Spectral fatigue damage assessment of tanker deck structural detail subjected to time-dependent corrosion

International Journal of Fatigue 48 (2013) 147–155

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Spectral fatigue damage assessment of tanker deck structural detail subjected to time-dependent corrosion K. Tran Nguyen, Y. Garbatov, C. Guedes Soares ⇑ Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e

i n f o

Article history: Received 10 June 2012 Received in revised form 18 September 2012 Accepted 8 October 2012 Available online 9 November 2012 Keywords: Fatigue damage Spectral analysis Operational conditions Corrosion wastage

a b s t r a c t The paper presents a spectral fatigue damage analysis of a double hull tanker structural detail accounting for corrosion wastage over time. The cyclic load of the wave-induced vertical bending moment, analysis using a strip theory on the frequency domain, is considered for two loading conditions. The influence of sea environment parameters and operational profiles including the use of different scatter diagrams, and wave variance spectra have been analyzed. The effect on the time-dependent cumulative fatigue damage as a function of corrosion deterioration is calculated. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Most often, ocean-going ships are sailing in waves of different regions in different loading conditions, leading to a challenging problem in the estimate of the wave-induced loads accounting for the interaction between ship and waves. The wave-induced loads induce hogging and sagging responses. Moreover, the complex ship hull structures contain numerous geometrical discontinuities and because of that the stress levels become higher and the fatigue strength assessment is an important criterion in ship structural design. In the analysis of ship structural details, such as the connections of deck longitudinal to transverse web, the interlaced hogging with sagging cause the repeated loading and unloading are considered to be the main source of fatigue damage. Fatigue cracking of structural details in the ship because of cyclic loading has been gaining attention during the last decades and much effort has been spent to develop different methods for predicting the fatigue strength/ life of welded joints [1]. However, fatigue life predictions for structural components subjected to random time varying loads continue to be the challenging problems in engineering. Strictly adhering to the experimental data of S–N curves and the Palmgren–Miner’s rule, the fatigue analysis based on the direct calculations of spectral method can predict the cumulative damage with a high degree of accuracy. The spectral-based fatigue analysis approach is a procedure for determining a complex stress transfer function, at a structural ⇑ Corresponding author. E-mail address: [email protected] (C. Guedes Soares). 0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2012.10.014

location of interest for the particular load conditions. The structural analyses are performed for specified ranges of wave frequencies and heading distributions, established through a direct wave load analysis combined with stress response analyses. Spectral analysis is used to develop a lifetime fatigue and an extreme load spectrum by considering the operational conditions of a ship at a sea state to be divided into different operational modes accounting for speed, heading, and significant wave height for a specific wave scatter diagram. Therefore, the spectral method accounts for various sea states as well as their occurrence probabilities. An effort to estimate the fatigue life associated with the wave loads on the tanker, have already been applied to calculate the fatigue damage for ship structures in [2,3]. The main steps in fatigue assessment of ship structures based on direct calculations of linear spectral analysis of the stress range response have been presented by Guedes Soares et al. [4]. The effect on ship main characteristics, operational profiles and wave climatic data for the spectral fatigue analysis, in combination with an assessment of the uncertainties induced by the different fatigue cumulative damage models for ship structural details have been analyzed by Garbatov and Guedes Soares [5]. Moreover, the fatigue life estimation and the probabilistic evaluation of the different steps to arrive at a safety index or time dependent reliability of a tanker hull structure are studied in [6]. Kukkanen and Mikkola [7] performed a spectral fatigue analysis in the ISSC comparative study of a hatch cover bearing pad on the container ship. The only important cyclic loading, wave induced vertical bending moment, is considered in the fatigue predictions. Recently, the practical case study of spectral fatigue analysis of a barge structural detail has been performed by Wang [8].

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The ship has to operate in a harsh humid environment and the aggressive atmosphere of the oceans as well as the corrosive effect of the sea and ballast water with rich chloride content, oxygen and other corrosive mineral degrade structures. The corrosion problems are considered to be one the most important factors leading to fatigue–age-related structural degradation and it is assumed in the present study here as a nonlinear one as a function of time. Several studies have been performed to evaluate the influence of corrosion wastage to aging tankers over time, including various environmental conditions [9–11], which is measured by the loss of hull girder section modulus to the deck [12,13] and the fatigue damage accumulation to the side structural details by Tran Nguyen et al. [14]. The present study deals with the spectral fatigue damage assessment of a structural detail at the center main deck of an oil tanker in the connection between the longitudinal stiffener and transverse web. The aim is to investigate the applicability of spectral methods for the estimation of the fatigue damage to the ship hull girder during her voyage in the different harsh environments. The influence of the sea state parameters, operational variations, and loading conditions including the use of different scatter diagrams, wave spectra are analyzed in terms of the short-term and long-term fatigue damage. Time-dependent cumulative fatigue damage analysis of tanker structural detail as a function of corrosion deterioration is also performed.

2. Spectral fatigue analysis Spectral analysis based on the frequency domain approach is applied for the assessment of fatigue damage in the present study. For this approach, several analytical formulas expressing fatigue damage are available using different spectral formulations. Typically, ocean wave-induced loads are the main source of fatigue damage of the structural system. A ship in open sea is subjected to many kinds of load effects: vertical and horizontal bending moments, torsional moments, side shell lateral pressures, etc. The sagging condition causes tensile stresses in the ship bottom, while the hogging condition may growth fatigue cracks in the ship deck. The lateral side shell and torsional loads are not considered in the present study, and only the vertical bending moments amidships as the loads on the ship is applied. The vertical bending moment RAOs (Response Amplitude Operators), which are used to calculate the stress transfer functions, Hr(x|h), can be obtained in various ways. The fundamental task of a spectral fatigue analysis is to define the stress transfer function, Hr(x|h), at a particular structural location as a function of the unit amplitude wave of wave frequency (x), wave heading angle (h), ship velocity (U) and loading condition (x). The resulting wave-induced bending moments, RAOM,ver, have been calculated using the in-house sea keeping code based on the linear strip theory developed by Salvesen et al. [15], which has been reported in [16,17]. The ship is assumed to be rigid and respond to the wave loads in heave and pitch degrees of freedom. The present formulation deals explicitly with the low frequency wave induced loads, thus neglecting the high frequency vibratory stresses associated with springing or with transient response to wave impact [18]. If these stresses were included the stresses would no longer be a narrow band process and for this case Jiao and Moan [19] have proposed a method in which correction factors are established to modify the calculated cumulative damage for the case in which the narrow-banded assumption is not fulfilled. Limiting the treatment to wave induced high cycle fatigue the stress range is described by the Weibull distribution. In general, it gives a satisfactory fit to full-scale measurements as well as numerical predictions of long term distribution [20].

If the ship response to wave excitation is linear the total response in a seaway is described by a superposition of the responses to all regular wave components that constitute the irregular sea, which can be performed in a frequency domain analysis. Given the linearity, the response is described by a stationary and ergodic but not necessarily narrow-banded Gaussian process. Linear strip theory is the established method of predicting wave-induced load effects despite the emerging availability of computer codes based on the discretisation of the hull in panels and on the application of three dimensional diffraction theory. The linear model assumption is generally adequate, but in severe seas, the response may not be linear and a non-linear analysis should be conducted. There are several non-linear theories, which result in an appropriate modeling of the non-linear dependency of the wave-induced vertical bending moment and wave amplitude. The theory of Jensen and Pedersen [21] is based on a perturbation approach, which keeps terms beyond the linear ones. Other approximation theory model the non-linearity in the hydrostatic restoring force and examples are the method of Guedes Soares and Schellin [22] and Fonseca and Guedes Soares [16]. Another line of work uses the panel method associated with time simulation, which however is very time consuming and still not applicable for design work. Long-term distributions have been developed for nonlinear wave induced loads, which showed a clear difference between hogging and sagging stresses [22]. However, since for fatigue it is the stress range that is important, this can still be calculated by linear theories in that the stress range shows a very small degree of nonlinearity [23]. For the fatigue analysis here, a linear model assumption of load is generally adequate and the non-linear effects can be neglected. The vertical bending stress, RAOr,ver, at the considered point in the cross section is given as:

RAOr;v er ¼ SCF 

z  z0 RAOM;v er Iyy

ð1Þ

where z is the vertical distance from the structural detail to the baseline and z0 is the vertical distance from the neutral axis to the baseline, Iyy is the moment of inertia with respect to the horizontal axis of the amidships section, SCF is the stress concentration factor to be decided for the considered typical detail in ship structures. More information about how to define the stress concentration factor in a corrosive environment can be found in [14,24]. Irregular waves are described by the modified wave energy spectrum, Sg(x|Hs, Tz), for the ocean surface elevation, which represents the distribution as a set of environmental parameters of the significant wave height Hs and the zero-up-crossing period Tz [25]. The occurrence probability, p(Hs, Tz), of sea states are obtained from the various waves scatter diagrams by the long-term statistical analysis of wave data. Each stress transfer function, Hr(x|h), is then used to calculate the stress response spectrum, Sr(x|Hs, Tz, h), by scaling the wave energy spectrum in the following manner [26]:

Sr ðxjHs ; T z ; hÞ ¼ jHr ðxjhÞj2 Sg ðxjHs ; T z Þ

ð2Þ

The nth spectral moments, ln, of the response process for applying a given wave spreading modifies may defined as [26]:

ln ¼

Z

hþ p=2 X

fs ðh0 Þ  xn Sr ðxjHs ; T z ; hÞdx

ð3Þ

x hp=2

where the confused short-crested sea conditions in a kinetic energy spread, which is generally approximated by a cosine-squared spreading function of the selected wave heading as [27]:

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( fs ðh0 Þ ¼

2

p

cos2 ðh0 Þ; h  p2 6 h0 6 h þ p2 0;

ð4Þ

otherwise

It is assumed that the short term sea characterization is a stationary zero mean Gaussian and narrow bandwidth process. Thus, the load response of ship structures FSij can be approximated to the Rayleigh distribution within each short term condition, in which the spectral moment of order zero l0 is equal to the square of the standard deviation of the stress process in a specific sea state, for a given sea state i and heading direction j leads to [28]:

F Sij ðDrk Þ ¼ 1  exp

f0;ij

1 ¼ 2p

Dr2k 8l0;ij

! ð5Þ

sffiffiffiffiffiffiffiffiffi

l2;ij l0;ij

ð6Þ

The long-term stress range predictions are determined by summing up the results of the short-term distributions into a weighting the expected overall sea states of the scatter diagram and heading directions with the probability of different operation modes and environmental conditions. An adequate approximation for the long-term distribution of wave-induced stress range can be fitted by the Weibull distribution as [29]:

"   # Drk h F L ðDrk Þ ¼ 1  exp  q

ð7Þ

where the shape parameter h and the scale parameter q are obtained through a least square technique of the Weibull probability plotting. 3. Fatigue damage cumulative models

D¼

nb X nk k¼1

Aa Sm a

ð8Þ

where ma and Aa are the empirical constants of the slope and the intercept parameter in the fitted S–N curve respectively with a = 1 for N 6 107 cycles and a = 2 for N > 107 cycles. Fatigue damage accumulated from each stress cycle can be calculated through the relevant S–N curve based on the assumption of

fσ(Δσ)

ð9Þ

Nk

where nb is the number of stress blocks, nk is number of constant amplitude stress range cycles Sk in block k, and Nk is the number of cycles to failure at constant stress range Sk. Fatigue damage is assumed to occur when the damage ratio D exceeds unity. For most marine structures, however, the service loading is a random variable, which can be described by a probability density function, fr(Drk), as shown in Fig. 1. To use the S–N fatigue test data with respect to the calculation of the cumulative fatigue damage due to a stochastic load process under the Palmgren–Miner’s rule, it is necessary to find a relationship between the characteristic value of the wave-induced random stress and the constant amplitude stress of the S–N curves. 3.1. Discrete approach The probabilistic descriptors of the long-term stress range distribution are obtained from recorded stress histories or estimated from the ship structural response by wave spectra analyses. An expression for the stress histogram of an equivalent stress range Se can be easily found by dividing the random distribution into many narrow stress blocks nb of width d(Dr), as illustrated in Fig. 1. The total reference number of cycles Nd of a ship structural component in a given duration service lifetime Td with respect to the different fraction time rx in the xth loading condition due to a long-term average zero-up-crossing frequency m0 per second, is approximated as:

Nd ¼

The fatigue strength is estimated based on the S–N curves and the Palmgren–Miner summation rule, due to a constant amplitude stress range S over a number of cycles N of failure, providing the following relation [28]:

N¼

the linear damage accumulation by a structural element expressed as [30]:

nx X

rx v 0 T d

ð10Þ

x¼1

The number of stress cycles in each block is given by:

nk ¼ Nd fr ðDrk Þ  dðDrÞ

ð11Þ

Thus, according to the linear damage accumulation hypothesis, the fatigue damage can be calculated as follows [31]:

D¼

nb X nk

Nk k¼1

¼

Nd Aa

Z

1

ðDrk Þma fr ðDrk ÞdDr ¼

0

Nd E½ðDrk Þma  Aa

ð12Þ

The number of blocks nb should be large enough to insure reasonable numerical accuracy, and normally should not be less than 20 as recommended in [28]. 3.2. Closed form expression approach Considering the random instantaneous ocean wave elevation process as a sequence of waves, the Weibull distribution is commonly used to describe the long term wave height and long term stress range processes. For ocean structures, the probability density function of the stress range can be presented by a two-parameter Weibull distribution. The cumulative damage complied with the Palmgren–Miner approach for a one-slope S–N curve is defined based on an earlier work for the closed-form mathematical expression by Nolte and Hansford [32] as:

fσ(Δσk)

δ(Δσ)

D¼

Δσk Fig. 1. Probability density function (PDF) of wave-induced stress.

Δσ

  m1 1 r x qm x C 1þ hx x¼1

nx m0 T d X

A1

ð13Þ

where nx is the total number considered load conditions with x = 1 for the full loading and x = 2 for the ballast loading respectively, C(1 + m1/hx) is the Gamma function.

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When bi-linear S–N curves are used, the fatigue damage is defined by the following closed form solution [28]:

( "   # 1 Drslope hx qm m x D ¼ v 0 T d rx C 1þ 1; A1 hx qx x¼1 "  hx #) m2 Drslope qx m c 1þ 2; þ A2 hx qx nx X

ð14Þ

The long-term loading is modeled based on a series of shortterm stationary sea states and heading directions, through the assumption of a Gaussian and narrow band process, where the amplitudes are fitted to the Rayleigh probability density function. The Rayleigh distribution is a special form of the Weibull distribution having the parameters [33]:

qffiffiffiffiffiffiffiffiffiffiffi h ¼ 2; q ¼ 2 2l0;ij

ð15Þ

As a result, based on the short-term stress range calculation, the fatigue damage accumulation for a linear S–N curve within the different loading conditions is defined as [28]:

all seastates A1

C 1þ

nx m1 X rx 2 x¼1

all headings X

 qffiffiffiffiffiffiffiffiffiffiffiffiffim1 lijx 2 2l0;ijx

ð16Þ

j¼1 i¼1

where the relative number lijx of stress cycles in a short-term condition i, j for each xth load case is defined as follows:

lij ¼

pij f0;ij

v0

pij :f0;ij ¼P pij :f0;ij

ð17Þ

where pij is the probability of occurrence of each sea state i combined with the heading direction j, which is obtained from the wave scatters. The fatigue damage expression, by applying a two slope S–N curve, is given as [28]:

all seastates D ¼ v 0Td

all headings nx X X rx lijx fB þ Cg x¼1 j¼1 i¼1

0 12 3  qffiffiffiffiffiffiffiffiffiffiffiffiffim2 2 2 2l0;ijx m2 B Drslope C 7 C¼ c6 ; @ qffiffiffiffiffiffiffiffiffiffiffiffiffiA 5 41 þ A2 2 2 2l

ð20Þ

0;ijx

4. Application of the spectral analysis

3.3. Spectral approach

m0 T d 

ð19Þ

0;ijx

where Drslope is the stress range at which a change in the slope occurs, C(;) and c(;) is the complementary incomplete and the incomplete Gamma function respectively.

D¼

0 12 3  qffiffiffiffiffiffiffiffiffiffiffiffiffim1 2 2 2l0;ijx m1 B Drslope C 7 6 B¼ C41 þ ; @ qffiffiffiffiffiffiffiffiffiffiffiffiffiA 5 A1 2 2 2l

ð18Þ

4.1. Detailed calculations and operational conditions A typical connection of bracket of an oil tanker at the intersection between a main deck longitudinal stiffener and a transverse web frame at the center line of the mid-ship section is considered in this study (see Fig. 2). The wave-induced vertical bending moment is the main load source causes fatigue. The fatigue damage is accumulated around the weld toes on the flange of the longitudinal stiffener attached to the bracket, which of the stress concentration factor for the hotspot is 1.47. The oil tanker (VLCC – very large crude carrier) has already been designed, successfully built and spent some years in-service, with the principal dimensions given in Table 1. The tanker is analyzed with respect of fatigue damage considering full and ballast loading conditions. Irregular waves may be described by a spectrum that shows the quantity of wave energy at different wave frequencies. Various formulations of wave spectrum have been developed from field measurements [25]. The two most widely used forms are the Pierson– Moskowitz (P–M) spectrum [34], which applies to fully developed waves in deep water, and the JONSWAP (Joint North Sea Wave Project) one [35], which has a sharper peak and applies to the growing waves in continental – shelf waters. Both of these wave energy spectra are employed and compared with the analysis presented here. The uniform distribution of the heading angles from 0o to 180o has been assumed in 22.5o intervals, which results in equal probabilities for each considered heading directions. The target operational life of the oil tanker is assumed to be 25 years, with the duration of 85% at sea, considered as the unrestricted service classification in the harsh environments of oceans. Long-term descriptions of the wave environment in the form of the wave scatter diagram are required for the long-term statistical analysis of ship structural response and the fatigue assessment. The probability of the occurrence of various sea states on her route

Fig. 2. Structural detail of the intersection main deck of oil tanker.

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K. Tran Nguyen et al. / International Journal of Fatigue 48 (2013) 147–155 Table 1 Vessel hull characteristics.

Corrosion depth

Coating Transition life time

Principal particulars of VLCC Length overall Length between perpendiculars Breadth Depth Fully draft Ballast draft Block coefficient Design speed Sectional properties of mid-ship section: Height of neutral axis above baseline Vertical moment of inertia of hull cross section

Loa Lpp B D Tf Tb CB U

332 m 320 m 58 m 31 m 20.8 m 14 m 0.82 16 kn

zo Iyy

13.525 m 1479.42 m4

τc

τt Long-term corrosion depth

0

Time

Fig. 4. Schematic corrosion wastage model as a function of time.

85.0

Section modulus [m3]

84.0

83.0

82.0

81.0

80.0 0

Fig. 3. Speed profile at the different sea states.

service are obtained from different wave scatter diagrams including the WWT-1 (World Wide Trade) the WWT-2 and the IACS [28]. In general, the VLCC tanker trade in the worldwide on long haul voyages over the oceans and try to avoid the bad weather as well as high waves. The design average speed of the ship is 16 knots. However, the increase or the reduction of speed depends on the sea climate conditions. Hence, the operational profiles with six modes from 0 to a maximum of 17 knots are analyzed. The assumption of the speed profile as a function of significant wave heights at the different sea states is shown in Fig. 3.

4.2. Effect of corrosive environment Ship structures also operate in a complex environment with the high humidity and an aggressive marine atmosphere. As a result, corrosion is recognized to be one of the most important phenomena leading to a degradation of strength and fatigue life of ship structures. The conventional models of corrosion wastage assumed a constant corrosion rate, leading to a linear relationship between the material lost and time. General corrosion, which is the most common form of corrosion, takes place over the entire surface of the metal and is the one addressed here. Starting with a linear equation proposed by Southwell et al. [36] to estimate the corrosion wastage thickness, which is considered as appropriate for design purposes and overestimate the corrosion depth at the initial phases of corrosion progress. Guedes Soares and Garbatov [37] developed a nonlinear model, that describes the time-dependent growth of corrosion wastage (see Fig. 4), which becomes widely used to predict the corrosion wastage [10,38,39].

5

10

15

20

25

30

Time [yrs] Fig. 5. Hull girder section modulus as a function of time.

It is noteworthy that the nonlinear corrosion wastage model is assuming no corrosion in the first phase with good corrosion protection. Failure of protection system will occur at a random point of time and the corrosion wastage will start a nonlinear process of a growth over time, which level asymptotically tends at the longterm corrosion wastage. Indeed, ship structures will only be exposed to corrosive environments after the effective coating period, and the hull girder section modulus will decrease over time. The loss of hull girder section modulus is an important parameter to investigate in the fatigue damage assessment. Based on the statistical study of Guo et al. [13], where the sets of corrosion data of deck plates of tankers, were fitted to an equation for predicting the mean values of the hull girder section modulus to the deck loss of aging tanker has as:

Rm ðtÞ ¼

0:62ðt  6:5Þ0:67 ; 100

t > 6:5 year

ð21Þ

For the studied VLCC tanker, the lifetime change of hull girder section modulus to the deck is derived as a function of time, which is shown in Fig. 5. It can be seen that the loss of hull girder section modulus after the coating life is approximately 4.4% for 25 years and 5.1% for 30 years. 5. Fatigue damage assessment The fatigue damage accounting for corrosion is the outcome of the interaction between the environment, material microstructure, and cyclic loads over the time. In this study, the S–N curves I for

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Fatigue Analysis

Hull section modulus, W(tn)

Direct spectral analysis

Simplified analysis, Rules

Bending moment RAOs [linear strip theory] Stress transfer function, Hσ(ω|θ) Stress energy spectrum, Sσ(ω|Ηs,Τz,θ) Spectral moment, μn Pierson-Moskowitz wave spectrum, Sη(ω|Ηs,Τz) Scatter diagram WWT-1, Case 1

Scatter diagram WWT-2, Case 3

JONSWAP wave spectrum, Sη(ω|Ηs,Τz)

Scatter diagram IACS, Case 5

Scatter diagram WWT-1, Case 2

Scatter diagram WWT-2, Case 4

Scatter diagram IACS, Case 6

Fatigue damage calculations, Δσo, Δσ(tn) Discrete approach

Closed-form approach

No

Corrosion degradation

Spectral approach

Yes Coating life, τc No

tn ≤ τc

S-N curve

Yes

No

corrosive environment

Yes Δσο

Δσο

Δσο

Δσ(tn)

S-N curve In Air

S-N curve In Air

S-N curve Corrosive Env.

S-N curve In Air

Fatigue damage Fatigue damage Dcoated(Δσo) Dcorrosion(Δσo)

Fatigue damage Dcorrosion(Δσ(tn))

No

tn ≥ Td

tn = tn-1 + Δt

tn ≥ T d Yes

Fatigue damage without corrosion, D = Din air(Δσo)

Fatigue damage with implicit corrosion, D = Dcoated + Dcorrosion(Δσo)

No

tn = tn-1 + Δt

Yes Fatigue damage with explicit corrosion, D = Dcoated + Dcorrosion(Δσ(tn))

Fig. 6. Fatigue damage assessment flowchart.

welded joints in corrosive and non-corrosive environment are used for the estimation of fatigue damage [28]. For the non-corrosive condition in air or with cathodic protection in sea water, the linear and bi-linear S–N curve I are applied, where the curve slope parameters are defined as m1 = 3 and m2 = 5 corresponding to two fatigue strength coefficients

A1 = 1.46  1012 and A2 = 4.04  1015 respectively. For the corrosive environment in sea water, the single linear S–N curve I is applied in the calculations with the slope parameter m1 = 3 and the strength coefficient A1 = 4.86  1011. It is important to investigate the influence of the different environmental, operational conditions as well as the sensitivity to

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2.4

Full load

Two slope-Spectral approach

Two slope-Spectral approach

Two slope-Closed form approach

2.0

One slope-Spectral approach

Cumulative Damage

Cumulative Damage

2.0

Ballast load

Two slope-Closed form approach

One slope-Closed form approach

1.6

1.2

0.8

0.4

One slope-Spectral approach One slope-Closed form approach

1.6

1.2

0.8

0.4

0.0

0.0 1

2

3

4

5

1

6

2

3

Cases

4

5

6

Cases

Fig. 7. Cumulative fatigue damage without corrosion, full load (left) and ballast load (right).

5

5

Implicit analysis

Cumulative Damage

4

Explicit analysis Case 2 Case 4 Case 6

Case 1 Case 3 Case 5

4

Cumulative Damage

Case 1 Case 3 Case 5

3

2

1

Case 2 Case 4 Case 6

3

2

1

0

0 0

5

10

15

20

0

25

5

10

15

20

25

20

25

Time [years]

Time [years]

(a). Full loading condition. 5

5

Implicit analysis

Cumulative Damage

4

Explicit analysis Case 2 Case 4 Case 6

Cumulative Damage

Case 1 Case 3 Case 5

3

2

1

Case 1 Case 3 Case 5

4

Case 2 Case 4 Case 6

3

2

1

0

0 0

5

10

15

20

25

Time [years]

0

5

10

15

Time [years]

(b). Ballast loading condition. Fig. 8. Cumulative fatigue damage with implicit (left) and explicit (right) corrosion, spectral approach.

wave-induced hull girder stress over time due to the corrosion wastage. The flowchart to perform the fatigue damage assessment based on the spectral analysis for the considered structural detail is shown in Fig. 6, where the parameter set up of the six studied cases, case 1–6 are identified. 5.1. Fatigue damage without corrosion Fig. 7 shows the results of the cumulative fatigue damage without corrosion degradation using the one-slope and two-slope S–N curves at the speed profile for full and ballast loading conditions.

It can be observed that the ballast load condition always leads to fatigue damage higher than in the full load one for all cases. However, the difference is slight in the case 1. In addition the effect of using different S–N curves for the fatigue predictions is also investigated. The bi-linear S–N curve yields the cumulative fatigue damage rather lower than the one calculated by the linear S–N curve for all cases. It is noticeable that in the cases 1 and 2 the fatigue damage prediction is very low for both loading conditions, but it increases rapidly in the cases 3–6. The fatigue damage accumulation is relatively uniform with respect to the spectral and closed-form approaches in the case 1–4.

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Pierson-Moskowitz

JONSWAP

1.2

0.8 25

0.4 15 0.0 0.5

4.5

8.5

12.5

5 16.5

Time [yr]

Fatigue Damage

Fatigue Damage

1.2

0.8 25

0.4

15 Time 0.0 0.5

4.5

8.5

12.5

5 16.5

[yr]

Hs [m]

Hs [m]

Fig. 9. Fatigue damage as a function of significant wave heights and time, the North Atlantic scatter diagram with P–M spectra (left) and JONSWAP spectra (right), explicit corrosion analysis.

It should be noted that in the case 6, the spectral approach demonstrates higher fatigue damage than the closed-form approach in full and ballast conditions. All of the calculated results of fatigue damage presented in Fig. 7, are assumed to be without corrosion for the ship structural details during the service lifetime adhering to the S–N curve in non-corrosive environment.

5.2. Time-dependent corrosion fatigue Corrosion has a harmful consequence, viewed from safety and leads to thickness reduction and consequently induces loss of the hull girder section modulus, decreasing the fatigue life. Therefore, it is necessary to consider the time-dependent corrosion for increasing the degree of accuracy in the fatigue damage assessment. The explicit corrosion degradation effect, as defined here, is related to a direct reduction of plate thickness due corrosion wastage, while the implicit corrosion is related to S–N curve developed in a corrosive environment, where the thickness reduction is accounted for during the fatigue tests. Fig. 8 shows the time-dependent cumulative fatigue damage with an implicit and explicit corrosion degradation modeling for all studied cases at the speed profile in full and ballast loading conditions. It is found that after the coating life, the fatigue damage with the explicit corrosion model is significantly lower than the one with the implicit corrosion model for all loading cases. It is also shown that the difference of fatigue damage over time, which applying the implicit corrosion model is fairly large for the studied cases in two loading conditions. It can be observed that the fatigue damage is very slight in the initial interval, which is identified with a good corrosion protection, but increases significantly over the time after the coating life for both of corrosion models and loading conditions. Similar to the fatigue analysis without corrosion, the estimated fatigue damage to the ballast loading condition is higher than the one of the full load in all cases. The cumulative fatigue damage is the lowest in the case 1, while the highest in the case 6. Clearly, for cases 3 and 5, the fatigue damage with implicit and explicit corrosion are nearly uniform in all loading conditions. In some cases the fatigue damage is bigger than one, leading to a fatigue damage before the end of service life. The contribution relative to the total cumulative fatigue damage accounting for the corrosion of the operational profile as a function of the significant wave heights and time, for sea state parameter of the North Atlantic described by the P–M and JONSWAP wave energy spectrum respectively, with respect to

the explicit corrosion analysis, is shown in Fig. 9. It is recognized that the difference in cumulative damage, using the JONSWAP spectra, is about double with respect to the P–M one. Integrating the area below the plotted surfaces in the considered year will give the total corrosion accumulated fatigue damage at that time. It can be observed that the peak of the cumulative fatigue damage described by the JONSWAP wave spectrum is rather narrow, sharper and higher than the P–M one. Similar corrosion fatigue analysis for the sea state of worldwide trade over time had also been performed.

6. Conclusions The application of the spectral fatigue damage approach to evaluate a main deck longitudinal stiffener, considering the timedependent corrosion effect in a double hull oil tanker subjected to a vertical wave-induced bending moment has been presented in this article. The fatigue analyses are performed for two basic loading conditions, full and ballast, of ship accounting for the expected operational time in the open sea with respect to an unrestricted service. The frequency domain analysis for the estimation of the vertical wave induced bending moment is based on the strip theory. The influences of different environmental and operational profiles were investigated including speeds, wave spectrums, and scatter diagrams. Time-dependent corrosion degradation over the entire service life of the ship structural detail has been analyzed based on the predicted corrosion degradation of hull girder section modulus to the deck of an oil tanker. As a result of the performed analysis, the cumulative corrosion–fatigue damage increases in all calculated cases after the failure of the corrosion protection system. It has been found that the fatigue damage in ballast loading condition is higher than the one in the full load case. In addition, using the explicit corrosion modeling after the coating peroid for the fatigue analysis over time leads to significantly lower cumulative damage than the one of the implicit model. The results from the spectral fatigue analysis have also shown a significant difference in fatigue damage between the Pierson– Moskowitz wave spectrum and the JONSWAP one, as well as among the used scatter diagrams. The biggest fatigue damage is achieved using the JONSWAP spectra and the IACS wave scatter diagram. The spectral fatigue damage analysis has been employed accounting for the corrosion effect during the service lifetime of the ship. It has been shown that the accuracy of the input data, as the sea state parameters, environmental and operational conditions, speed ranges, loading conditions as well as S–N curve

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