Statistical properties of bulk carrier longitudinal strength

Marine Structures 39 (2014) 438e462

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Marine Structures journal homepage: www.elsevier.com/locate/ marstruc

Statistical properties of bulk carrier longitudinal strength A. Campanile a, V. Piscopo b, *, A. Scamardella b a b

The University of Naples “Federico II”, Department of Industrial Engineering, Italy The University of Naples “Parthenope”, Department of Sciences and Technology, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 July 2014 Received in revised form 5 September 2014 Accepted 3 October 2014 Available online

The paper focuses on time-variant longitudinal strength of bulk carriers under corrosion wastage, applying both Taylor series expansion method and Monte Carlo simulation. Hull girder section modulus and ultimate bending moment capacity are determined, by classical beam theory and the commonly applied incrementaliterative approach, respectively. Two main aspects are fully discussed and investigated, according to the final recommendations of last ISSC Report. The former regards the covariance between annual corrosion rates of individual structural members, commonly assumed as uncorrelated. Really, as it is conceivable that a certain correlation exists, three different cases are investigated: uncorrelated variables, fully correlated variables, full correlation between structural members belonging to the same category of compartments. The latter investigates the probability density function of both hull girder section modulus and ultimate bending capacity, because the Central Limit Theorem cannot be applied when correlation among random variables exists. Finally, as a test example, the bulk carrier section, presented in the last ISSC benchmark study, is analysed. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Longitudinal strength Bulk carrier Corrosion Taylor series expansion method Monte Carlo simulation

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

http://dx.doi.org/10.1016/j.marstruc.2014.10.007 0951-8339/© 2014 Elsevier Ltd. All rights reserved.

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1. Introduction Ship structures are continuously exposed to age-related damages such as corrosion, fatigue cracking and localized dents, which may affect the ship structural capacity up to loss of life, property damage and environmental pollution [14]. Corrosion wastage effects can be monitored, or at least mitigated and delayed, by means of ad hoc detail design and proper preventing measures [4], although appropriate thickness reductions have to be always considered, especially for those vessels operating beyond the expected service life, such as bulk carriers, tankers and floating production storage and off-loading units, designed for continuous offshore operations [16]. Corrosion wastage has not been explicitly considered up to the 1980s, since the scantlings of merchant ships were determined by more or less empirical formulas, implicitly accounting for corrosion safety margins, derived by experience and individually developed by each classification body. To harmonize relevant standards and safety margins for double hull oil tankers and bulk carriers, in June 2003 IACS decided to prioritize the development of common structural rules (CSR), that became effective on 1st April, 2006 [9,10]. The CSR are based on the so-called “net scantling approach”, as far as corrosion additions for each structural element are derived independently from the net scantling requirements, to address corrosion wastage that is likely to occur during the ship-in-operation phase. In this respect, the method takes into account the corrosion effect, based on explicitly defined corrosion additions for one side of each structural element, depending on both compartment category and structural member (platings, ordinary stiffeners or primary supporting members). Scantling compliance in relation to rules depend on considered structural requirements (i.e. local strength, hull girder strength, fatigue assessment) and analysis type (thickness, buckling and collapse capacity, hull girder section properties). In the review period of CSR, industry stakeholders urged IACS to harmonize the key technologies used to derive the rules [14], developed independently for double hull oil tankers and bulk carriers. In order to harmonize relevant standards, IACS was committed to create the common structural rules for bulk carriers and oil tankers (CSR-H), issued on 1st January 2014 and expected to become effective on 1st July 2015. In any case, the “net scantling approach” has been entirely preserved with no relevant variations, as regards the CSR approach. The “net scantling approach” is very easy to be applied in the ship design process, but it doesn't consider explicitly the corrosion probabilistic behaviour in terms of mean value and variance of the annual corrosion rate. In this respect, more refined corrosion models were developed in the last years [17]. proposed a corrosion wastage model with some initial nonlinearities due to the activity of the anaerobic sulphate-reducing bacteria (SRB), adopting five phases of corrosion, namely kinetic (actual), kinetic (linearized), oxygen diffusion, SRB growth and SRB steady state [19]. developed a timedependent corrosion wastage model, presenting three different phases, namely coating life, transition time and corrosion wastage, furnishing a wide variety of data, in terms of expected annual corrosion rate and coefficient of variation [7]. proposed a corrosion model based on three subsequent phases, namely no-corrosion period (coating life), signal damage of corrosion coating period (corrosion occurrence) and decrease in the corrosion rate period [21]. presented a similar corrosion model, but different in the second phase, assuming that the coating gradually deteriorates, thus allowing pitting corrosion. The last ISSC Final Report 2012 [12] has also emphasized the need for a corrosion wastage model, based on a full statistical approach, evaluating the ship ultimate strength capacity by means of a probabilistic format, accounting for uncertainties due to thickness wastage. Only recently, the impact of ageing effects on the ultimate strength of hull girder and its components has been considered in the condition assessment of aged ships and offshore structures. In this respect, few advances have been reached in the main degradation modes of ship structures due to corrosion, fracture and mechanical damages. Wang et al. [27] carried out a statistical investigation of time variant hull girder strength of ageing ships and coating life, based on data from ships in service, taken from the measurement of section belts during CAP surveys. Although some difficulties in identifying plate renewals that probably slightly affected the analysis, the authors identified uncertainties in the decrease of tankers' hull girder section modulus and derived some formulas for the relevant strength assessment. Ivanov [13] focused on both hull girder loads and section modulus degradation due to corrosion, determining relevant geometric properties in a probabilistic format as annual distribution for any given lifespan. Probabilistic models of corrosion were combined with as-

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built plate thickness variation models. Saad-Eldeen et al. [22] performed several corrosion deterioration tests for three box girders under ultimate bending moment, to simulate different corrosion levels of the hull and compare relevant results with numerical ones. The box girder, not representing a scale model of an existing ship, was designed, built and tested to simulate the behaviour of a tanker or FPSO midship section, subjected to vertical bending moment and corrosion deterioration. The main objective of the study was to validate the actual approach and support the development of new ones for the ultimate strength assessment of box girder type structures, accounting for relevant deterioration effects along the entire service life. Finally Kim et al. (2014b) [15] investigated various types of industry corrosion addition practices, with regard to the ultimate strength performance of several bulk carriers, namely 37 K (Handysize), 57 K (Supramax), 82 K (Kamsarmax), and 181 K (Capesize), obtaining useful results concerning the impact of different corrosion additions on the ultimate strength capacity. The probabilistic format suggested by the ISSC Committee for the ultimate strength capacity assessment, is based on the first-order Taylor series expansion method and easily allows evaluating both mean value and standard deviation of any required performance function. Anyway, two main aspects, regarding the correct estimation of both standard deviation and probability density distribution of time-variant hull girder ultimate bending capacity, have to be further investigated. In this sense, the correct estimation of co-variation terms among random variables is fundamental to correctly estimate the standard deviation of any performance function that could be sensibly underestimated respect to the correlated variable case. Furthermore, when a certain correlation exists among random variables, the Central Limit Theorem cannot be applied, which implies that it is not possible to assure that the performance function follows the normal distribution. In this respect, the first aim of the present work is to estimate the influence of co-variation terms among basic random variables on standard deviation of hull girder section modulus and ultimate bending moment capacity, accounting for corrosion wastage effects by the time-variant corrosion model proposed by Paik et al. [20]. The classical beam theory and the incremental-iterative approach [11] are applied to evaluate hull girder section modulus and ultimate bending moment capacity, respectively. Both Taylor series expansion method and Monte Carlo simulation are applied to three different scenarios: uncorrelated, fully correlated and partially correlated variables, assuming for the latter one that full correlation exists between members belonging to the same category of compartments, as it will be subsequently specified, for which it is conceivable to assume the same probability level of annual corrosion rate. The incidence of correlation coefficients among random variables, on standard deviation of hull girder section modulus and ultimate bending capacity, is fully discussed and then the three considered scenarios and two proposed methods are compared. Subsequently the shape of time-variant probability density function of both hull girder section modulus and ultimate bending moment capacity is investigated. As previously said, in case of partial or full correlation among random variables, the Central Limit Theorem cannot be applied in its standard form, so Monte Carlo simulation becomes the simplest method to derive the probability density distribution of both hull girder section modulus and ultimate bending moment capacity. Furthermore, it reveals useful in investigating non-linear effects on strength function variance, as Taylor series expansion method is generally applied under the hypothesis of a first-order approximation. Finally the bulk-carrier midship section, presented in the benchmark study shown in the ISSC Final Report 2012 [12], has been assumed as test example and relevant statistical properties have been determined for a 30 year ship service life. 2. Rule and time-variant corrosion wastage models In the last decade classification bodies proposed several standards and rules to establish corrosion additions for ship construction and maintenance of both oil tankers and bulk carriers. In the “Pre-CSR period”, i.e. before 1st April 2006 [15], each individual classification body developed own rules, producing a variety of different standards. To homogenize structural and maintenance requirements and reduce classification societies' competition, after a three-year extensive work, by 1st April 2006 IACS members adopted the common structural rules (CSR) for bulk carriers [9] and oil tankers [10]. CSR rules prescribe general corrosion additions, which mainly depend on structural members' type and location, based on a 25-year ship lifetime. The corrosion thickness is determined as the sum of two values, each one relative to one side exposure, plus a reserve thickness of 0.5 mm.

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Anyway as CSR for bulk carries and oil tankers have been developed independently, in the last years the IACS Council devoted to the harmonization of these standards [11], that will supersede the previous ones and will become effective on 1st July 2015. The new harmonized standard will consist of two parts, the former providing requirements common to both double hull oil tankers and bulk carriers, mainly related to general arrangement design, hull girder loads and scantling, buckling and fatigue analysis, the latter providing additional specific requirements. As concerns the rule corrosion wastage model, the CSR-H adopt the “net-scantling approach”, as well as CSR ones. Rule corrosion additions have been determined based on a corrosion propagation model, developed by probabilistic theory. The estimation of corrosion wastage was made considering 25 years at the cumulative probability of 90% and the corrosive environment for different structural members. In this way, it was ensured that the corrosion additions are in direct relation to measured wastage data, accounting for the corresponding corrosive environment and different structural members. The “net-scantling approach” is useful in practical scantling procedures, but it furnishes no information about corrosion statistical properties, at least in terms of mean value and standard deviation of annual corrosion rate, which implies that global responses, such as hull girder section modulus and ultimate bending moment capacity, can be determined only on the basis of the net scantling properties, roughly obtained by the gross offered thickness, reduced by one half corrosion addition for each structural element. In order to predict likely corrosion damage tolerance, a real estimate of corrosion rate for each structural area is necessary [20]. In this sense, the easiest alternative to the “net-scantling approach” is the statistical analysis of past data for comparable situations, by which mean value and standard deviation of annual corrosion rate of each longitudinal strength member are derived. The time-variant corrosion wastage model by Paik et al. [19] allows to determine the reduction thickness tr(T) at time T (in years) on the basis of both coating life Tc (in years) and annual corrosion rate C1 (in mm/year) for each structural element:

tr ðTÞ ¼ C1 ðT  Tc Þ

(1)

The model is based on a corrosion rate that linearly varies with time, even if some past studies indicate that the corrosion rates apparently decrease or stabilize over time for statically loaded structures [29,18]. On the contrary, for dynamically loaded ones, the corrosion rate may accelerate with time, as the mechanical stress may increase the rate of the anodic reaction and the corrosion environment may change over time, due to either different operating environments or different cargo, as influenced by ship operation [20]. Because of these uncertainties in the behaviour of corroded structures, the annualized corrosion rate can be assumed constant regardless of time for practical design purposes. Both expected value mC1 and standard deviation sC1 of annual corrosion rate have been determined [20] for 23 types of primary structural member groups of bulk carrier, on the basis of a large amount of thickness measurements, carried out on 109 in service ships. The best-fit probability density function was determined for both average and severe corrosion rates, founding that while for young ships the corrosion depth relative frequency follows the normal distribution, for old structures it follows the two parameter Weibull one, whose scale (a) and shape (l) parameters can be easily determined from:

  8 1 > > ¼ aG 1 þ m > C1 < l      > 2 1 > > : sC1 ¼ a2 G 1 þ  G2 1 þ l l

(2)

Denoting by tg the gross thickness of any structural element, the net one x(T) at time T is given by:

  xðTÞ ¼ tg  tr T

(3)

The net thickness statistical characteristics at time T > Tc can be easily found, denoting by COV the coefficient of variation of the annual corrosion rate:

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mX ¼ tg  mC1 ðT  Tc Þ sX ¼ mC1 ðT  Tc ÞCOV

(4)

In Table 1 average and severe annualized corrosion rates mC1 and coefficient of variation COV, for 23 different structural members of bulk carriers, subdivided by location and category, are reported [20]. Relevant data refer to a coating life of 7.5 years; the average corrosion was determined based on all measured data, while the severe one by the corrosion measurement corresponding to 95% and upper band. 3. Statistical methods Statistical properties of hull section modulus and ultimate bending moment capacity can be determined by either Taylor series expansion method or Monte Carlo simulation. The former is normally less time-consuming, but it doesn't allow deriving the target function probability distribution, except in some cases when the Central Limit Theorem can be applied. On the contrary, the latter method is more time consuming, as it requires many simulated events, depending on the number of basic random variables. In the following, a brief review of the applied methods is presented. 3.1. Taylor series expansion method Let x1, …,xn be random variables with mean mX1, …,mX1 and standard deviation sX1, … sXn. Any function G(x1,x2, …,xn) can be expanded into the first-order Taylor series around its mean value mX¼(mX1, …, mX1):

GðxÞyGðmX Þ þ

 n  X vG i¼1

vxi

ðxi  mXi Þ

(5)

mX

Table 1 Average and severe corrosion rates for bulk carriers [20]. Member type

Outer bottom plates Lower wing tank side shells Side shells Upper wing tank side shells Upper deck plates Upper sloping plates Lower sloping plates Inner bottom plates Bottom girders Outer bottom longitudinals e web Outer bottom longitudinals e flange Lower wing tank side longitudinals e web Lower wing tank side longitudinals e flange Upper wing tank side longitudinals e web Upper wing tank side longitudinals e flange Upper deck longitudinals e web Upper deck longitudinals e flange Upper sloping longitudinals e web Upper sloping longitudinals e flange Lower sloping longitudinals e web Lower sloping longitudinals e flange Inner bottom longitudinals e web Inner bottom longitudinals e flange

Acronym

OBP LWTSS SS UWTSS UDP USP LSP IBP BG OBLW OBLF LWTSLW LWTSLF UWTSLW UWTSLF UDLW UDLF USLW USLF LSLW LSLF IBLW IBLF

Average corrosion

Severe corrosion

mC1 (mm/year)

COV

mC1 (mm/year)

COV

0.0497 0.0698 0.0670 0.0532 0.1188 0.0530 0.0952 0.1265 0.0580 0.0404 0.0488 0.0374 0.0466 0.0627 0.0861 0.0937 0.0454 0.0604 0.0859 0.0465 0.0735 0.0472 0.0731

0.9557 1.0931 0.7597 1.0751 0.9217 0.9452 0.7794 0.6751 1.2741 1.0148 1.0819 1.0000 1.0107 0.9425 0.9140 0.9413 0.7136 0.8526 0.7892 0.9698 1.0312 1.0508 1.1969

0.1882 0.1845 0.2020 0.2060 0.3681 0.1748 0.2901 0.3295 0.2235 0.1298 0.1782 0.1267 0.1713 0.1853 0.2360 0.3135 0.0857 0.1848 0.2413 0.1444 0.1652 0.1638 0.2408

0.2387 0.2185 0.1974 0.2055 0.1866 0.1942 0.2080 0.2055 0.1827 0.1827 0.1974 0.1649 0.1539 0.1827 0.1866 0.2055 0.2185 0.1974 0.1974 0.1649 0.1974 0.1709 0.1758

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Denoting by rij the correlation coefficient between any random variables xi and xj (0 for uncorrelated variables and 1 if full correlation exists), the mean value mG and the standard deviation sG of G(x) can be determined as follows [25]:

8 m ¼ Gðm Þ G X > > < 2 31=2 ! 2 n  X X  vG  vG vG > > s2 þ 2 sXi sXj rij 5 : sG ¼ 4 vxi m Xi vxi m vxj i>j i¼1 X X mX

(6)

If all random variables are uncorrelated and identically distributed, G(x) follows the normal func
vy Central Limit Theorem, while when all variables are uncorretion, according to the LindebergeLe lated but not identically distributed, the LindebergeFeller Central Limit Theorem is applied [8], according to which the sum of independent random variables, properly standardized but not necessarily identically distributed, converges to the normal distribution, as long as the Lindeberg condition is satisfied:

 n h i 1 X 1 if jxi mxi j  εsG 2 E ðx m Þ Ifjx m  εs ¼ 0 c ε>0 ; If:::g ¼ j g G i xi i xi n/∞ s2 0 if jxi mxi j<εsG G i¼1 lim

(7)

having denoted by I the indicator function. The above equation is in turn implicitly satisfied if the Lyapunov condition exists:

d d > 0 : lim

n/∞

1

n

X E jxi  mxi j2þd ¼ 0

s2þd i¼1 G

(8)

In the case of interest it can be proved (see Appendix A), that Eq. (8) can be resembled by the following inequality:

( max

i¼1:::n

)   1 vG ðT  Tc ÞgCi sCi < 1 GðmX Þ vxi mX

(9)

having denoted by gci the skewness of the two-parameter Weibull distribution:

gCi ¼

  G 1 þ l3i a3i  3mCi s2Ci  m3Ci s3Ci

(10)

Anyway, when a certain correlation exists among random variables, it is quite hard to prove the statement of the Central Limit Theorem for weakly dependent processes, so that a simpler technique, such as Monte Carlo simulation, even if more time consuming, becomes preferable. Furthermore, Taylor series method is commonly applied under a first-order expansion, so that Monte Carlo simulation may reveal possible non-linear effects on the target function variance. 3.2. Monte Carlo simulation The application of Monte Carlo simulation to ship structural analysis is quite recent. Garbatov et al. [6] estimated the ultimate strength of stiffened panels by applying classical FEM models, including the effects of different structural parameters on the uncertainty of results by a Monte Carlo simulation and an analysis of variance (ANOVA) methodology. Vhanmane and Bhattacharya [26] determined the effects of randomness in yield strength and initial imperfections on hull girder ultimate strength capacity, founding that correlation has no effect on the mean value of ultimate strength, but the

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uncertainty significantly increases with a higher correlation between yield strength and initial imperfections. Furthermore, they found that variations in hull girder strength are lower when both sources of uncertainties were considered together, rather than separately. Moving from hull girder to hull structural components Silva et al. [23], applied Monte Carlo simulation to steel plates subjected to non-uniform corrosion wastage, under axial compressive loads, to simulate the effects of randomly distributed corrosion on the plating ultimate strength capacity. Based on the analysis of the surface geometry of 570 plates, they developed some empirical formulas to predict the plating strength reduction due to corrosion. In the present paper, Monte Carlo simulation is applied considering as random variables the thickness reduction of any structural member contributing to hull girder section modulus or ultimate bending moment capacity. Three different cases have been simulated: (i) all random variables are independent; (ii) random variables are fully correlated; (iii) full correlation exists among groups of variables belonging to the same category of compartments, as it will be subsequently discussed. After fixing the time interval at which hull girder section modulus and ultimate bending capacity are determined, Monte Carlo simulation can be easily carried out for the first case, generating random annual corrosion rates for each input variable, based on relevant Weibull distribution for each structural element. Instead, if correlation exists among all variables or groups of them, as assumed in cases (ii) and (iii), a different procedure, based on two subsequent steps, is necessary. A random number p is firstly generated in the interval [0, 1] for the entire structural section in case (ii), or for each category of compartments in case (iii). Annual corrosion rates of each structural element are subsequently determined, based on relevant cumulative probability distribution function. As concerns the number of simulations, it has to be properly chosen in order to obtain consistent results, in terms of both mean value and standard deviation of hull girder section modulus and ultimate bending moment capacity, founding that in the case of interest 1000 iterations are always sufficient. 4. Statistical properties of ISSC bulk carrier 4.1. General data and preliminary analysis The bulk carrier section scheme, reported in the benchmark study carried out by the ISSC Ultimate Strength Committee [12], is analysed. Gross scantlings and material properties are shown in Fig. 1. A dedicated programme developed in Matlab MathWorks has evaluated time-variant longitudinal strength statistical properties. The hull girder section modulus has been determined by classical beam theory, while the ultimate bending moment capacity by the incremental-iterative approach, according to Common Structural Rules for Bulk Carriers and Oil Tankers [11]. The Matlab Code has been preliminarily tested, comparing actual gross scantling sectional properties with the ones in the ISSC Final Report [12], where the ultimate hogging MH and sagging MS bending moment capacities have been determined by several Working Organizations and candidate methods. Although a certain variability between candidate methods and Working Organizations is recognized, actual values are in accordance with the FE ones [24,28]. As regards overall strength, it is noteworthy that bulk carriers are sensitive to torsional [2] and shear [3] effects, as well as to longitudinal bending ones, although they are out of the aim of the present study. 4.2. Longitudinal strength statistical properties According to the final recommendations issued by the last ISSC Final Report [12], current analysis of time-variant hull girder section modulus and ultimate bending moment capacity investigates two different aspects: (i) the influence of covariance between random variables; (ii) the target function probability density distribution. As concerns the point (i), three different cases are analysed: (I) uncorrelated variables, (II) fully correlated variables, (III) full correlation among elements belonging to the same category of compartments. Cases (I) and (II) represent the least and the most conservative scenarios respectively, while case (III) is intermediate, as it assumes that all structural members, belonging to the same category of compartments, are subjected to the same environmental and operational conditions, hence showing

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Fig. 1. Bulk carrier section scheme.

corrosion rates with the same probability of exceedance. In this respect bulk-carrier structural members can be subdivided into four categories of compartments, namely double-bottom, lower wing tank, upper wing tank and side, for which it is conceivable that the annual corrosion rate probability level is almost the same. The point (ii), instead, investigates some aspects concerning the Central Limit Theorem and its applicability. As previously said, in fact, the LindebergeFeller Central Limit Theorem can be applied in case (I), but if there is correlation among random variables, as for cases (II) and (III), it becomes hard to prove that the target function is normally distributed. So in the following Monte Carlo simulation is used to derive the best-fit probability density function of both hull girder section modulus and ultimate bending moment capacity. 4.3. Taylor series expansion method Thickness reductions have been evaluated according to Table 1 for severe annual corrosion rates. Partial derivatives, to be inserted in Eq. (5), have been determined by central differencing the target function, as this method reveals superior respect to the forward or backward differencing ones [5]:



vG vxi

 mX

GðmX þ 2dÞ þ 8GðmX þ dÞ  8GðmX  dÞ þ GðmX  2dÞ z 12d

(11)

In Table 2 both mean values and coefficients of variation (COV) of hull girder section modulus at keel ZK and bottom ZD are reported. As predictable, COVs increase with time and strictly depend on covariance among random variables [1]. While for case (I) COV reaches a maximum value of 0.671%, when there is full correlation among all variables or groups of them, as assumed in cases (II) and (III), it

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Table 2 Mean values and COVs of hull girder section modulus e Taylor series expansion. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Mean value

Case (I)

Case (II)

Case (III)

E(ZK) m3

E(ZD) m3

COV(ZK) %

COV(ZD) %

COV(ZK) %

COV(ZD) %

COV(ZK) %

COV(ZD) %

62.325 60.243 58.160 56.078 53.995 51.912 49.829 47.746 45.663 43.579

44.625 43.213 41.801 40.388 38.976 37.564 36.152 34.740 33.327 31.915

e 0.042 0.087 0.136 0.188 0.245 0.306 0.373 0.445 0.525

e 0.055 0.114 0.177 0.244 0.317 0.395 0.480 0.571 0.671

e 0.709 1.469 2.285 3.164 4.114 5.144 6.265 7.487 8.827

e 0.638 1.319 2.047 2.828 3.668 4.574 5.553 6.616 7.772

e 0.501 1.039 1.617 2.240 2.914 3.645 4.441 5.311 6.266

e 0.578 1.194 1.854 2.561 3.321 4.141 5.027 5.988 7.034

reaches a maximum value of 8.827% and 7.034%. Table 3 shows the same results for the hull girder ultimate bending moment capacity in hogging MH and sagging MS conditions: similar conclusions can be withdrawn, in terms of both mean values and coefficients of variation. Fig. 2 shows the mean momentecurvature curves and neutral axis height above keel at various times. Finally, the inequality (9) is always satisfied, reaching a maximum value of 104, so much less than one. In this respect, the LindebergeFeller Central Limit Theorem, applicable to case (I) only, assures that hull girder section modulus and ultimate bending moment capacity are normally distributed. On the contrary, when correlation exists Monte Carlo simulation is useful to determine the best-fit probability density function of longitudinal strength capacity. 4.4. Monte Carlo simulation A similar analysis has been carried out by Monte Carlo simulation; 1000 samplings revealed sufficient to obtain consistent results, in terms of both mean value and coefficient of variation. Relevant results, shown in Tables 4 and 5, are in accordance with the Taylor ones, as predictable. Some slight differences on the target function coefficient of variation, increasing with time T, may be probably due to some non-linear effects, not accounted by the first-order Taylor series expansion method. Monte Carlo simulation also allowed obtaining the best-fit probability density function of both hull girder section modulus and ultimate bending moment capacity. In this respect Tables 6 and 7 show the correlation coefficients of the normal distribution, assumed as best-fit curve. Actual values for case (I) confirm the LindebergeFeller Central Limit Theorem. The surprising result is that even when correlation exists, as assumed in cases (II) and (III), where the Central Limit Theorem is not directly applicable, relevant correlation coefficients are very high, which implies that the probability density function is very close to the normal one. Figs. 3e5 show the best-fit curves for cases (I), (II) and (III) at various time intervals. Table 3 Mean values and COVs of hull girder ultimate bending moment capacity e Taylor series expansion. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5

Mean value

Case (I)

Case (II)

Case (III)

E(MS) GNm

E(MH) GNm

COV(MS) %

COV(MH) %

COV(MS) %

COV(MH) %

COV(MS) %

COV(MH) %

14.708 14.069 13.497 12.839 12.275 11.660 11.052 10.460 9.885

18.364 17.681 16.959 16.252 15.548 14.810 14.099 13.346 12.595

e 0.056 0.116 0.183 0.253 0.331 0.415 0.503 0.601

e 0.044 0.090 0.141 0.195 0.255 0.319 0.391 0.469

e 0.804 1.670 2.609 3.618 4.715 5.912 7.158 8.557

e 0.765 1.597 2.515 3.494 4.592 5.748 7.092 8.546

e 0.589 1.223 1.926 2.668 3.492 4.400 5.249 6.300

e 0.485 1.008 1.574 2.186 2.855 3.577 4.394 5.277

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Fig. 2. Momentecurvature and centre of mass position-curvature at various times.

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Table 4 Mean values and COVs of hull girder section modulus e Monte Carlo simulation. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Mean value

Case (I)

Case (II)

Case (III)

E(ZK) m3

E(ZD) m3

COV(ZK) %

COV(ZD) %

COV(ZK) %

COV(ZD) %

COV(ZK) %

COV(ZD) %

62.325 60.240 58.160 56.080 53.990 51.910 49.830 47.740 45.660 43.580

44.625 43.210 41.800 40.390 38.980 37.570 36.160 34.740 33.340 31.920

e 0.042 0.089 0.137 0.179 0.233 0.292 0.368 0.426 0.518

e 0.053 0.111 0.190 0.243 0.315 0.394 0.459 0.569 0.643

e 0.696 1.457 2.203 3.144 4.124 5.097 6.076 7.496 8.917

e 0.621 1.308 1.965 2.802 3.631 4.547 5.348 6.601 7.729

e 0.504 1.063 1.678 2.225 2.987 3.651 4.721 5.449 6.264

e 0.570 1.190 1.851 2.673 3.460 4.182 5.104 6.271 6.982

5. Longitudinal strength of ISSC bulk carrier Monte Carlo simulation allowed verifying that hull girder section modulus and ultimate bending moment capacity follow the normal distribution, even if there is correlation among random variables. Relevant lower bound values, corresponding to a 5% probability of non exceedance, can be now determined, for any normally distributed function with mean value E[G] and coefficient of variation COV%, by the following formula:

  COV% G5% ¼ E½G 1  1:645 100

(12)

In Tables 8 and 9 mean and lower bound values of hull girder section modulus and ultimate bending moment capacity are shown for the three analysed cases. Lower bound values are significantly influenced by covariance terms among random variables. If all variables are uncorrelated, as assumed in case (I), mean and lower bound values are almost coincident; on the contrary when full correlation exists, as assumed in cases (II) and (III), lower bound values are significantly lower. It follows that the correct estimation of covariance among input variables is a focal point in actual analysis. In this respect cases (I) and (II) give the maximum and minimum lower bound values, which implies that an intermediate hypothesis, such as the one assumed in case (III) could be the most reasonable one. Anyway, it is demonstrated that covariance is one of the major aspects to be further investigated. The same results are plotted in Figs. 6 and 7, too. It is interesting to highlight that the incidence of covariance on lower bound values increases with time, as it could be predictable by the applied corrosion wastage model. Fig. 6 plots the ratio between actual and gross hull girder section modulus at

Table 5 Mean values and COVs of hull girder ultimate bending moment capacity e Monte Carlo simulation. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Mean value

Case (I)

Case (II)

Case (III)

E(MS) GNm

E(MH) GNm

COV(MS) %

COV(MH) %

COV(MS) %

COV(MH) %

COV(MS) %

COV(MH) %

14.708 14.100 13.490 12.890 12.290 11.700 11.110 10.510 9.926 9.311

18.364 17.680 16.960 16.260 15.550 14.810 14.100 13.360 12.610 11.860

e 0.073 0.153 0.262 0.338 0.440 0.562 0.679 0.856 1.031

e 0.044 0.094 0.157 0.202 0.295 0.345 0.461 0.576 0.646

e 0.805 1.738 2.589 3.769 4.940 6.228 7.330 9.442 11.592

e 0.803 1.567 2.687 3.743 4.959 6.326 7.879 9.821 11.912

e 0.777 1.595 2.559 3.760 4.939 6.000 7.280 9.244 11.226

e 0.517 1.099 1.625 2.396 3.159 4.086 5.619 6.395 7.899

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449

Table 6 Correlation coefficients of hull girder section modulus e normal distribution. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Case (I)

Case (II)

Case (III)

r(ZK)

r(ZD)

r(ZK)

r(ZD)

r(ZK)

r(ZD)

e 0.980 0.988 0.992 0.995 0.995 0.994 0.976 0.995 0.981

e 0.986 0.979 0.934 0.985 0.985 0.986 0.985 0.985 0.982

e 0.967 0.979 0.985 0.966 0.989 0.979 0.987 0.967 0.989

e 0.980 0.958 0.986 0.962 0.987 0.958 0.985 0.962 0.988

e 0.991 0.968 0.952 0.967 0.962 0.958 0.981 0.963 0.989

e 0.968 0.958 0.967 0.956 0.953 0.988 0.979 0.950 0.972

keel (in black) and bottom (in red) vs. time; continuous, dashed and pointed-dashed lines represent cases (I), (II) and (III), respectively. Fig. 7 similarly plots the ratio between actual and hull girder bending moment capacity vs. time. It must be finally pointed out that the correct estimation of strength function variance influences the scheduling of maintenance and repair operations that are necessary when actual sectional properties become lower than the minimum rule value. 6. Conclusions The paper focuses on statistical properties of time-variant hull girder section modulus and ultimate bending moment capacity, determined by Taylor series expansion method and Monte Carlo simulation. According to the recommendations of last ISSC Final Report, two main aspects have been discussed and analysed. The former investigates the influence of correlation among random variables on the target function coefficient of variation. Three different cases are examined: uncorrelated variables, fully correlated variables, full correlation among members belonging to the same category of compartments. The latter aspect investigates the probability density distribution of the required strength function as, in the case of correlation among random variables, it is not possible to directly apply the Central Limit Theorem. In this respect, Monte Carlo simulation allowed to overcome this problem, furnishing useful information about the required probability density function. After an extensive study on a bulk carrier section, analysed in the last ISSC benchmark study on hull girder ultimate strength, two main results have been achieved: (i) The coefficient of variation of hull girder section modulus and ultimate bending moment capacity is sensibly influenced by covariance among random variables. The commonly applied hypothesis of uncorrelated variables may lead to overestimate relevant lower bound values, while the hypothesis of full correlation may lead to underestimate them. Although it is a questionable assumption, it seems that the proposed hypothesis of full correlation among Table 7 Correlation coefficients of hull girder ultimate bending moment capacity e normal distribution. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Case (I)

Case (II)

Case (III)

r(MS)

r(MH)

r(MS)

r(MH)

r(MS)

r(MH)

e 0.980 0.971 0.964 0.975 0.984 0.981 0.983 0.976 0.985

e 0.985 0.987 0.993 0.992 0.988 0.987 0.970 0.987 0.981

e 0.969 0.945 0.985 0.970 0.981 0.959 0.984 0.965 0.985

e 0.980 0.962 0.973 0.967 0.980 0.972 0.973 0.962 0.989

e 0.966 0.963 0.975 0.965 0.932 0.988 0.976 0.955 0.974

e 0.991 0.987 0.961 0.953 0.958 0.979 0.985 0.975 0.987

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A. Campanile et al. / Marine Structures 39 (2014) 438e462

Fig. 3. Monte Carlo simulation e Uncorrelated variables e Case (I).

A. Campanile et al. / Marine Structures 39 (2014) 438e462

Fig. 3. (continued).

451

452

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Fig. 3. (continued).

A. Campanile et al. / Marine Structures 39 (2014) 438e462

Fig. 4. Monte Carlo simulation e Fully correlated variables e Case (II).

453

454

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Fig. 4. (continued).

A. Campanile et al. / Marine Structures 39 (2014) 438e462

Fig. 4. (continued).

455

456

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Fig. 5. Monte Carlo simulation e Partial correlation among variables e Case (III).

A. Campanile et al. / Marine Structures 39 (2014) 438e462

Fig. 5. (continued).

457

458

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Fig. 5. (continued).

A. Campanile et al. / Marine Structures 39 (2014) 438e462

459

Table 8 Mean and lower bound of hull girder section modulus e Taylor series expansion method. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Mean value

Case (I)

Case (II)

Case (III)

E(ZK) m3

E(ZD) m3

ZK5% m3

ZD5% m3

ZK5% m3

ZD5% m3

ZK5% m3

ZD5% m3

62.325 60.243 58.160 56.078 53.995 51.912 49.829 47.746 45.663 43.579

44.625 43.213 41.801 40.388 38.976 37.564 36.152 34.740 33.327 31.915

e 60.201 58.077 55.953 53.828 51.703 49.578 47.453 45.329 43.203

e 43.174 41.723 40.270 38.820 37.368 35.917 34.466 33.014 31.563

e 59.540 56.755 53.970 51.185 48.399 45.613 42.825 40.039 37.251

e 42.759 40.894 39.028 37.163 35.297 33.432 31.567 29.700 27.835

e 59.747 57.166 54.586 52.005 49.424 46.841 44.258 41.674 39.087

e 42.802 40.980 39.156 37.334 35.512 33.689 31.867 30.044 28.222

Table 9 Mean and lower bound of hull girder ultimate bending moment capacity e Taylor series expansion method. T (years)

up to 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

Mean value

Case (I)

Case (II)

Case (III)

E(MS) GNm

E(MH) GNm

MS5% GNm

MH5% GNm

MS5% GNm

MH5% GNm

MS5% GNm

MH5% GNm

14.708 14.069 13.497 12.839 12.275 11.660 11.052 10.460 9.885 9.307

18.364 17.681 16.959 16.252 15.548 14.810 14.099 13.346 12.595 11.868

e 14.059 13.478 12.810 12.237 11.613 10.996 10.396 9.813 9.227

e 17.665 16.927 16.205 15.486 14.733 14.007 13.241 12.477 11.737

e 13.905 13.171 12.356 11.636 10.871 10.117 9.382 8.668 7.956

e 17.495 16.591 15.705 14.825 13.916 13.038 12.127 11.224 10.351

e 13.953 13.266 12.497 11.823 11.101 10.389 9.696 9.021 8.348

e 17.513 16.626 15.756 14.893 14.001 13.139 12.242 11.354 10.495

Fig. 6. Time-variant lower bound hull girder section modulus vs. time. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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A. Campanile et al. / Marine Structures 39 (2014) 438e462

Fig. 7. Time-variant lower bound hull girder ultimate bending moment capacity vs. time.

structural elements belonging to the same category of compartments may be considered a suitable choice. (ii) Hull girder section modulus and ultimate bending moment capacity follow the normal distribution, not only under the assumption of uncorrelated variables, to which the LindebergeFeller Central Limit Theorem applies, but also when full correlation exists and the above theorem cannot be applied. The correct knowledge of probability density function of hull girder longitudinal strength capacity is, in this respect, fundamental to correctly estimate its lower bound value with 5% percentage of exceedance. Actual analysis may be considered the first step of a more comprehensive evaluation of hull girder statistical properties. In this respect, other hull girder sections may be analysed, in order to further study the incidence of correlation among random variables on the required target function and extend actual results to other ship typologies.

Appendix A Hull girder section modulus and ultimate bending moment capacity can be developed into firstorder Taylor series, so becoming the sum G(… ) of n random variables x1, x2,...,xn having mean value mx1 ; mx2 ; :::; mxn and representing the net thickness of any structural element at time T:

Gðx1 ; x2 ; :::xn Þ ¼ GðmX1 ; mX2 ; :::mXn Þ þ

 n  X vG i¼1

vxi

ðxi  mXi Þ

(A.1)

mX

The net thickness xi depends on the gross one tg,i, as well as on relevant annual corrosion rate Ci and coating life Tc, according to the corrosion wastage model proposed by Paik et al. [19]:

xi ¼ tg;i  Ci ðT  Tc Þ

(A.2)

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461

Eq. (A.1) can be rewritten in a non-dimensional form as follows:

gðx1 ; x2 ; :::xn Þ ¼ 1 

n X

Yi

i¼1

;

Yi ¼

    1 vG C  mCi ðT  Tc Þ GðmX Þ vxi mX i

(A.3)

having denoted by mCi the mean value of relevant annual corrosion rate. If all variables are independent (but not necessarily identically distributed) the LindebergeFeller Central Limit theorem states that the sum converges in distribution to the standard normal, as long as the Lindeberg condition is satisfied [8]:

lim

n/∞

n h  i 1 X E ðYi Þ2 I jYi j  εsg ¼ 0 c ε > 0 s2g i¼1

(A.4)

denoting by I the indicator function and sg the standard deviation of g(… ) that for uncorrelated variables is:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n uX sg ¼ t s2Yi

(A.5)

i¼1

The Lindeberg condition is in turn satisfied if the Lyapunov condition holds:

d d > 0 : lim

n/∞

1 s2þd g

n

X E jYi j2þd ¼ 0

(A.6)

i¼1

Eq. (A.6) is generally proved for d ¼ 1 [8], finally becoming:

lim

n/∞

n

1 X E jYi j3 ¼ 0 3 sg i¼1

(A.7)

As in the case of interest all variables follow the two parameter Weibull distribution, the following equality holds:

E jYi j3 ¼

"

#3   1 vG ðT  Tc Þ gCi s3Ci GðmX Þ vxi mX

(A.8)

denoting by gCi and sCi the skewness and standard deviation of Ci, respectively. Denoting by sYi the standard deviation of Yi, Eq. (A.8) can be so rewritten:

E jYi j3 ¼

  1 vG ðT  Tc ÞgCi sCi s2Yi GðmX Þ vxi mX

(A.9)

If the inequality (A.10) holds:

(

)   1 vG max ðT  Tc ÞgCi sCi < 1 i¼1:::n GðmX Þ vxi m X

(A.10)

the Lyapunov condition is in turn satisfied, as it follows:

lim

n/∞

n n

1 X 1 X 1 E jYi j3 < lim 3 s2Yi ¼ lim ¼0 3 n/∞ s n/∞ sg sg i¼1 g i¼1

(A.11)

In the case of interest (A.10) inequality was by far numerically satisfied in all the worked cases.

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