A study on reliability-based inspection planning – Application to deck plate thickness measurement of aging tankers

Marine Structures 25 (2012) 85–106

Contents lists available at SciVerse ScienceDirect

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

A study on reliability-based inspection planning – Application to deck plate thickness measurement of aging tankers Jinting (Jeffrey) Guo b, *, Ge (George) Wang a, Anastassios N. Perakis b, Lyuben Ivanov a a b

American Bureau of Shipping, 16855 Northchase Drive, Houston, TX 77060, USA Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 July 2009 Received in revised form 26 October 2011 Accepted 18 December 2011

In order to conduct a timely and effective inspection to avoid the ultimate failure of deck plate, a reliability-based procedure for inspection planning is introduced. A review of the uncertainties in deriving the capacity of deck plate and through life degradation effects on tanker structural integrity is given. Using the Latin Hypercube sampling method to perform the Monte Carlo Simulation, the time-variant failure probabilities of deck plate are computed, then compared with target values. Based on the comparison, the time for thickness measurement of deck plate is predicted. A total of 1080 cases for nine sample tankers are analyzed to illustrate the procedure, including sensitivity and parametric studies. This paper is a sequel to the preceding paper [6], where a semi-probabilistic approach is presented to assess the time-variant ultimate strength of aging tanker’s deck plate considering corrosion wastage. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Inspection planning Time-variant reliability Thickness measurement Aging tanker Deck plate Ultimate strength Uncertainty Monte Carlo simulation Latin hypercube sampling

1. Introduction The increased pressure for continued economical operation of aging tankers has extended the service life toward the safety limits to which the vessels were originally designed. Age-related

* Corresponding author. Presently at ExxonMobil Development Company, 12450 Greenspoint Drive, Houston, TX 77060, USA. Tel.: þ1 281 654 0960. E-mail address: [email protected] (J.J. Guo). 0951-8339/$ – see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2011.12.002

86

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Nomenclature Symbols f g K kw L Ms MT Mw Pf P*f r s t0 tc tn T Za Zd

a az b l d h m s sy sE scr su

probability density function limit state function buckling factor load combination factor ship length still water bending moment (SWBM) total bending moment wave-induced bending moment (WBM) failure probability target failure probability random number generated in MCS stiffener spacing as-built thickness of the plate panel corrosion wastage net thickness of the plate panel vessel’s age as-built hull girder section modulus (HGSM) HGSM to the deck aspect ratio reduction factor of HGSM slenderness ratio reference degree of slenderness standard deviation modeling uncertainty factor mean value nominal compressive stress specified minimum yield stress of the plate reference stress critical buckling stress of the plate ultimate strength of the plate

structural degradations such as corrosion wastage and cracks, if neglected, have the potential to lead to catastrophic failures. Properly planned inspection is one of the keys to prevent structural failures or cargo oil leakage in the aging tankers. The experience-based inspection planning may result in a major structural failure being missed due to long inspection intervals and bring increased costs to unnecessary inspections. A strategy to accurately determine a timely and effective inspection plan for longterm maintenance of ship structures is highly desirable. In recent years there has been an increased interest in research of risk-based inspection (RBI) planning for ship structures [1–6]. Structural reliability analysis is a key component in the RBI process. As ships age, the integrity of the structural capacity not only decreases due to the effects of timedegradation, but also the uncertainties associated with it grow over time. In the maintenance and operation of ship structures, it may no longer be adequate to consider the uncertainties determined at the time of construction. In order to provide useful results to the owner or operator, it is therefore important to carry out a time-variant reliability analysis which provides a measure of the failure probability of ship structures over a vessel’s life.

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

87

There are many publications about structural reliability analysis of ship structures. Only a few of them focus on the reliability of unstiffened plate [7–12] in contrast to thousands devoted to hull girder reliability. The local failures of ship structures are more likely to occur. As one of the fundamental building components in the construction of ships, the deck plate panel is often highly stressed. Although the failure of deck plate panel may not lead to the structural collapse of the vessel, it will affect the hull girder reliability. A failure of a stiffened panel can be partially controlled by the strength of the plate between stiffeners, thereby making the failure probability of the deck plate a very important measure of a vessel’s structural integrity. Thickness measurement is one of the mandatory surveys to monitor the condition of ship structures during the entire lifecycle. Timely thickness measurement should be conducted to avoid potential structural failure. The cost of unnecessary inspections and repairs can be reduced by accurately predicting the reliability of ship structures. In this paper, a procedure to determine the intervals of necessary thickness measurement for the aging tankers’ deck plate panel based on the reliability analysis is introduced. A fleet of nine sample tankers is analyzed to illustrate the procedure. 2. Overview of the procedure for planning the thickness measurement of the deck plate Ship structures need to be inspected on a regular basis. The proposed reliability-based procedure for planning the thickness measurement of deck plate is illustrated in Fig. 1. It is clear that evaluation of the time-variant failure probabilities of the deck plate is the key to planning an effective and timely

Fig. 1. A reliability-based procedure for planning thickness measurement of the deck plate over the vessel’s lifetime.

88

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Define limit state (Ultimate Strength) Uncertainty models Responses and loads

Strength modeling

Material properties

Structure and wastage

Calculate the failure probability of deck plate panels Define target levels of the failure probability Determine when to take thickness measurement Fig. 2. Projection of failure probability of deck plate panels.

inspection. As shown in Fig. 2, each step of time-variant reliability analysis of deck plate panel is described in the following sections. Although it is not discussed in this paper, the reader should be apprised of the importance of the data collection and execution of inspection. 2.1. Limit state function Deck plates experience large in-plane compression primarily in the ship’s longitudinal direction and small lateral pressure. When the applied compressive stress exceeds the ultimate strength of the deck plate panel, the ultimate failure occurs. The ultimate failure of the deck plate can be defined by the following limit state function:

g ¼ su  s

(1)

Then, the probability of failure can be computed from:

Z Z Pf ¼ Pf ½g  0 ¼

fsu ;s ðsu ; sÞdsu ds

(2)

g0

where fsu ;s ðsu ; sÞ is the joint probability density function of ultimate strength su and nominal compressive stress s, and the domain of integration is over all values of su and s where g is not positive. The formula introduced in IACS Common Structure Rule (CSR) [13] was used for calculating the ultimate strength of the deck plate. The critical buckling/ultimate strength of wide plate with uniaxial compression and in-plane bending is

scr ¼

8 < s y :c

1

l



0:22

l

2



for l  lc

sy for l > lc

where:

c ¼ 1:13; lc ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi  2  sy c 0:88 tn ; l ¼ ; K ¼ 2 ; sE ¼ 0:9E 1þ 1 K sE s 2 c

(3)

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Since b ¼

s tn

89

rffiffiffiffiffi

sy E

Then we can have

scr

8 sy <  2:14 0:89 ¼  2 sy :

b

b

for b  1:58 (4)

for b > 1:58

Neglecting the local bending of the secondary structures due to the action of the lateral load applied, it is considered that the nominal compressive stress of the deck plate s is induced by the hull girder vertical bending only. It can be calculated by the following equation:

s ¼ MT =Zd

(5)

where the total bending moment MT can be presented by:

MT ¼ Ms þ kw Mw

(6)

The still water bending moment (SWBM) Ms and wave-induced bending moment (WBM) Mw should satisfy the rule minimum requirements. Strictly speaking, Ms and Mw influence each other and the two random variables are statistically dependent. However, in a practical sense, they are often treated as statistically independent variables [14]. In general, there are two types of methods adopted for combining the moments. The stochastic method combines the two stochastic processes directly, while the deterministic method combines the characteristic values of the stochastic processes. In ship design, International Association of Classification Societies (IACS) [15] and American Bureau of Shipping (ABS) [16] consider the worst-case scenario following the peak coincidence method, which assumes that the maximum still water and wave-induced bending moments occur at the same instance. This paper follows the same assumption. Permissible SWBM and design WBM in the classification rule are applied in this study. The vertical wave-induced bending moment at amidships is



Mw ¼

0:19CL2 BCb ðkN  mÞ; 0:11CL2 BðCb þ 0:7Þ ðkN  mÞ;

for hogging for sagging

(7)

where the wave coefficient C is given by the following equation as a function of the ship length:

  8 300  L 1:5 > > 10:75  > < 100 C ¼ 10:75 1:5  > > > : 10:75  L  350 150

150  L  300 m 300 < L  350 m

(8)

350< L  500 m

2.2. Uncertainties and their measures One of the keys to a reliability analysis is the quantitative measure of involved uncertainties. The uncertainties of the deck plate’s capacity prediction include the following.    

Variation in physical properties of materials including yielding stress and Young’s modulus. Variation in thickness and geometry of the structures. Corrosion wastage. Modeling uncertainties due to simplifying assumptions in analytical and prediction models, simplified methods, and idealized representations of real performances.

Literature was reviewed to estimate the uncertainties of these variables using mean value, standard deviation, coefficient of variation (COV) and probability distribution. The detailed study will be presented in Section 3.

90

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

To develop the uncertainty model of the load, an understanding of the following is required.  Vessel’s reference cargo loading condition to analyze the uncertainties of the SWBM.  Environmental condition and wave data, operational status and analytical computation model to analyze the uncertainties of the WBM. The uncertainties of the loadings are large and will be included in the next paper. The methods described in Section 2.1 will be used to obtain the extreme value other than probabilistic value. The result is considered conservative. 2.3. Corrosion wastage and time-variant formulas In general, whenever a critical failure model involves a time-variant strength such as fracture, fatigue cracking, thermal effect, and corrosion, the variations with time should be taken into account. Only the corrosion wastage was considered in this study. Other impacts on structural safety and their interaction could be another research topic in the future. Because of corrosion, the thickness of the deck plate as well as the hull girder section modulus (HGSM) decrease with ship aging. Eq. (3) can be presented with time-variant variables as follow:

8 sy for bðTÞ  1:58 <  2:14 0:89 su ðTÞ ¼ h s for bðTÞ > 1:58  : bðTÞ bðTÞ2 y

(9)

where the time-dependent slenderness ratio of the plate is

bðTÞ ¼

s tn ðTÞ

rffiffiffiffiffi

sy E

¼

s t0  tc ðTÞ

rffiffiffiffiffi

sy

(10)

E

The time-dependent HGSM to the deck Zd(T) is determined by taking into account the corrosion wastage information known in every location of the hull structure. By introducing reduction factor of HGSM az(T), it can be obtained by

Zd ðTÞ ¼ Za ð1  az ðTÞÞ

(11)

Not only the net plate thickness tn and HGSM Zd, but also the SWBM Ms and WBM Mw can be functions of time. To simplify the methodology, they are assumed to be independent on time in this paper. Combining Eqs. (5), (6) and (11), the time-dependent nominal compressive stress of the deck plate is as follows:

sðTÞ ¼

Ms þ kw Mw Za ð1  az ðTÞÞ

(12)

In order to consider the modeling uncertainties of the simplified formula for representing the ultimate strength of the deck plate, a modeling uncertainty factor was introduced:

h ¼ su ðexperimental resultÞ=su ðprediction by calibrated formulaÞ

(13)

Combining Eqs. (1), (9) and (12), the time-variant limit state function of the deck plate’s ultimate strength failure can be presented as follows:

gðTÞ ¼

8 < hsy 

for bðTÞ  1:58

:

for bðTÞ > 1:58

M s þ kw M w Za ð1  az ðTÞÞ   2:14 0:89 M þ kw M w h s  s  bðTÞ bðTÞ2 y Za ð1  az ðTÞÞ

(14)

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

91

2.4. Projection of time-variant failure probabilities As a ship ages, the local carrying capacity of its structural components decreases and the uncertainties associated with its strength grows. It is no longer adequate to consider the uncertainties determined at the time of construction. It is therefore important to predict the failure probabilities of ship structures at any year. The following categories of probability approaches are widely used in marine-related reliability analysis.     

First-Order Reliability Method (FORM) Second Order Reliability Method (SORM) Mean Value First-Order Second-Moment Method (MVFOSMM) Response Surface Method (RSM) Monte Carlo Simulation Method (MCS)

FORM and SORM give approximate results on the limit state functions around the most probable points (MPPs). MVFOSMM and RSM make approximations on the limit state functions around the mean point. Although MVFOSMM and RSM are usually more effective, they can only provide rough prediction of failure probability. FORM and SORM are asymptotically correct with respect to the reliability index while MCS provides “exact” failure probability because true limit state functions are used. One drawback to using MCS is that it is time consuming. Considering the accuracy aspect, MCS is chosen to be applied in the present work. 2.5. Target levels of failure probability In the marine industry, in-service inspections are required to ensure the safety and operability of the marine structures and prevent sudden failure due to degradation of the ultimate strength. Selecting a target failure probability level P*f is required in order to determine reliability-based inspection intervals for ship structures such as the deck plate panel. In general, target failure probabilities are usually calibrated from existing structures having a history of successful service depending on type of structures, consequence of the failure, reliability analysis method, type of uncertainties, and maintenance plan. In the past twenty years, different target reliability levels have been calibrated by several authors. However, most of the publications focus only on the failure of hull girder [17–19]. The study on target failure probability of unstiffened plate is rare. A target failure probability of 2.3  102 (reliability index of 2) was suggested in [10] for unstiffened plate (tertiary) failure mode. A range of target reliability indices from 3 to 4 was recommended to derive partial safety factors for the reliability-based design for the unstiffened panels [10,12]. In order to monitor the condition of the deck plate panels and plan follow-up actions accordingly, three levels of target failure probability are considered based on professional judgment applied to the available data (Table 1). Please note that these values are recommended in this paper to illustrate the feasibility of the approach. It is not intended to reach a conclusion on the target reliability values in this study. 2.6. Planning thickness measurement As shown in Fig. 1, the proposed procedure for inspection planning includes two stages – predicting the time for the first thickness measurement and updating the following inspection intervals according to available gauging data. The inspection intervals can be schematically depicted in Fig. 3. The first Table 1 Suggested target failure probabilities. Level of target failure probability Trigger Ultimate strength of deck plate panel

1 5-year interval 1.8  102

2 2.5-year interval 2.3  102

3 Plate renewal 2.8  102

92

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Fig. 3. Schematic 1st and 2nd inspection intervals in terms of deck plate’s ultimate strength failure.

thickness measurement is to be conducted when projected time-variant failure probability of the deck plate panel Pf becomes equal to or greater than the “level 1” target value P*f1 suggested in Table 1. At the scheduled time, thickness measurements of deck plate panels will be taken and the updating phase starts. The updating scheme includes three steps. (1) Updating the corrosion model. Generally, the quantitative corrosion models applied in the marine industry represent the worst scenarios. After the first gauging survey, the corrosion models can be updated based on the application of Bayes’ rule using the gauging data. After updating, the corrosion model should predict corrosion wastage as close as possible to the actual degradation process rather than the worst scenario along the service life. (2) Evaluating the failure probabilities of deck panels in the consequent years. The failure probability will be re-evaluated based on the as-gauged scantlings. (3) Determining the interval by comparing the projected failure probability with the target values. As an example, the second inspection interval can be determined by the following.  If the calculated failure probability Pf0 using the as-gauged scantling is lower than P*f1, the next inspection can be predicted using the updating scheme. The updating scheme includes three steps – updating the corrosion model, evaluating the failure probabilities of deck plate panels in the consequent years, and determining the interval by comparing the projected failure probability with the target value P*f1.  If Pf0 is equal to or greater than P*f1, but lower than the “level 2” target value P*f2, the next inspection will be conducted in 5 years.  If Pf0 is equal to or greater than P*f2, but lower than the “level 3” target value P*f3, the next inspection will be conducted in 2.5 years.  If Pf0 is equal to or greater than P*f3, the deck plate panel is considered not to have adequate reliability and the plate should be renewed to bring the failure probability back to a lower level.

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

93

Following the same procedure, all the inspection intervals for the deck plate panels over the lifetime of the vessel can be obtained. 3. Uncertainties of capacity prediction The ultimate strength of the deck plate su is mainly governed by the slenderness, aspect ratio, initial imperfections, residual stresses, boundary conditions, material properties and type of loading. 3.1. Uncertainties of the material properties of steel Table 2 shows the statistical information of Young’s modulus listed in [20–26]. As indicated in [22] and [23], the COV value is determined by the variation due to different types, heat, mills, etc. Since ships are built worldwide, it should be noted that there is a wide range in the quality of the steel used making it difficult to accurately estimate the probabilistic characters of E based on a small sample size of testing. To study the effect of the largest variation of the random variables, the calculated average mean and COV of E in [25] together with the assumption of normal distribution are applied in this study. Statistical studies have been conducted on the yielding stress by many researchers and institutes in the past several decades. Some researchers suggested that numerical or statistical analysis is probably worthless since the measurement methods vary [22]. There is a large amount of statistical summary available in the public literatures and internal database, as listed in Table 3. The goodness-of-fit tests suggest that the normal or lognormal distribution is the applicable choice for describing the yield strength of steel. Since the present study is focused on the aging ships, the data sets with largest COV were chosen. 3.2. Uncertainties due to manufacturing tolerance Concentrated efforts were made by many researchers in an effort to quantify the variations in the geometries of the plate since 1970s (some statistical data are available in the public literatures [21,24,26–32]). Based on selected statistical information, the calculated averages for the mean and the COV of the plate thickness and stiffener spacing are summarized in Tables 4 and 5. Based on the same reason for choosing E and sy, the calibrated equation of Data Set 5 for plate thickness and the calibrated equation of Data Set 3 for stiffener spacing are chosen here to describe the uncertainty. 3.3. Uncertainties of corrosion wastage in deck plates Over time, corrosion wastage of structural members in ships causes variability in structural properties. When ships are loaded in a sagging condition, deck plates are under compression. Buckling/ ultimate strength is often the governing criteria. The assessment of the uncertainty of the corrosion wastage is a challenge because it is affected by many variables. Table 2 Statistical information of Young’s modulus E. References

Mean/rule valuea

COV

Notes

Johnson and Opila (1941) [20] Caldwell (1972) [21] Galambos and Ravindra (1978) [22] Mansour et al. (1984) [23] Guedes Soares (1988) [24]

0.992 1 0.967 1.002 / / 0.963 0.990

/ 0.026 0.06 0.031 0.04 0.06 0.105 0.0179

Normal distribution is assumed For both tension and compression behaviors Weighted average mean is calculated Samples are taken from one steel mill Samples are taken from different steel mills Normal distribution is fitted Lognormal or normal distribution are assumed

Atua et al. (1996) [25] Hess et al. (2002) [26] a

Rule value of E ¼ 2.06  107 N/cm2 (2.1  106 kgf/cm2, 30  106 lbf/in2).

94

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Table 3 Yielding stress sy of shipbuilding steels. References

Steel type

Mean/rule valuea

COV

Notes

Caldwell (1972) [21] Russian shipyard (1979)

/ OS HT 36 /

/ 1.298 1.155 /

0.066 0.079 0.05 0.089

/ / OS

/ / 1.097

0.10 0.08 0.068

HT 32

1.078

0.089

AH 32 AH 36 DH 32 DH 36 EH 32 EH 36 A AH 32 AH 36 AH 40

1.222 1.148 1.223 1.196 1.434 1.300 1.230 1.206 1.189 1.197

0.058 0.082 0.061 0.075 0.044 0.058 0.044 0.042 0.038 0.042

Normal distribution is assumed / / More than 60,000 samples with weighted COV, lognormal distribution is suggested Samples are taken from different steel mills Samples are taken from one steel mill Based on the data from [22,23,25], lognormal distribution is assumed Based on the data from [23], lognormal distribution is assumed 325 samples, Normal distribution 165 samples, Normal distribution 72 samples, Normal distribution 178 samples, Normal distribution 188 samples, Normal distribution 220 samples, Normal distribution 8365 samples 7038 samples 3041 samples 367 samples

Mansour et al. (1984) [23] Guedes Soares (1988) [25] Hess et al. (2002) [26]

Steel maker 1 (2007)

Steel maker 2 (2007)

a Rule value of yielding stress: 235 Mpa (24 kgf/mm2, 34  103 lbf/in2) for ordinary steel (OS); 315 Mpa (32 kgf/mm2, 46  103 lbf/in2) for high tensile (HT) 32 steel; 355 Mpa (36 kgf/mm2, 51  103 lbf/in2) for high tensile (HT) 36 steel; 390 Mpa (40 kgf/mm2, 57  103 lbf/in2) for high tensile (HT) 40 steel.

Statistical analysis of the experienced data is often believed to be reliable for predicting corrosion wastage. The simplified deterministic representation was first introduced by Southwell et al. [33] and then extended by Melchers [34–36]. Over the last 10 years, there have been intensive efforts in the collection of corrosion wastage measurement data from commercial ships [6,37–49]. Table 6 shows the corrosion wastages of tanker’s deck plate at 20 years calculated using these published studies. The calculated corrosion wastage varies over a wide range. This could be due to the fact that the available corrosion data are different and so are their assumptions. The same corrosion data were applied in the reference [46] and [6], but the latter predicted more severe corrosion. For conservative consideration, the corrosion model from [6] is used in this paper to predict the corrosion wastage of the deck plate and Weibull distribution is applied for presenting the corrosion wastage over a ship’s life. 3.4. Uncertainty and sensitivity analysis on the slenderness ratio From Eq. (9), it is obvious that the important parameters for assessing the ultimate strength of the deck plate are material-specified minimum yield stress sy and the time-dependent plate slenderness Table 4 Statistical information of the plate thickness t. Data Set

Average mean (mm)

Average COV

References

1 2 3

/ 0.9652t t

0.04 0.2301/t 0.6299/t

Wierzchowski (1971) [27] Caldwell (1972) [21] Hess et al. (2002) [26]

4

t

0.3531/t

Hess et al. (2002) [26]

5

t

0.4369/t

Hess et al. (2002) [26]

Note: t ¼ nominal plate thickness.

Notes

Calculation based on data from [31][32] only including the measurement taken after construction Calculation based on data from [31][32] only including the measurement taken before cut Calculation based on all data from [31][32]

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

95

Table 5 Statistical information of the stiffener spacing s. Data Set

Average mean (in)

Average COV

References

Notes

1 2 3

/ / s  0.013

0.01 0.0019 0.093/(s  0.013)

Wierzchowski (1971) [27] Caldwell (1972) [21] Barsa and Stanley (1978) [29] Daidola and Barsa (1980) [31]

4

s  0.2514

/

Hess et al. (2002) [26]

The uncertainty of the plate width is represented by the uncertainty of stiffener spacing Calculation based on data from [31]. Sample size of 261

Note: s ¼ nominal stiffener spacing.

ratio b(T). The slenderness ratio is a function of five variables: the as-built thickness t0, the spacing s of the longitudinals or stiffeners, the corrosion wastage tc(T), the yielding strength sy and Young’s modulus E of the material. All of these variables are treated as statistically independent. The sensitive analysis is also performed to measure the importance of the uncertainty associated with each random variable. With the use of first-order second-moment methods, the uncertainty was calculated using a linear approximation to the functional relations [24]. The mean of a multi-parameter function F is given by the value of the function at the mean value of the variables:

mF ¼ Fðx1 ; x2 ; .xn Þ

(15)

Neglecting the correlation of the variables, the standard deviation of the function is obtained from:

(

dF ¼

n  X



dx i

i¼1

vF vxi

2 )12



(16) xi ¼xi

where dxi is the standard deviation of xi. The uncertainty is characterized by dF or by the coefficient of variation (COV) which is given by:

COV F ¼ dF =mF

(17)

The uncertainty of b, according to second-moment methods is obtained from Eq. (18):

 COV 2b ¼ COV 2s þ COV 2t0

mt0

2

mt0  mtc

 þCOVt2c

mtc mt0  mtc

2 þ

COV 2sy 4

þ

COV 2E 4

(18)

Summarized in Table 7 are the uncertainties of the variables and the relative contribution to the variance of the slenderness ratio b for the deck plate panel of the 20 years aging tanker 80-B. Because the mean value and COV of the corrosion wastage of the deck plate varies with the vessel’s age, the Table 6 Calculated corrosion wastages of the tanker’s deck plate at 20 year from selected publications. References

Mean (mm)

COV

Tank type

Notes

Paik et al. (2003) [42] Garbatov et al. (2005) [46]

1.3550 0.7263 0.7712

0.8183 0.8262 0.4404

Ballast Cargo oil Ballast

1.0140

0.6513

Cargo oil

1.0196

0.7538

Ballast

1.2124

1.1779

Cargo oil

Weibull distribution for corrosion rate; corrosion starts at 7.5 year Weibull distribution for corrosion rate; corrosion starts at 7.5 year Corrosion starts at 10.54 year; long-term corrosion wastage is 1.85 mm Corrosion starts at 11.5 year; long-term corrosion wastage is 1.91 mm Weibull distribution for corrosion wastage; corrosion starts at 6.5 year Weibull distribution for corrosion wastage; corrosion starts at 6.5 year

Guo et al. (2008) [6]

96

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Table 7 Uncertainties of the basic variables and relative contribution to the variance of the slenderness ratio (Taking Tanker 80-B Ballast tanks as an example). Variables

Mean

COV

Contribution to the variance of b (%)

As-built plate thickness, t0 Stiffener spacing, s Corrosion wastage, tc Young’s modulus, E Yielding strength, sy

19 mm 850 mm 1.0196 mm 198,373 MPa 339.65 Mpa

0.023 0.00278 0.7539 0.105 0.089

8.25 0.11 25.51 38.48 27.65

relative contribution of each basic variable’s uncertainty to the variance of b varies as well. The trends for sample tanker 80-B are indicated in Fig. 4. From the analysis results, we summarize the following.  The contribution of tc increases with the tanker’s age, while others decrease.  Because of the significant contribution to the variance of b, it is necessary to consider yield stress sy and Young’s modulus E as random variables to evaluate the ultimate strength.  Considering the contribution of s and t0, with the vessel aging, it is possible to treat them as deterministic variables without significantly affecting the failure probability of the deck plate. Therefore, in this paper, only corrosion wastage tc, the yield stress sy and Young’s modulus E are treated as probabilistic variables to evaluate the ultimate strength of plates. 3.5. Uncertainty of ultimate strength prediction In classification rules, several different formulas are applied to predict the ultimate strength of a plate. Although the predictions are made by using theories based on the principles of mechanics, the different assumptions of loading types or boundary conditions, and the approximations taken during the computations will result in the differences between these formulas.

0.8

Cotribution to the variance of the slenderness ratio

0.7

Stiffener Spacing As-built Plate Thickness Corros ion Wastage Yield Stress Young's Modulus

0.6

0.5

0.4

0.3

0.2

0.1

0

10

15

20

25

30

Ves sel's age (year) Fig. 4. Time-variant relative contribution of variables uncertainty to the variance of slenderness ratio (using the deck plates of the sample tanker 80-B as an example).

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

97

Table 8 Mean/COV of modeling uncertainty of h the plate uniaxial compression along short edges (modified from ABS [51]). Critical buckling stress

Mean COV

Ultimate strength

ABS buckling guide [52]

API bulletin 2V [53]

API bulletin 2V [53] and ABS buckling guide [52]

1.1149 0.2845

1.0747 0.3050

0.9559 0.1481

The modeling uncertainty factor h can be obtained by the ratio of the test results over the predicted results [50]. According to this definition, the strength prediction is conservative if modeling uncertainty is greater than 1.0. A plate test dataset was analyzed by ABS [51] to evaluate the modeling uncertainties of some existing formulas. Table 8 shows the statistical characteristics of the modeling uncertainty for critical buckling stress and ultimate strength of the long plate under uniaxial compression along short edges. Since the dataset was collected from numerous publications and all the tests were performed for different research purposes, the geometry of the testing plates and the boundary conditions vary in a wide range. However, for the deck plate of the ship structure, some index such as aspect ratio a and slenderness ratio b stay within a certain range, which are 2 < a < 9 and 1.2 < b < 3.5, respectively. For the critical buckling stress and ultimate strength prediction formulas for deck plates simply supported all around and subjected to a uniform uniaxial compression along short edges, the sample size is reduced and the statistical characteristics are shown in Table 9. Figs. 5 and 6 show the relationship between critical buckling stress/ultimate strength and slenderness ratio for the deck plate. From the statistical results, the predictions of the critical buckling stress and ultimate strength of the plate panels vary in a wide range. Although h for some formulas are greater than 1, they are not necessarily considered as conservative approaches because some corrosion wastage values are introduced in these formulas. Therefore, simply applying any formula to evaluate the critical buckling stress and ultimate strength of the plate may not be accurate according to the above analysis. The projected results will be close to actual values if the model uncertainty effect is taken into account. This conclusion will make the argument about the formula application less important. In this study, as mentioned in Section 2.1, CSR formula is applied. The mean value and COV of h are 0.87 and 0.0982, respectively. 3.6. Uncertainties of HGSM due to corrosion wastage From Eq. (12), as the corrosion takes place on the structure, HGSM decreases over time. The timevariant reduction factor becomes a very effective tool in measuring the loss of HGSM during the entire life of the vessel. During the last two decades, some analytical studies on HGSM of corroded aging ship hulls were carried out by assuming corrosion wastage of individual structural members [55–59]. Based on a dataset of as-gauged hull structures, a statistical study for the loss of HGSM was performed by Wang et al. [60]. The dataset demonstrated a high variation of HGSM over time, and Weibull distribution is assumed to represent the time-variant reduction factor. It was also found that most of the Table 9 Mean/COV of modeling uncertainty h of the buckling strength/ultimate strength prediction formulas for deck plates. Organization

IACS API ABS

Critical buckling strength

Ultimate strength

Mean

COV (%)

Mean

COV (%)

1.07 / 1.03 1.00 1.07 1.00

12.62 / 13.91 12.02 12.62 12.02

/ 0.87 0.98 0.90 / 0.98

/ 9.82 9.41 9.95 / 9.41

References

IACS S11 [15] IACS CSR [13] API bulletin 2V [53] ABS steel vessel part 5C [16] ABS steel vessel part 3 [54] ABS buckling guide of offshore structure [52]

Note: 2 < a < 9 and 1.2 < b < 3.5; plates are simple supported; plate panels are between angles or tee stiffeners.

98

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Fig. 5. Comparison of the formulas with test data for critical buckling stress prediction.

publications overestimated the loss of HGSM. The comparisons for 20-years old vessels are listed in Table 10. However, there is not an easy conclusion to find a better way to calculate the reduction factor with the many uncertainties involved. In this paper, the calibrated formulas [60] are applied to predict the loss of HGSM.

4. Calculation of failure probability by MCS The analytical solution of Eq. (2) is difficult to obtain because it involves multidimensional probability integration. The MCS method was used for this integration. This method uses an iterative

Fig. 6. Comparison of the formulas with test data for ultimate strength prediction.

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

99

Table 10 Calculated reduction factor az of the ship structure at 20 years from selected publications (modified from Wang et al. [60]). Reference

Mean (%)

COV

Notes

Ivanov [55] Guedes Soares and Garbatov [56] Wirsching et al. [57] Ayyub et al. [58] Paik et al. [59] Paik et al. [59]

4.2 20

/ 0.3

8 7% 12 7.5

0.25 / / /

Calculated for a bulk carrier based on assumed corrosion wastage Calculated for a single hull very large crude carrier (VLCC) based on assumed corrosion wastage Calculated for single hull tankers based on assumed corrosion wastage Calculated for a single hull tanker based on assumed corrosion wastage Calculated for a double hull tanker based on average corrosion wastage Calculated for a conversion single hull FPSO based on assumed corrosion wastage Statistical result of hundreds single hull tankers Calibrated result for hundreds single hull tankers based on statistical results

Wang et al. [60] Wang et al. [60]

3.2 3.6

0.593 0.694

scheme for a deterministic model using sets of random numbers as inputs. It can be used for the complex, nonlinear or implicit function. The drawback is that calculation is quite time consuming. In the case of the time-variant failure probability calculation, the MCS technique samples each s u ðTÞ, sb ðTÞ. Then, the limit state function g will be random variable randomly to give sampling values b checked. After N trials, the probability of failure is approximated by:

  s u ðTÞ; sb ðTÞ  0 n g b Pf ðTÞz N

(19)

b u ðTÞ; s b ðTÞÞ  0Þ denotes the number of n trials for which gð s b u ðTÞ; s b ðTÞÞ  0. Obviously, the where nðgð s required number of trials N is influenced by the desired degree of accuracy of Pf(T). 4.1. Random number generation A key part of MCS is generating random numbers. Most MCS methods use pseudorandom numbers, which are obtained on the basis of a formula. The commercial software Crystal Ball [61] was applied to perform the simulation. Crystal Ball uses Multiplicative Congruential Generator to generate the random number:

 r)ð62089911$rÞmod 231  1

(20)

The generator has a period of length 231–2, or 2,147,483,646. This means that the generated numbers will not repeat after 2  1010 trials. This would be adequate for many marine-related problems where the failure probability often falls in the range of 102 to 108. 4.2. Latin hypercube sampling The probability of failure is estimated as the ratio of the number of failures to the total number of simulation trials. If failure probability were small, such as 104, Monte Carlo direct sampling would request a large number of trials to estimate the probability of failure within an acceptable level of statistical error. In addition, direct simulation requires binary definition of failure according to the limit state equation. These drawbacks make MCS with direct sampling not feasible. Latin hypercube sampling (LHS), a variant of MCS, was applied for the simulation because it ensures that the ensemble of random numbers is representative of the real variability, whereas traditional random sampling is just an ensemble of random numbers without any guarantees. LHS was first developed by McKay et al. (1979) [62] to generate a distribution of plausible collections of parameter values from a multidimensional distribution. It was further elaborated upon by Ronald et al. (1981) [63,64]. A comparison of Latin hypercube sampling with other techniques is given in Iman and Helton [65]. With Latin hypercube sampling, each parameter’s probability distribution is divided into several non-overlapping segments on the basis of equal probability, and a value for the parameter would be

100

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

generated at random from each segment. LHS is generally more precise when calculating simulation statistics than conventional Monte Carlo direct sampling because the entire range of the distribution is sampled more evenly and consistently. With Latin hypercube sampling, the number of trials is remarkably reduced to achieve the same level of statistical accuracy as with Monte Carlo sampling. 4.3. Convergence of simulation MCS relies on sampling of random variables. The calculation accuracy depends on the sampling number and is not affected by the distribution type and the number of basic variables. In general, as more trials are calculated, the statistical error decreases and the results become more accurate. Fig. 7 shows an example of the convergence. It is clear that in the instance where the number of simulations reaches 106, the calculated failure probability was converged. A million trials of MCS were applied to calculate the failure probability of the deck plate. 5. Sample calculations 5.1. Sample tankers and statistical properties of input variables The same set of nine sample tankers [6] is selected for demonstrating this probabilistic approach. The statistical data of the random variable together with the basic information of the 20-year-old sample vessels are given in Table 11. 5.2. Analysis results The time-variant probability density functions of tc(T) and az(T), together with probability distribution of the sy, E and h, were applied to Eq. (13) to calculate the failure probabilities of the corroded aging tanker’s deck plate. The detailed results are shown in Fig. 8, including deck plate’s failure probabilities for both ballast tank and cargo oil tank. It is obvious that the failure probability increases with tankers’ age and the possibilities of failure in the cargo oil tank are slightly higher than those in the ballast tank. Meanwhile, the calculated failure probabilities are high. The reason may be because the extreme value of loadings on the hull girder is considered. In order to investigate the impact of MT to the failure probability of the aging tankers, case two is calculated with the adjusted load combination factor kw (m ¼ 0.75 and COV ¼ 0.1). The calculated Cargo Oil Tank Ballast Tank

Failure Probability

0.022

0.02

0.018

0.016

0

5 10

5

6

1 10

1.5 10

6

2 10

Number of Simulations Fig. 7. Test of MCS convergence (deck plate of the 20-year-old tanker 70-A).

6

Table 11 The properties of each parameter for 20-year-old sample tankers (deck plate panels in way of cargo oil tanks). Year built

1970s

Ship ID L (m) Deadweight (tonne) t0 (mm) s (mm) Msw (109 N-m) Mw (109 N-m) rz Q

1990s

70-A

70-B

70-C

80-A

80-B

80-C

90-A

90-B

90-C

362.0 423,700 27.5 1000 9.53 16.13 1.081 0.78

327.0 273,900 27.5 940 5.98 10.41 1.035 0.78

331.0 – 25.5 1100 5.98 11.44 1.060 0.72

236.8 88,500 18 850 2.21 3.85 1.065 0.78

277.6 149,200 19 850 3.78 6.63 1.007 0.78

173.2 – 13 750 1.00 1.48 1.074 1

315.8 298,300 20 910 6.16 10.42 1.004 0.78

230.4 104,800 16 820 2.27 3.82 1.095 0.78

177.7 40,000 14.5 800 1.47 1.58 1.142 1

Year built

1970s

Ship ID

70-A

70-B

70-C

80-A

80-B

80-C

90-A

90-B

90-C

339.65 0.089 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

339.65 0.089 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

410.02 0.05 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

339.65 0.089 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

339.65 0.089 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

305.03 0.079 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

339.65 0.089 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

339.65 0.089 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

305.03 0.079 Lognormal 198,373 0.105 Normal 1.2124 1.1779 Weibull 0.87 0.0982 Normal 3.5327% 0.7193 Weibull

Probabilistic variables

sy (MPa) E (MPa)

tc (mm)

h az

Mean COV Distribution Mean COV Distribution Mean COV Distribution Mean COV Distribution Mean COV Distribution

1980s

1990s

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Deterministic variables

1980s

101

102

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Fig. 8. Time-variant failure probability of the corroded aging tanker’s deck plate.

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

103

failure probabilities together with the original results from case one are presented in Table 12. From Table 12, it is found that the failure probabilities drop dramatically compared with the original case. In case two, the range of failure probabilities is about 104–102. The uncertainties of the loads will be included in the next stage of study. 5.3. Application to inspection planning The first required inspection intervals for the sample tankers are listed in Table 13, based on the calculated failure probabilities from case two and suggested target failure probability indicated in Table 1. From the table, it can be concluded that the sample tankers 70-A to 70-C have more adequate reliability in deck plate panels than those in the 80s and 90s. The failure probabilities of deck plate panels in cargo oil tank are higher than those in ballast tank because of the higher corrosion wastage. The same conclusions have been drawn in [6] based on a semi-probabilistic approach. The reliability-based analysis shows that no thickness measurements of deck plates are necessary for nearly a vessel’s entire life. However, beyond inspections planned using the proposed procedure, additional inspections may be needed for confirmation or other purposes. Such inspections are outside the scope of the current study. After the International Conventions for the Prevention of Marine Pollution from Ships (MARPOL) requirements became mandatory in 1983, with the change in length to depth proportions, ship builders tried to reach optimum arrangements for ballast and cargo in order to comply with the convention’s requirements. As a result, due to the increased ship depths involved, the scantlings for 80s and 90s tankers dropped dramatically compared with 70s. See also Table 11 for the scantlings of the sample tankers. Based on the same predicted corrosion wastage over the vessels’ life and the same target failure probability, the required inspections are expected to be more frequent. The results in Table 13 show this information. The common practice of ship design is not reliability based. For this reason, the integrity of each vessel differs over time. Hence, reliability-based inspection planning is a more rational approach in monitoring the condition of a vessel. This paper describes reliability-based inspection planning for local structures. Following the same procedure, inspection intervals can be estimated for each compartment of vessels, such as side shell, bottom, transverse frames, etc. By putting together the

Table 12 Failure probabilities of the corroded aging tanker’s deck plates in selected year. Ship ID

Case

Vessel age 10

70-A 70-B 70-C 80-A 80-B 80-C 90-A 90-B 90-C

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

15

20

25

Cargo

Ballast

Cargo

Ballast

Cargo

Ballast

Cargo

Ballast

9.9  103 6.0  104 1.9  102 1.3  103 1.6  102 1.0  103 1.3  101 1.4  102 1.5  101 1.7  102 9.1  102 1.3  102 1.8  101 2.1  102 1.9  101 2.6  102 2.2  102 3.4  103

9.5  103 5.4  104 1.9  102 1.2  103 1.4  102 7.8  104 1.1  101 1.1  102 1.4  101 1.4  102 7.3  102 6.9  103 1.7  101 2.8  102 1.8  101 2.0  102 1.4  102 1.3  103

1.4  102 9.2  104 2.6  102 1.8  103 3.3  102 2.3  103 1.9  101 2.8  102 2.1  101 3.2  102 1.6  101 3.5  102 2.4  101 3.9  102 2.7  101 5.3  102 5.0  102 1.2  102

1.3  102 7.7  104 2.4  102 1.6  103 2.6  102 1.4  103 1.6  101 1.8  102 1.9  101 2.3  102 1.3  101 1.5  102 2.2  101 2.9  102 2.5  101 3.6  102 2.9  102 3.2  103

2.1  102 1.9  103 3.3  102 2.9  103 5.4  102 5.9  103 2.5  101 5.1  102 2.7  101 5.4  102 2.3  101 6.8  102 3.0  101 6.2  102 3.5  101 8.7  102 9.0  102 3.0  102

1.7  102 1.1  103 3.0  102 2.2  103 3.9  102 2.3  103 2.1  101 2.9  102 2.4  101 3.5  102 1.8  101 3.0  102 2.8  101 4.3  102 3.2  101 5.6  102 5.0  102 7.2  103

3.2  102 4.1  103 4.6  102 5.2  103 8.1  102 1.2  102 3.0  101 7.8  102 3.3  101 8.3  102 2.9  101 1.0  101 3.6  101 9.2  102 4.1  101 1.2  101 1.3  101 5.3  102

2.2  102 1.6  103 3.7  102 3.0  103 5.6  102 4.1  103 2.7  101 4.5  102 3.0  101 5.1  102 2.5  101 5.4  102 3.3  101 6.1  102 3.8  101 8.2  102 8.1  102 1.6  102

104

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

Table 13 Required time for the first thickness measurement.

Ballast tank Cargo oil tank

70-A

70-B

70-C

80-A

80-B

80-C

90-A

90-B

90-C

>30 >30

>30 >30

>30 28.0

14.8 11.8

12.5 10.4

16.0 11.5

<10 <10

<10 <10

26.2 17.0

inspection plans for plate panels, stiffener panels and hull girder, a rational lifetime inspection plan can be achieved. 6. Conclusion and discussion This paper presents a procedure for planning the inspection of ships by accessing the time-variant failure probability. The main purpose of this study includes the following.  Determining the failure probability level for corroded aging tankers.  Providing ship owners and operators with needed tools and processes for rationalized decision on when to take thickness measurement. The failure probability was calculated by Monte Carlo Simulation with Latin Hypercube Sampling. The method considers the through life degradation of the structure due to corrosion and probabilistic models of geometry and materials. The uncertainty model for each variable was discussed and a sensitivity study was performed. The following are the conclusions from the study of the reliability of corroded deck plate.  The failure probability of deck plate by ultimate strength increases with vessel’s aging and varies in a wide range.  Uncertainties of the following random variables need to be considered in structural reliability analysis: loads, modeling uncertainty, corrosion wastage, yield stress and the Young’s modulus of the material.  Latin Hypercube Sampling is a practical method for structural reliability analysis. It is more efficient than traditional Monte Carlo Simulation especially for the marine-related problem.  The required time to take thickness measurements for 70s tankers is much longer than for the 80s and 90s. The following topics may be considered as the further development of the present study.    

Analysis of time-dependent loads acting on the ship structure. Determination of the probabilistic model of loads. Calibration of the target probability. Completion of inspection plan updating based on the gauging results.

Acknowledgments The authors appreciate the valuable input from many colleagues, including H. Sun, N. Chen, A. Lee, E. Khoo and C. Serratella. We also would like to thank J. Speed and D. Browning for improving the manuscript. References [1] Onoufriou T. Reliability based inspection planning of offshore structures. Mar Struct 1999;12:521–39. [2] Ma KT, Orisamolu IR, Bea RG. Optimal Strategies for inspection for fatigue and corrosion damage. Ship Structure Committee report; SSC-407. Washington D.C.: Ship Structure Committee, U.S. Coast Guard; 1999.

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

105

[3] Ayyub BM, Akpan UO, Rushton PA, Koko TS, Ross J, Lua J. Risk-informed inspection of marine vessels. Ship Structure Committee report; SSC-421. Washington D.C.: Ship Structure Committee, U.S. Coast Guard; 2002. [4] Ku A, Spong RE, Serratella C, Wu S, Basu R, Wang G. Structural reliability application in risk-based inspection plans and their sensitivities to different environmental conditions. Offshore Technology Conference (OTC’05), Houston, USA; 2–5 May 2005. [5] Lee A, Serratella C, Basu R, Wang G, Spong RE. Flexible approaches to risk-based inspection of FPSOs. Offshore Technology Conference (OTC’06), Houston, USA; 1–4 May 2006. [6] Guo J, Wang G, Ivanov L, Perakis AN. Time-varying ultimate strength of aging tanker deck plate considering corrosion effect. Mar Struct 2008;21:402–19. [7] Nikolaidis E, Hughes O, Ayybu BM, White GJ. A methodology for reliability assessment of ship structures. In: Ship structures symposium, Arlington, USA; 16–17 November 1993. [8] Assakkaf IA, Ayyub BM. Reliability-based design of unstiffened panels for ship structures. In: Proceedings of the Joint ISUMA-NAFIPS, College Park, USA; April 1995. [9] Mansour AE, Wirsching PH, Luckett M, Plumpton A. Assessment of reliability of ship structures. Ship Structure Committee Report; SSC-398. Washington D.C.: Ship Structure Committee, U.S. Coast Guard; 1997. [10] Mansour AE, Wirsching PH, White GJ, Ayybu BM. Probability-based ship design. Ship Structure Committee report; SSC392. Washington D.C.: Ship Structure Committee, U.S. Coast Guard; 1996. [11] Assakkaf IA, Cárdenas-García JF. Reliability-based design of doubler plates for ship structures. In: Proceedings of the fourth international symposium on uncertainty modeling and analysis (ISUMA), College Park, USA; 20–24 September 2003. [12] Assakkaf IA, Ayybu BM, Hess PE, Knight DE. Reliability-based load and resistance factor design (LRFD) guidelines for unstiffened panels of ship structures. Nav Eng J ASNE 2002;114(2):69–88. [13] Common structural rules for double hull oil tankers. Available from: http://www.iacs.org.uk/; 2007. [14] Mansour AE, Hovem L. Probability-based ship structural safety analysis. J Ship Res 1994;38(4):329–39. [15] IACS (International Association of Classification Societies) UR S11. Longitudinal strength standard, Rev. 5. IACS; January 2006. [16] ABS. Rules for building and classing steel vessels. Part 5C. ABS; 2007. [17] Casella G, Dogliani M, Guedes Soares C. Reliability-based design of the primary structure of oil tankers. J Offshore Mech Arct Eng 1997;119:263–9. [18] Horte T, Wang G, White N. Calibration of the hull girder ultimate capacity criterion for double hull tankers. In: Tenth international symposium on practical designs of ships and other floating structures, 30 September–5 October 2007, Houston TX. [19] Moan T, Shu Z, Drummen I, Amlashi H. Comparative reliability analysis of ships – considering different ship types and the effect of ship operations on loads. SNAME annual meeting, Ft. Laudendale, FL; 2006. [20] Johnson BG, Opila F. Compression and tension tests of structural alloys. In: ASTM Proceedings. American Society for Testing and Materials; 1941. [21] Caldwell JB. Design – construction. De Jngenieur December 1972;No. 49. [22] Galambos TV, Ravindra TV. Properties of steel for use in LRFD. J Struct Div ASCE 1978;104(No. ST9). [23] Mansour AE, Jan HY, Zigelman CI, Chen YN, Harding SJ. Implementation of reliability methods to marine structures. Trans Soc Nav Architect Mar Eng 1984;92:11–20. [24] Guedes Soares C. Uncertainty modeling in plate buckling. Struct Saf 1988;5:17–34. [25] Atua K, Assakkaf IA, Ayyub BM. Statistical characteristics of strength and load random variables of ship structures. In: Seventh specialty conference of probabilistic mechanics and structural reliability, Worcester, Massachusetts, USA; 1996. [26] Hess III PE, Bruchman D, Assakkaf IA, Ayyub BM. Uncertainties in material strength, geometric, and load variables. Nav Eng J 2002;2. [27] Wierzchowski EP. Statistical investigation of the sectional dimensions of ship structural members. Res. Report No. 018-BR/ BW3-71. Gdansk, Poland: Ship Design and Research Center; 1971. [28] Mansour AE, Faulkner D. On applying the statistical approach to extreme sea loads and ship hull strength. Trans Roy Inst Nav Architects 1973;115. [29] Barsa NS, Stanley RF. Survey of structural tolerances in the United States commercial shipbuilding industry. Ship Structure Committee Report; SSC-273. Washington D.C.: Ship Structure Committee, U.S. Coast Guard; 1978. [30] Ivanov LD, Rousev SG. Statistical estimation of reduction coefficient of ship’s hull plates with initial deflections. Nav Architect 1979;4:158–60. [31] Daidola JC, Barsa NS. Probabilistic structural analysis of ship hull longitudinal strength. Ship Structure Committee Report; SSC-301. Washington D.C.: Ship Structure Committee, U.S. Coast Guard; 1980. [32] Ayyub BM. Uncertainty modeling and analysis in civil engineering. Boca Raton, FL: CRC Press; 1998. [33] Southwell CR, Bultman JD, Hummer Jr CW. Estimating service life of steel in seawater. In: Schumacher M, editor. Seawater corrosion handbook. New Jersey: Noyes Data Corporation; 1979. p. 374–87. [34] Melchers RE. Probabilistic modeling of marine corrosion of steel specimens. In: Proceedings Conference ISOPE-95, Hauge, Netherlands; June 12–15, 1995. [35] Melchers RE. Probabilistic modeling of immersion marine corrosion. In: Shiraishi N, Shimozuka M, Wen YK, editors. Structural safety and reliability. Balkema; 1998. p. 1143–9. [36] Melchers RE. Corrosion uncertainty modeling for steel structures. J Constr Steel Res 1999;52:3–19. [37] Paik JK, Kim SK, Lee SK. Probabilistic corrosion rate estimation model for longitudinal strength members of bulk carriers. Ocean Eng 1998;25:837–60. [38] Guedes Soares C, Garbatov Y. Reliability of maintained, corrosion protected plates subjected to non-linear corrosion and compressive loads. Mar Struct 1999;12:425–45. [39] Harada S, Yamamoto N, Magaino A, Sone H. Corrosion analysis and determination of corrosion margin. Part 1 and 2, IACS discussion paper; 2001. [40] Wang G, Spencer J, Sun HH. Assessment of corrosion risks to aging oil tankers. In: 22nd international conference on offshore mechanics and arctic engineering, Cancun, Mexico; 8–13 June 2002.

106

J.(Jeffrey) Guo et al. / Marine Structures 25 (2012) 85–106

[41] Qin S, Cui W. Effect of corrosion models on the time-dependent reliability of steel plated elements. Mar Struct 2003;16:15– 34. [42] Paik JK, Lee JM, Hwang JS, Park YI. A time-dependent corrosion wastage model for the structures of single- and double-hull tankers and FSOs and FPSOs. Mar Technol July 2003;40(No. 3):201–17. [43] Paik JK, Thayamballi AK, Park YI, Hwang JS. A time-dependent corrosion wastage model for bulk carrier structures. Int J Marit Eng Roy Inst Nav Architect 2003;61–87. [44] Paik JK, Thayamballi AK, Park YI, Hwang JS. A time-dependent corrosion wastage model for seawater ballast tank structures of ships. Corros Sci 2004;46:471–86. [45] Hu Y, Cui W, Pedersen PT. Maintained ship hull girder ultimate strength reliability considering corrosion and fatigue. Mar Struct 2004;17:91–123. [46] Garbatov Y, Guedes Soares C, Wang G. Non-linear time dependent corrosion wastage of deck plates of ballast and cargo tanks of tankers. In 24th international conference on offshore mechanics and arctic engineering, Halkidiki, Greece; June 12–17, 2005. [47] Sun H, Guedes Soares C. A corrosion model and reliability-based inspection for ship-type FPSO hulls. J Ship Res 2006. [48] Melchers RE. Development of new applied models for steel corrosion in marine applications including shipping. In: Tenth international symposium on practical design of ships and other floating structures, Houston, USA; Sept. 30–Oct. 5, 2007. [49] Guedes Soares C, Garbatov Y, Zayed A, Wang G. Corrosion wastage model for ship crude oil tanks. Corros Sci 2008;50: 3095–106. [50] Hoadley PW, Yura JA. Ultimate strength of tubular joints subjected to combined loads. In: Offshore technology conference (OTC), Houston, USA; 1985. [51] ABS. Commentary on the guide for buckling and ultimate strength assessment for offshore structures. ABS; 2005. [52] ABS. ABS guide for buckling and ultimate strength assessment for offshore structures. ABS; 2004. [53] API. Bulletin on design of flat plate structures. API Bull 2000;2. [54] ABS. Rules for building and classing steel vessels. Part 3. ABS; 2007. [55] Ivanov L. Statistical evaluation of the ship hull cross section geometrical characteristics as a function of her age. Int Shipbuild Prog 1987;198–203. [56] Guedes Soares C, Garbatov Y. Reliability of maintained ship hulls subjected to corrosion and fatigue. J Struct Saf 1998;20: 201–19. [57] Wirsching PH, Ferensic J, Thayamballi A. Reliability with respect to ultimate strength of a corroding ship hull. Mar Struct 1997;10:501–18. [58] Ayyub B, Akpan UO, DeSouza GF, Koto TS, Luo X. Risk-based life cycle management of ship structures. Report no. SSC-SR 416. Ship Structure Committee; 2000. [59] Paik JK, Wang G, Thayamballi AK, Lee JM. Time-variant risk assessment of aging ships accounting for general/pit corrosion, fatigue cracking and local dent damage. In: SNAME annual meeting, San Francisco, USA; 2003. [60] Wang G, Lee AK, Ivanov LD, Lynch TJ, Serratella C, Basu R. A statistical investigation of time-variant hull girder strength of aging ships and coating life. Mar Struct 2008;21:240–56. [61] Crystal ball release 5, reference manual; 2001. [62] McKay MD, Conover WJ, Beckman Rj. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 1979;21:239–45. [63] Iman RL, Helton JC, Campbell JE. An approach to sensitivity analysis of computer models, Part 1. Introduction, input variable selection and preliminary variable assessment. J Qual Technol 1981;13(3):174–83. [64] Iman RL, Helton JC, Campbell JE. An approach to sensitivity analysis of computer models, part 2. Ranking of input variables, response surface validation, distribution effect and technique synopsis. J Qual Technol 1981;13(4):232–40. [65] Iman RL, Helton JC. A comparison of uncertainty and sensitivity analysis techniques for computer models. Technical Report SAND84-1461. Albuquerque, NM: Sandia National Laboratories; 1985.