Reliability assessment of ship hull girders considering pitting corrosion and crack

The current study aims to investigate the combined effect of cracking and pitting damage on the ultimate strength of ships. The well-known Smith’s approach is modified considering the random number and distribution of cracked-pitted plates in the ship cross-section. Using the Monte Carlo approach, the structural reliability index of the cracked-pitted ship is determined. A single-bottom oil tanker’s ultimate strength is computed, and it turns out that the reliability indices for various damage scenarios are nearly identical when the ship is at its early age. When the ship ages, its reliability index rises to its maximum if the damage is concentrated at the bottom under sagging conditions and at the sides and longitudinal bulkheads in hogging conditions. The reliability indices in the hogging conditions are often greater than those in the sagging conditions. Furthermore, it is determined that, while the ship is at its early age, the detrimental effect of pitting, cracking, or a combination of both on the reduction of the ship’s hull girder ultimate strength is equal. The lowest reliability index is seen in aged ships when cracking and pitting are combined, followed by cracking and pitting damage separately. It is shown that pitting corrosion has a lower reliability index than the general type of corrosion.


Introduction
Ultimate strength analysis is one of the limit state design approaches of the structures in which collapse load is determined.The ultimate strength of a complex structure as a ship depends on the ultimate strength of its structural components, i.e. plates and stiffened plates.The ultimate strength determination of plates and stiffened plates has been the subject of much research in recent years.Average stress-average strain curves, also known as load-end shortening curves, are the output of these investigations.They are provided for various boundary conditions, plate aspect ratios, loading conditions (uniaxial, biaxial in-plane loading with/without lateral loading), and material characteristics.
Numerous studies have examined the negative impact of corrosion or cracks on the ultimate strength of plates and stiffened plates.Rahbar Ranji [1] used numerical methods to determine the average stress-average strain curves for plates with undulated surfaces due to general corrosion.Rahbar Ranji and Zarookian [2] determined the average stress-average strain curves for stiffened plates with a crack using numerical methods.Hu et al [3] numerically investigated the ultimate strength of the cracked stiffened plates under cyclic loads.Poknam et al [4] studied the ultimate strength of continuous stiffened panels with a crack under combined lateral pressure and in-panel compression.Silva et al [5] performed numerical analyses to evaluate the ultimate strength of the corroded plates with random location and form of the corrosion.
The effect of corrosion on the initiation and/or propagation of cracks was the subject of some studies.Song et al [6] studied pit-to-crack transition under simultaneous electrochemical and mechanical effects.Zijie et al [7] studied the fatigue life of pressure vessels with multiple pits and cracks.
The ultimate strength of damaged plates/stiffened plates with both cracks and corrosion has been studied by some researchers.Ahmadi and Rahbar Ranji [8] and Ahmadi et al [9] developed average stress-average strain curves for degraded plates with both pitting corrosion and cracking.The ultimate strength of plates with both crack and corrosion damage was investigated by Feng et al [10].Woloszyk and Garbatov [11] studied the ultimate strength of plates with cracks and corrosion.
Harsh environments and operational conditions induce many deteriorations in ships, in which corrosion and cracking are the most frequent.Due to recycling challenges and the high cost of new construction, attempts have been made to prolong the life of older ships.This has led to numerous research on the assessment of the ship's residual strength.Tekgoz et al [12] provided an overview of the ultimate strength assessment of the aging and damaged ship structures.Feng et al [13] studied the ultimate torsional strength of large deck opening stiffened box girder subjected to pitting corrosion.Vu and Dong [14] studied the effect of corrosion in the web and flange of stiffeners on the hull girder's ultimate strength.Van and Yang [15] studied the effect of corrosion wastage on the ship hull of a double-hull tanker.
The reliability-based method provides a logical framework to consider the uncertainties that exist in the analysis of structures.The main point is basically to make a logical connection between the probability of structural failure and uncertain parameters.Several reliability-based studies are carried out to determine the residual strength of the corroded plates and ships.Guo-qing et al [16] numerically analyzed the ultimate strength of the corroded stiffened panels using the Monte Carlo simulations to define statistical descriptions of the ultimate strength of the corroded stiffened panels.Woloszyk and Garbatov [17] carried out the reliability assessment of the ultimate strength of a tanker ship subjected to corrosion degradation.Zayed et al [18] investigated the reliability of ship structures degraded by corrosion.Paik and Frieze [19] determined the reliability of ship structure.
Some researchers have studied the reliability of structures degraded by a combination of corrosion and other flaws.Huang et al [20] developed a method to evaluate the reliability of welded structures subjected to corrosion and fatigue loading.Akpan et al [21] calculated the reliability of aged ship hull structures degraded by corrosion, and corrosion-enhanced fatigue cracks.Kwon and Frangopol [22] studied the reliability of ship hull structures under corrosion and fatigue.
As seen, a lot of research has been done on the ultimate strength of ship hull girders and the reliability of ship structures considering corrosion and fatigue cracks.To the best of the authors' knowledge, no research has been done on the reliability of ship hull girders when pitting corrosion and/or cracks occur randomly.This study investigates the detrimental effect on a ship's ultimate strength reliability resulting from the combined effects of cracking and pitting damages which are randomly distributed across the transverse cross-section of the ship.The well-known Smith's approach has been modified to consider the average stress-average strain curves of crackedpitted plates randomly distributed in the midship section.The stress-strain curves of plates degraded by crack and pitting corrosion are derived based on an extensive non-linear Finite Element Analysis (FEA) [8,9].Having the load-end shortening curves of cracked-pitted plates, the ultimate strength of the ship hull girder is determined, and the corresponding reliability index is calculated.

Ultimate strength of ship hull girder
The vertical bending moment and shear force are two internal forces imposed on the ship's transverse crosssection because of weight and buoyancy.The maximum vertical bending moment a ship can withstand before failure is defined as the ultimate strength of the ship hull girder, and the determination of it is necessary according to the International Association of Classification Society (IACS) for ships larger than 150 m [23].The ship hull girder's ultimate strength is determined using the widely used incremental-iterative approach or progressive collapse method, which is based on Smith's principle [24].The method assumes that the ship's transverse section remains plane during each curvature increment.The ship's transverse section is divided into a set of elements composed of stiffeners and attached plating, which are considered to act according to corresponding load-end shortening curves.The bending moment applied on the ship transverse section at the assumed/determined curvature is obtained by summing up the contribution given by the stresses acting on each element.The stress corresponding to the strain of each element is determined for each curvature increment from that element's non-linear load-end shortening curves.The peak point of the bending moment curve versus curvature is defined as the ultimate strength of the ship hull girder.
A computer code developed in MATLAB based on Smith's method to consider the random number and distribution of plates degraded by corrosion and/or cracks in the ship cross-section.The input data includes the main dimensions of the ship, effective elements in the hull girder, the geometric center, the cross-sectional area, and the average stress-average strain curve of each element.
Figure 1 shows the algorithm of Smith's method, where, ̅ ( ̅ ) s e i i is the average stress-average strain curve of i-th element; s i is the stress in i-th element; e i is the strain in i-th element; z i is the distance of the center of i-th element from the neutral axis of cross section; Z e is the distance from the neutral axis of cross section to the baseline ( ) e ult i is the ultimate strain of i-th element; e y is the yield strain of material; c is the curvature of ship hull girder; c 0 is initial curvature of ship hull girder; Z n is maximum elastic curvature of ship hull girder; A ei is the effective area of i-th element; and M k is calculated bending moment in k-th iteration.

Reliability analysis
Reliability always represents a kind of probability that establishes a relation between the performance of a system and what is expected ideally.To evaluate the probability of failure, it is necessary to determine the relationship between the ultimate strength of the structure and the maximum load applied to the structure.The corresponding limit function is expressed as follows: where L is the maximum applied load and Q is the structure's ultimate strength.The structure collapses when the maximum applied load is greater than the ultimate strength, as indicated by a negative value for the limit function ( ) g x .Following the determination of the limit function, the following formula can be used to get the failure probability: where the joint probability distribution function of the main variables is denoted by ( ) f X x and the vector of random variables is represented by Due to the difficulty of estimating the probability of failure, an approximation method based on the reliability index notion was developed.The limit function, ( ) g x is normally distributed, if Q and L are normally distributed.The probability of failure in this case would be as follows [25]: where F is the cumulative distribution function of standard normal distribution, m g is the mean value of ( ) g x , s g is the standard deviation value of ( ) g x , and b is the Cornell reliability index [25].The Cornell reliability index is the simplest reliability index, which is accurate when the ( ) g x is normally distributed, otherwise, it may give inadequate results.In the case Q and L are normally distributed and correlated, the reliability index is calculated as follows: where, m Q is the mean value of the strength function, m L is the mean value of the applied load function, r QL is the correlation coefficient between two random variables of load and strength, s Q is the standard deviation of strength function, and s L is the standard deviation of the applied load function.

Reliability of ship hull girder
The Monte Carlo Simulation (MCS) method is a simulation experiment with numbers to quantify the response to uncertainties.First, a set of random numbers is obtained to express statistical uncertainty in the structural parameters.Then, these random numbers are replaced in the response equation to obtain a set of random numbers that indicate the uncertainty in the structural response, and this set is analyzed for the quantitative and qualitative uncertainty of the response.To analyze the reliability of structures consisting of a random vector, . n with a joint probability distribution function ( ) f X , x the probability of failure defined by the limit state function, ( ) g X can be estimated from N sets of independent samples { x x x ..... n 1 2 } generated from the probability distribution function ( ) f X .
x If there is N f number of failure samples, which satisfies the condition of ( )  g X 0, then the probability of failure is approximated as: and reliability is state as: is the simplest Monte Carlo method for reliability and failure probability problems.To improve the convergence rate, some modifications are introduced for MCS using quasi-random points.Methods such as the Importance Sampling and Latin Hypercube Sampling (LHS) were developed to reduce variance [26].The limit state function governing the ultimate strength of ship hull girders at time t is considered as: where, ( ) M t u is the ship's ultimate strength, and ( ) M t L is the maximum applied bending moment on the ship hull girder, and is expressed as follows: where, ( ) M t sw is the still water bending moment, and ( ) M t w is the wave bending moment.A simple linear model for the limit state function of ultimate strength of ship hull girder is given by Mansour [27] as follows: where, x x , u sw and x w are random variables of the model uncertainties related to the ultimate strength, still water bending moment, and wave bending moment, respectively.The probabilistic characteristics of these random variables are given by Han et al [28].The International Association of Classification Societies (IACS) [23] recommends the following equations for the calculation of the still water and wave bending moments: where L is the length between perpendiculars, C b is block coefficient, B is the breadth of the ship and C w is the wave factor and is determined as follows:

Validation of the code for calculation of the ship's ultimate strength
A single-bottom tanker [29] has been selected for the ultimate strength calculation of the ship hull girder with main particulars depicted in table 1.The midship section consists of 230 components (figure 2) built of two types of steel, i.e., high tensile steel and mild steel (table 2). Figure 3 depicts the ultimate strength of the ship hull girder  calculated for hogging conditions using Smith's method.As seen, good agreement was observed between the results of the present study and the literature [30].

Ultimate strength of the single tanker ship [29] considering the combined effect of random distribution of pitting and cracking
The normal distribution with a coefficient of variation (COV) = 0.07 is considered for the ultimate bending moment of the tanker ship [29].Still water and wave bending moments are calculated using equation (12) with the corresponding statistical distribution (table 3).Corrosion data is considered based on 5-year period, and the reliability index is calculated after 5, 10, 15, and 20 years.The crack position, crack length, depth and number of pits, and spatial distribution of pits are considered as randomly varied parameters.The crack length is assumed to be 30, 60, 180, or 300 mm [8,9].Different relations have been proposed to calculate the depth and the radius of the pits as a function of time [28,31,32].In this work, the ratio of the diameter to the depth of pits is assumed as 10 [33].The Poisson distribution is used to characterize the number of pits as follows [28]: n At where t is the time in year, N(t) is the number of pits, A is the considered area of pitting, and λ is the intensity of pits.The depths of the pits as a function of time, p(t) (in mm), are calculated as follows [28]: where i corr is the intensity of the corrosion flow (μA cm −2 ) and R is a proportional constant.Table 4 depicts the range of parameters used in equations ( 14) and (15) and table 5 shows the number of pits calculated by equation ( 14) and degree of pitting (DOP), which is defined as the ratio of the total pitting area to the panel area, namely: where, A i is the area of the i-th pit, and a and b are the width and length of the plate panel, respectively.
For the random distribution of pits in the corroded plates in the ship cross-section, the methodology proposed by Wang et al [34] is used: where ( ) x z , i i represents the coordinates of the center of each pit in the xz plane.In other words, the coordinates of the center of each pit are randomly generated, and at that point a pit with random depth and radius, is inserted.After randomly generating pits, cracks with random lengths and positions are added.For each cracked-pitted plate, the corresponding average stress-average strain curve is determined.These curves were developed based on an extensive Non-Linear Finite Element Analysis (NLFEA) [8,9]., and different degree of pitting (DOP).
Having average stress-average strain curves for all plates and stiffened plates, Smith's method is used to determine the ultimate strength of the ship hull girder.The ultimate strength of the ship hull girder is calculated for five different damage scenarios (figure 6) and different assumed percentages of damages at each time interval of the age of the ship (table 6).
The damaged elements and type of damage (crack/pit) are randomly distributed in the transverse crosssection of the ship.For example, if the damage percentage after 5 years is 15%, 34 elements out of 230 should be damaged.After randomly selecting 34 elements out of 230 elements, the damage of these 34 elements could be cracking, pitting, or combined cracking and pitting, with variable damage intensity, i.e., random values of crack length, depth, diameter, and position of pits.As seen, many random parameters are involved, necessitating numerous calculations.For each damage ratio and time, the results are expressed as an absolute minimum and maximum value for ultimate strength.The ship reliability indices, b is calculated as follows:  where the probability of failure, P f is determined from equation (7) for 10 7 Monte Carlo Simulation trials.Figures 7 and 8 depict the reliability indices of the considered tanker for five different scenarios and four different damage percentages.
As seen, the reliability indices for different damaged scenarios are the same when the ship is at its early age.This means that when the percentage of damage is low, say between 10 to 20 percent, the distribution of damage does not affect the reliability index.When the ship's age increases, the damage scenario starts to influence the reliability index.For example, for the hogging condition, in which the deck is under tension and the bottom in compression, scenario No. 3 in which damages are concentrated at sides and longitudinal bulkheads, has the highest reliability index followed by scenario No. 1 in which damages are concentrated at the deck.For sagging conditions, scenario No. 2 has the highest reliability index in which damage is concentrated at the bottom.Additionally, it can be concluded that for the sagging condition with a damage percentage of 40 to 50 percent all scenarios, except scenario No. 2 in which damages are concentrated at the bottom, have the same low value of reliability indices.For the hogging conditions, scenarios No. 4 and 5 have the same low value of reliability indices.Therefore, scenario No. 4, in which damage is concentrated at deck and bottom, and scenario No. 5 in which damages are distributed all over the cross-section, have the low value of reliability index regardless of hogging or sagging conditions.It can be concluded that the reliability index decreases for every damage scenario as the ship ages, and reliability indices in the hogging conditions are higher than those in sagging conditions.Thus, the loading conditions of the aged ship should be in such a way that the still water bending moment is in the hogging condition.Reliability indices are calculated for scenario No. 5 at different ages of the ship when all damages are in the form of cracking, combined cracking and pitting, and pitting for hogging condition (figure 9) and sagging condition (figure 10).
As seen, at the early age of the ship, there is no difference between different forms of damage for hogging and sagging conditions.By increasing the ship's age, the combined cracking and corrosion mode of damage has the lowest reliability indices regardless of hogging or sagging condition, and pitting damage is more detrimental than cracking damage.
General corrosion, in which corrosion occurs in a large area, usually is considered by uniform thickness reduction of plates.Figures 11 and 12 compare reliability indices of the ultimate strength of the ship hull girder for hogging and sagging conditions, respectively for the same amount of degradation at different ages of the ship in scenario No. 5 for pitting corrosion with general corrosion [35].As seen regardless of the age of the ship and sign of bending moment, pitting corrosion leads to a lower reliability index than the general form of corrosion, which could be due to a stress concentration around pits leading to early yielding of plate.

Conclusion
The available stress-strain curves for cracked-pitted plates with random position of cracks, crack length, depth of pits, degree of pitting corrosion, and spatial distribution of pits are used in an in-house computer code based on Smith's method.To demonstrate the applicability of the developed computer code, the ultimate strength of the ship hull girder of a single-bottom oil tanker is calculated for five different damage scenarios every five years of the ship's age.It is found that at the early age of the ship, the reliability indices for different damage scenarios are the same regardless of hogging or sagging conditions.As the age of the ship increases, the reliability index  decreases for all damage scenarios and loading conditions.For hogging conditions, the highest reliability index corresponds to damages concentrated at sides and longitudinal bulkheads; for sagging conditions, the reliability index is the highest when damages are concentrated at the bottom.When damage is concentrated at deck and bottom or damages are distributed all over the cross-section, have a low value of reliability index regardless of hogging or sagging conditions.In general, the reliability indices in the hogging condition are higher than those in the sagging condition, and the loading conditions of the aged ship should be in such a way that the still water bending moment is in the hogging condition.Also, it is concluded that, at the early age of a ship, there is no difference between different damages in the form of pure cracking, mixed cracking and pitting, and pure pitting.As the ship ages, the mixed mode of damage has the lowest reliability indices regardless of the hogging or sagging condition, and pure pitting damage is more detrimental than pure cracking damage.It is also found that the pitting type of corrosion has a lower reliability index than general corrosion for the same amount of damage, regardless of the age of the ship and type of bending moment.

Figure 1 .
Figure1.The procedure of incremental-iterative method for ultimate strength evaluation of ship hull girder.

Figure 2 .
Figure 2. One-bay model of the single-hull VLCC oil tanker [29] is considered in this study.

Figure 3 .
Figure 3. Hogging ultimate strength versus curvature of the single tanker ship[29] and compared with the results of other studies[30].

Figure 4 .
Figure 4. Finite element model of a cracked-pitted plate.

Figure 4 ,
shows the FE model of a crackedpitted plate, and figure 5, shows average stress-average strain curves derived for different positions of one or two transverse cracks with different ratios of crack length to plate width ( ) / c b 2

Figure 5 .
Figure 5. Average stress-average strain curves of intact, pitted, cracked and cracked-pitted plates of thickness 8 mm.

Figure 6 .
Figure 6.Five different damage distributions at the midship section.

Figure 7 .
Figure 7. Reliability index in five different considering scenarios and damage percentage for a hogging moment.

Figure 8 .
Figure 8. Reliability index in five different considering scenarios and damage percentage for Sagging mode.

Figure 9 .
Figure 9. Reliability index of ultimate strength of ship hull girder in hogging condition for different types of damage in scenario number 5.

Figure 10 .
Figure 10.Reliability index of ultimate strength of ship hull girder in sagging condition for different types of damage in scenario number 5.

Figure 11 .
Figure 11.Reliability index for the ultimate strength of ship hull girder for hogging condition in scenario number 5.

Figure 12 .
Figure 12.Reliability index for the ultimate strength of ship hull girder for the sagging condition in scenario number 5.

Table 1 .
[29]main particulars of the single tanker (Energy Concentration[29]), is considered in this study.

Table 2 .
[29]tling of structural components of the single tanker[29], is considered in this study.

Table 3 .
[29]m variables in reliability assessment of ship hull girder of the considered single oil tanker[29].

Table 5 .
DOP of corrosion in stiffened plate.

Table 6 .
Number of deteriorated elements in the considered lifetime of the ship.