound values, for given non-exceedanceprobability levels, strictly depending on hull girder residualstrength COVs. Furthermore, in case of partial correlation

among input variables, minimum sagging and hogging lowerbound values have been recognized for side damage andsymmetrical grounding scenarios, after 25-year ship lifetime.

Actual results may be considered the rst step of a morereliable evaluation of hull girder residual strength statisticalproperties, as far as different hull girder sections and typologiesmay be analysed under several damage scenarios and the inci-dence of residual strength COVs on ship structural reliabilityanalysis needs to be further investigated.

References

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