A comparative study of fatigue assessments of container ship structures using various direct calculation approaches

A comparative study of fatigue assessments of container shipstructures using various direct calculation approaches

Zhiyuan Li a,n, Wengang Mao a, Jonas W. Ringsberg a, Erland Johnson b, Gaute Storhaug c

a Chalmers University of Technology, Department of Shipping and Marine Technology, Division of Marine Design, SE-412 96 Gothenburg, Swedenb SP Technical Research Institute of Sweden, SE-501 15 Borås, Swedenc Det Norske Veritas AS, Veritasveien 1, 1322 Høvik, Norway

a r t i c l e i n f o

Article history:Received 19 March 2013Accepted 22 February 2014Available online 15 March 2014

Keywords:Comparative studyContainer shipDirect calculationFatigueFull-scale measurement

a b s t r a c t

It is common practice today to carry out fatigue assessments of ship structures using direct calculationprocedures to compute fatigue loads. Many numerical codes are available for use in such fatigue loadanalyses. In addition to the various degrees of computation complexity associated with fatigueestimation methods, such methods also have large inherent uncertainties. In this investigation, acomparative study was carried out for two container ships using various typical direct fatigue calculationmethods. The fatigue damage amounts calculated using these methods were compared with thoseobtained from full-scale measurements. Most of the direct calculation approaches investigated yieldedsimilar fatigue damage estimates. The approach that employs nonlinear time-domain hydrodynamicanalysis and the finite element method yields reasonable and conservative fatigue damage results and istherefore recommended. In addition, the results of this study confirm that various measures of waveenvironments and of the variation in wave models are important sources of uncertainty in fatigue lifeprediction.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

As manufactured goods become increasingly containerised,container ships are becoming more important in the shippingindustry. The container ship fleet has expanded from 1.6% of thetonnage of the world fleet in 1980 to over 13% in 2011 (UNCTAD,2012). The role of container ships in global trade is even moreimportant than this tonnage share would suggest: in terms ofdollars, 52% of today's seaborne trade is containerised (UNCTAD,2012). Meanwhile, container ships are increasing dramatically insize. Since the emergence of the first container ship in the mid-1950s, five major generations of container ships have been devel-oped, and container transport capacity has increased from lessthan 1000 TEU (20-foot-equivalent units) to 18,000 TEU (Rodrigue,2013). The size of container ships is expected to continue toincrease because of the driving force of scale economics.

The rapid development of container ships has created chal-lenges with respect to fatigue design. As ships become larger, theirloads increase, and the hull strength must be enhanced as a result.To reduce material costs and maximise the deadweight for loadcapacity, high-tensile steel has been used extensively in ship hullstructures, and the steel strengths used are increasing. High-

tensile steels with yield strengths as high as 470 MPa have beenused in the construction of hull structures for container vessels.Given that the fatigue resistance of welded structures is notproportional to that of the base material but rather is limited bythe fatigue strength of the welded joints, higher cyclic stressesalong with the usage of high-tensile steel have resulted inincreased fatigue damage to previously acceptable ship structuraldetails. Moreover, although high-tensile steel is already used,thicker plates must be used in the upper hulls of container shipsbecause of large deck openings. Thus, the fatigue strengths of deckstructures with thick plating decrease due to the thickness effect.To address the increase in hull girder stress and the decrease infatigue strength, structural details should be optimised withrespect to stress concentration and fatigue loads must be evalu-ated with improved accuracy.

Fatigue cracks in container vessels have not been widelyreported in the public literature until recently. This can beexplained in part by the fact that the global container fleet isrelatively young, with an average ship age of 10.7 years, which isconsiderably lower than the average age of other major types ofships (UNCTAD, 2012). However, container vessels do have fatigueproblems. Fricke et al. (2010) reported a growing number offatigue cracks in the side-shell structures of 10-year-old Panamaxcontainer ships. Storhaug and Moe (2007) reported the presenceof serious deck cracks in container vessels after less than 8 years ofservice. Surveys indicated that the structural design of these ships

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Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2014.02.0220029-8018 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel. +46 31 772 2655;fax: +46 31 772 3699.E-mail address: [email protected] (Z. Li).

Ocean Engineering 82 (2014) 65–74

followed regulations and that good welding workmanship wasobserved at the cracked joints. Fig. 1 provides examples of fatiguecracking for both of these cases. These findings justify the need forfatigue assessments of individual vessels under actual operatingconditions using more sophisticated approaches with respect towave loading representation and fatigue stress analyses.

Fatigue assessments of ship structures are typically conductedusing direct calculation procedures to compute fatigue loads. Incontrast to the conventional-rule design approach, the directcalculation approach considers the structural and operationalfeatures of individual vessels, which is expected to lead to animproved fatigue assessment. Numerical codes are widelyemployed in analyses of wave loads and ship structural responses.These codes are based on various theories with different levels ofcomplexity. However, there is no consistent evidence that oneapproach is optimal in all cases. The use of different methods forthe same vessel typically yields different calculated fatiguedamages. Thus, a comparison is needed to rationally investigatethe reliability of the commonly used fatigue analysis methods.

Several authors have investigated the accuracy of fatigueassessment of ship structural details by direct calculations. In acomparative study of fatigue strength assessment procedurescoordinated by the International Ship and Offshore StructuresCongress (ISSC), a case study of a deck detail of a Panamaxcontainer vessel was investigated using spectral approach, inaddition to procedures used by the classification rules. The resultsof this comparative study were reported by Fricke et al. (2002) andKukkanen and Mikkola (2004). The spectral approach yielded apredicted fatigue life of 5.3 years, which is remarkably short forthis structural detail of known good design. The results of that casestudy suggest that significant uncertainties are associated withdirect calculation approaches. Researchers have also conductedcomparative studies of wave loads on ships. For example, Cariouand Jancart (2003) conducted a comparative study of wave-induced loads on barge ship models and found that the varioushydrodynamic codes yield significantly different results. Theyconcluded that the geometric nonlinear effects should not beneglected for high sea states. This conclusion agrees with thefindings of Watanabe and Guedes Soares (1999), who comparedmotion and load calculations for a realistic hull shape. Althoughthese studies focused on ship motions and loads on the hull as awhole, they provide insights into the reliability of fatigue assess-ment via direct calculation methodologies.

In this investigation, a comparative study of two containervessels was carried out using various typical direct calculationmethods. Both vessels operate along North Atlantic routes. Decklongitudinals amidships were the focus, and the results of the

fatigue damage assessments were compared with those from full-scale measurements. The objectives of this comparative studywere to obtain a better understanding of the factors that influencefatigue loads on ship structures and to evaluate the uncertaintiesassociated with the direct calculation procedures. Although thisparticular study concerns container vessels, it also providesinsights into direct calculation procedures for fatigue in general.

2. Methodology

The direct calculation of fatigue loads on ship structuresinvolves hydrodynamic analysis, stress response evaluation, andfatigue damage calculation. Many numerical codes are availablefor these types of analyses. The following sections discuss themost commonly used numerical approaches and a validationmethodology.

2.1. Wave modelling

The sea surface is irregular and changes constantly. An exactmathematical representation of the wave elevation as a functionof such variables as time, wind speed, and wind direction is impos-sible. Therefore, statistical methods must be used to quantify wavecharacteristics. Wave elevation records can be treated as recordsof a random process. For ocean waves and derived wave loadprocesses, measurements have shown that a stationary stochasticprocess can be assumed over short periods of time.

In ship structural analysis practice, it is typically sufficient touse linear (Airy) wave theory for the wave model, and this theorywas used throughout this study. The irregular wave time historycan be decomposed into a series of component regular wavesusing Fourier analysis:

ζ ¼ ∑N

i ¼ 1ζi sin ðωit�kixþεiÞ ð1Þ

where ζi, ωi, ki, and εi are the amplitude of the component wave,the wave frequency, the wave number, and the phase of the ithharmonic wave component, respectively.

A stationary sea condition is typically described by a wavespectrum. A wave spectrum S(ω|Hs, Tp) describes the amount ofwave energy at different wave frequencies and can be expressed asa function of the significant wave height Hs and, for instance, thepeak period Tp. With the spectral density known, the amplitude of

Fig. 1. (Left) Typical repaired crack on the upper inner side longitudinal below deck amidships (from Storhaug and Moe (2007)); (Right) Fatigue crack at the side longitudinalof a Panamax container vessel (from Fricke et al. (2010)).

Z. Li et al. / Ocean Engineering 82 (2014) 65–7466

the component waveζican be determined using Eq. (2).

ζi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2SðωiÞΔω

pð2Þ

where Δω is the frequency increment shown in Fig. 2.A wave is typically considered to be “long-crested” which

implies that all of the component waves propagate in the samedirection. This is an idealised situation; in seas, waves almostalways propagate in other directions as well and are thereforecalled “short-crested”. In ship design practice, the assumption oflong-crested waves may result in overly conservative analysisresults for head seas or following seas. Furthermore, the direc-tional wave components have a strong influence on the ship's rollmotion and should thus not be neglected when consideringoblique waves. A short-crested sea can be modelled by includingsome spreading functions in the wave spectrum. A “cosine square”spreading function is often assumed in practice. Section 7 dis-cusses the effects of various wave spectra with and without wavespreading on fatigue damage estimation in further detail.

2.2. Wave loads on ships

In the structural analysis of ships, hydrodynamic loads can becomputed by either the strip theory method or the panel method.In the strip theory method, the ship hull is treated as a series ofstrips that represent cross sections in the longitudinal direction.Many authors have reported that the results predicted by the striptheory method correspond well with experimental results in thevertical plane but less well with experimental results in the lateralplane; see, for example, Beck et al. (2009). In the panel method,the hull form is modelled as a number of panels. The wave loadsare represented by water pressure loads distributed on thesepanels along the entire hull of the ship. In contrast to the striptheory method, the panel method considers three-dimensional(3D) effects. Hence, the panel method is expected to yield betterresults with respect to ship motions and section loads in thelateral plane.

Wave loads on ships contain nonlinear terms, even if the waveitself is modelled as the sum of linear regular waves, as in thisinvestigation. This nonlinearity originates from the hull geometryof the ship and is called geometric nonlinearity. The non-verticalhull form at the mean water line both fore and aft of the hull tendsto promote more hull sagging moments than hogging moments.As a result, bending moments and shear forces may increase inboth amplitude and frequency. For example, Gu and Moan (2002)demonstrated that nonlinear wave loads result in significantincreases in fatigue damage compared to linear wave loads. Thisfinding was confirmed by Jensen (1991).

In practice, however, a linear relationship is often assumedbetween wave and ship responses. For most practical purposes,ocean waves can be assumed a Gaussian process that can becompleted described by the spectral density (Lewis, 1989). Theassumption of a linear relationship between wave and ship allowsthe responses to be described by the superposition of theresponses to all regular wave components that make up theirregular sea. Making use the linear assumption, hydrodynamicproblems can be solved in the frequency domain and shipresponses are derived based on transfer functions. Further, fatiguedamage can be calculated from the stress response spectrum. Thismethodology is termed as a spectral approach; see, for example,DNV (2010a).

2.3. Derivation of fatigue stress

An accurate computation of fatigue stress requires the con-struction of a detailed finite element (FE) model for all fatiguecritical locations, followed by FE analyses that demand intensivecomputational resources. Therefore, instead of using the actualhotpot stress, the common practice is to multiply the nominalstress by the corresponding stress concentration factor. Thisapproach is again an approximation that typically introducessignificant uncertainties into the direct calculation procedure.The stress concentration factor has a significant influence on theresults of a fatigue analysis and must be computed in a moreaccurate way in which realistic load combination is considered.Recently, Li et al. (2013) proposed a method to derive the localstress for use in fatigue calculations from the ship's global nominalstress. The idea behind this method is that a stress concentrationfactor should be calculated from realistic loadings. However, in thecurrent investigation, the stress concentration factor was set totwo for simplicity because it is the relative value of fatigue damagethat is of interest.

The nominal stresses in a ship hull can be derived eitherthrough beam theory or FE analyses. Both methods wereinvestigated in this study. In direct calculations using beamtheory, the nominal stress is often computed using the verticalbending moment (VBM), horizontal bending moment, torsionmoment as well as the vertical and horizontal shear forces. Forthe structural details of the deck longitudinal in container ships,however, it is common to consider only the VBM in the fatiguecalculation because the nominal stress in the investigatedlocations is mainly caused by the VBM. This fact indicates thatthe VBM dominates the longitudinal stress, not because thehorizontal bending and torsion are small but rather becausethey counteract each other. A detailed discussion of the relation-ships among the sectional forces in container vessels is providedby Li and Ringsberg (2011).

Nominal stresses calculated from FE analyses yield moreaccurate representations of a ship structure's response. Thisobservation is particularly true for fatigue damage estimationfor structural details close to the water line, where the non-linear effects due to the external water pressure variationbecome prominent. In this area, in addition to the global andlocal hull girder loads, the local pressure loads contribute amajor portion of the fatigue damage in the side-shell struc-tures. It is difficult to express this pressure loading in combi-nation with hull girder loads through beam theory. Global andlocal FE analyses are more appropriate for capturing thisfatigue phenomenon. However, for the target structural detaillocated in the upper deck, it is generally sufficient to obtain thefatigue stress from a global FE model. Details of the FE modelare presented in Section 3.3.

Fig. 2. Harmonic wave component amplitudes (energy) chosen from a wavespectrum.

Z. Li et al. / Ocean Engineering 82 (2014) 65–74 67

2.4. Estimation of fatigue damage

Fatigue in ship structures is typically considered using the S–Napproach. When the stress history is available, the stress cycles,denoted by Si, can be extracted by various counting approaches,the most frequently used of which is the rainflow counting (RFC)method. This approach identifies the local maxima that should bepaired with the local minimum to form a hysteresis loop, which isbelieved to yield the best representation of the cyclic stress–strainbehaviour of the material. The fatigue damage caused by theextracted stress cycles is calculated using the Palmgren–Minercumulative rule, in combination with S–N curves that are specificfor various structural details:

D¼∑iðNiS

mi Þ=α ð3Þ

In Eq. (3), Ni is the number of stress cycles for the stress range Siand m and α are the material parameters of the S–N curve. A one-slope hotspot stress S–N curve (m¼3, α¼1012.164) was employedin this study, which is similar to IIW FAT90 (Hobbacher, 2009).

Alternatively, followed by spectral analysis of hydrodynamicloads, fatigue damage can be approximated by assuming that thestress signals are narrow-band Gaussian processes. For a stationarysea state during period T, i.e., tA[0, T], the expected fatiguedamage accumulation is approximated by narrow band approx-imation (NBA) (Rychlik, 1993) using a linear S–N curve as follows:

EðDÞ ¼ α�1ð2ffiffiffiffiffiffiffiffi2λ0

pÞmTf zΓ 1þm=2

� � ð4Þwhere Γ() is the gamma function, λ0 is the variance of the stressresponse during period T, fz is the zero up-crossing frequency,and m and α are the material parameters of the S–N curve.The expected fatigue damage can be computed from the spectralmoment of the stress if the process is described by its powerspectrum.

A NBA tends to overestimate fatigue damage because a stressprocess is typically wide-banded in reality. Thus, some correctionfactors have been proposed; see, for example, Wirsching and Light(1980). However, the correction factor that they proposed was notemployed in this investigation because the commonly acceptedS–N methodology that is based on constant-amplitude test dataappears to predict non-conservative fatigue damage results underrandom loads (Agerskov, 2000). A NBA may compensate to somedegree for the bias.

3. Numerical models

In this study, full-scale measurements of two container ships,denoted as 2800 TEU and 4400 TEU, were employed in casestudies. The main characteristics of the two container ships arelisted in Table 1.

Numerical models were established for these two containervessels. For both ships, the ship motions and sectional loads were

calculated using strip models. For the 4400 TEU vessel, a panelmodel was developed as well, which allowed the wave loads to becomputed using a 3D time-domain methodology. Fatigue stresseswere calculated from the sectional loads using beam theorytogether with the empirical stress concentration factor. Thehydrodynamic loads were further transferred to the FE model forthe 4400 TEU ship, which made it possible to compute the stresshistories from FE analyses.

3.1. Strip model

Strip models were created, and the simulations were carriedout using the seakeeping programme WAVESHIP (DNV, 1993).WAVESHIP is a linear frequency-domain code without any non-linearity modification. However, the forward speed correction isconsidered in WAVESHIP, in contrast to conventional strip-theory-based codes. Moreover, the diffraction forces are derived from theslender body approach. This should be considered an approximatesolution for 3D effects in place of using full 3D methods. The shiphulls of the 2800 TEU and 4400 TEU vessels considered in thisstudy were represented by 55 and 58 strips, respectively. The wavespectrum comprised a total of 25 waves with wavelengthsbetween 0.1 and 3.25 times the ship length. The wave headingangles were evenly distributed in 101 increments.

3.2. Panel model

The Rankine panel method-based code WASIM (DNV, 2006)was also used in the current investigation, and the calculationswere carried out in the time domain. In WASIM waves can comefrom arbitrary directions, and ship motions are calculated in allsix degrees of freedom. The forward speed is included, andthe incident waves are modelled as irregular short-crested waves.WASIM handles full 3D effects, and the sectional loads are derivedby integrating pressures along the ship hull.

In WASIM, wave loads can be computed through either linearor nonlinear algorithms. For the linear solution, the Froude–Krylovand restoring hydrostatic forces are determined by integrating thepressure up to the mean wetted surface. The sea pressure abovethe mean wetted surface is always zero for the splash zone,whereas negative pressure occasionally occurs below the meanwetted surface, leading to an unrealistic stress distribution in thesplash zone. For the nonlinear solution, the Froude–Krylov andrestoring hydrostatic forces are integrated up to the instanta-neously wetted hull. Thus, the nonlinear load caused by seapressure near the mean free surface is modelled more realistically.By this means, the geometric nonlinearity due to the hull shape isalso considered, yielding more accurate values for the wave-induced hogging and sagging moments. However, this nonlineareffect is still simplified because the radiation and diffraction partsare linearised and solved for the mean free and mean wettedsurfaces. In this sense, WASIM must be considered a weaklynonlinear approach. Nevertheless, the existing literature indicatesthat this type of nonlinearity is essential for solving nonlinear shipmotions (ITTC, 2008). Research has demonstrated that nonlinearship motions and hull girder loads computed from hydrodynamiccodes similar to WASIM agree with experimental results (Songet al., 2011). Codes that incorporate more nonlinear terms areavailable, but these codes require much more computation effortand were therefore not utilised in the current study. Additionalcomparisons and discussion of various levels of nonlinearity inwave loads can be found in, for example, Singh and Sen (2007).

A total of 2872 panels were allocated along the hull of the4400 TEU vessel, with refined meshes at the bow and stern.The free surface mesh was set to be five ship lengths to meetthe longest radiated or scattered wave of the encountered waves.

Table 1Main characteristics of the two container ships used in the validation.

Max. TEU 2800 4400

LPP (m) 232 281LOA (m) 245 294Beam moulded (m) 32.2 32.2Depth, moulded (m) 19.0 21.5Design draught (m) 10.78 10.78Block coefficient 0.69 0.69Deadweight (tonnes) 40,900 47,754Service speed (knots) 21.3 23.0Year built 1998 2003

Z. Li et al. / Ocean Engineering 82 (2014) 65–7468

Two hundred component waves, ranging from 0.25 to 1.6 rad/s,were used to formulate the incident wave. The length of the timestep is an important parameter in time-domain hydrodynamicanalyses. Both the accuracy and numerical stability require asufficiently small time step, which is in turn determined by theship speed and panel size. For the 4400 TEU vessel, a time step of0.1 s was generally sufficient to provide approximately 100 panelsin the longitudinal direction. In the subsequent time-domain FEanalyses, the time interval was set to 0.5 s, which did not result inany great loss of accuracy. Detailed results and a discussion areprovided by Li et al. (2013).

WASIM can also perform calculations in the frequency domain.Using the linear solution, time-domain results are transformedinto the frequency domain using Fourier transforms. In this study,hydrodynamic analyses of the 4400 TEU vessel were performed inthe frequency domain in WASIM as well. The unit wave height wasassumed to be 0.1 m, and equal wave headings at 101 intervalsbetween 01 and 3601 were assumed. A selection of 25 wavefrequencies was used, with a minimum wave period of 4 s and amaximum period of 35 s.

3.3. FE model

The entire hull structure of the 4400 TEU vessel was modelled.The mesh size was at the level of the frame spacing. Structuraldetails, such as local stiffeners and brackets, were excluded fromthe FE model. Accordingly, the nominal stress determined from FEanalysis can be compared directly with full-scale measurements.The plates were represented by triangular and quadratic shellelements, whereas the girders, stringers, and main stiffeners were

represented by beam elements. Mass elements were alsoemployed to represent fuel tanks and ballast water tanks. Thecontainers were modelled with shell elements. The materialdensities of the shell elements were adjusted to represent theproper weight of containers. Young’s modulus of the shell ele-ments were also modified so that the containers were in fact softand contribute little to the structural rigidity of the hull eventhough they connected to the hull. Fig. 3 presents a section view ofthe FE model of the hull (containers removed).

4. Full-scale measurements

Full-scale measurements data from standard hull monitoringsystem HMON (DNV, 2005) are available for the two containervessels considered here. The HMON system comprises straingauges and other devices. Fig. 4 provides examples of the waveradar and strain sensors. There are four strain sensors installed atthe deck level along the hull of the 2800 TEU vessel and eightstrain sensors mounted at various locations in the midship sectionof the 4400 TEU ship. These strain sensors measure the ship'slongitudinal strain responses. Other components of the HMONsystem record the ship motions and environmental loads fromwave and wind. Operational profiles such as the heading angle andship speed, are also recorded; see Storhaug and Moe (2007),Storhaug et al. (2007) for details.

4.1. Wave environment and operational profile

To compare the direct calculation approaches, a case study wasconducted using the 4400 TEU container ship. The waves encoun-tered were measured by onboard directional wave radar. Experi-ence has shown that there is a high probability that the wave radaroverestimates the wave heights (Storhaug et al., 2007). In view ofthis tendency, instead of using the data from the wave radar, thesignificant wave height Hs and wave period Tp were obtained fromrelated hindcast data, which is considered to be more accurate.The hindcast data were retrieved from the ERA interim data,provided by European Centre for Medium-Range Weather Fore-casts (ECMWF). The hindcast data provided us with a 0.751�0.751grid resolution with four observations daily. A detailed discussionof the accuracy, reanalysis model, and input observations for theECMWF ERA interim data is provided in Dee et al. (2011).Fig. 3. Global FE model of the 4400 TEU vessel.

Fig. 4. Left: onboard wave radar that measures the encountered waves (from Storhaug and Moe (2007)); right: deck longitudinal strain sensor before coating at frame 179 onthe starboard side of the 2800 TEU vessel (from Storhaug et al. (2007)).

Z. Li et al. / Ocean Engineering 82 (2014) 65–74 69

Three actual sea states from a winter voyage were selected torepresent high-, medium, and low-wave environments. The mea-sured main wave heading angle from the onboard wave radar wasassumed to be unchanged within a 1 h period, as was the shipspeed, U. Table 2 presents the main characteristics of the seastates. Here, a 1801 heading corresponds to head sea, whereas a901 heading corresponds to beam sea from the starboard.

In all analyses, the specific forward speed was included, and theincident waves were modelled as irregular short-crested (cosinesquare) waves. The Pierson–Moskowitz (P–M) spectrum was usedto describe the sea state conditions. The mathematical formulationof the P–M wave spectrum can be expressed as follows:

SP�MðωÞ ¼ 516

H2Sω

4pω

�5exp �54

ωωp

� ��4 !

ð5Þ

where ωp¼2π/Tp is the angular spectral peak frequency and Hs isthe significant wave height. The P–M spectra of the three seastates are presented in Fig. 5.

4.2. Measurement of fatigue stress

The current investigation focused on the deck level in themidship cross section. One of the strain sensors, the deck long-itudinal (DL) on one side of the 4400 TEU vessel, is shownschematically in Fig. 6. Strain sensors were also located on theopposite side. Strain sensors were installed on the 2800 TEU vesselas well. In this paper, all measurements and calculations of fatiguedamage are carried out for these locations.

The onboard measurement records were composed of compo-nents of wave-frequency (WF) loads and high-frequency loads.However, in this study, the emphasis was on wave-frequencyloads, and thus, the high-frequency loads were excluded using alow-pass filter at 0.3 Hz, which is consistent with the hydrody-namic model. Fig. 7 provides an example of the total measuredstress in comparison to the WF stress.

Table 3 presents the coefficient of variation (CoV), skewness,and kurtosis of the WF stress histories on deck in the midshipsection under the three sea states considered. The WF stresseswere found to follow approximately normal distributions for the

midship section even when the waves were high, indicating thatthe linear algorithm in the hydrodynamic model yields reasonableresults amidships. The approximately normal distributions alsoimply that the NBA method is applicable.

5. Fatigue damage estimates of the 4400 TEU vessel accordingto various direct calculation methods

In the current study, short-term fatigue damages of the4400 TEU vessel were calculated using five direct calculationmethods (referred to as Approaches 1 to 5); see Table 4 for asummary:

� Approach 1 (Strip): Ship motions and sectional loads werecalculated using WAVESHIP. The fatigue stresses at the decklevel were represented by the VBM divided by the verticalsectional modulus. The fatigue damage was calculated via NBA.

� Approach 2 (Panel FD): The hydrodynamic loads and shipresponses were calculated using the frequency-domain WASIM.The stress ranges were derived from the VBM divided by thevertical section modulus, and fatigue damage was approxi-mated from NBA.

Table 2Main characteristics of the studied sea states.

Sea state Date and time Hs (m) Tp (s) U (m/s) Heading angle (1)

1 2008-12-17 08:00–09:00 6.1 13.7 8.5 1302 2008-12-17 18:00–19:00 4.7 12 9.7 1403 2008-12-16 21:00–22:00 3.5 10 10.2 160

Fig. 5. P–M wave spectra of the three sea states.

Fig. 6. Location of the DL strain sensor on one side of the 4400 TEU vessel.

Fig. 7. Example of measured total stress and WF stress.

Z. Li et al. / Ocean Engineering 82 (2014) 65–7470

� Approach 3 (Panel linear): The wave loads and ship motionswere calculated in time-domain WASIM. The stress recordswere obtained from the VBM history divided by the verticalsectional modulus. The rainflow analysis was carried out usingWAFO (WAFO-group, 2000).

� Approach 4 (Panel nl; nl¼nonlinear): This approach is the sameas Approach 3 except that nonlinearity is considered in thehydrodynamic analysis.

� Approach 5 (Panel FEA): The hydrodynamic simulations are thesame as in Approach 4. The hydrodynamic loads were trans-ferred to the FE model with a time interval of 0.5 s. The fatiguestress was computed by FE analysis, and the fatigue damagewas determined by the RFC method.

The three sea states listed in Table 2 were utilised for all of thedirect calculation approaches. Each sea state lasted for 20 min, andthe fatigue damage was calculated from strains measured bysensors at locations on the starboard and port sides, termed DLSand DLP, respectively. For Approaches 1–4 the fatigue damage onthe starboard side is identical to that on the portside because onlyVBM is considered. Table 5 lists the calculated fatigue damageamounts for all five approaches, together with the measuredfatigue damage amounts. For each sea state, the 1 h measuredrecord was divided into three consecutive 20 min blocks. Thefatigue damage for each individual block was calculated indepen-dently, and the mean was taken as the fatigue damage for this seastate. The uncertainty in the calculations is reflected by thestandard deviation of fatigue damage for the three blocks. It was

assumed that the 1 h sea states encountered were stationaryprocesses.

In Table 5, the mean values of the fatigue damage amountscalculated using the time-domain approaches (Approaches 3–5)were computed using a resampling approach. All of the time-domain analyses were based on the time record of a sea state thatlasted for 1 h. Within the 1 h time record, a period of 20 min wasrandomly extracted, and the fatigue damage was computed forthis period. In this study, a total of 100 realisations were carriedout for each sea state, and the average fatigue damage andstandard deviation were derived. In this manner, the randomnessin wave loads in irregular seas was largely considered withoutinvolving computationally intensive time-domain simulations. Thecalculated fatigue damage amounts were normalised with respectto the measured mean values and presented in Fig. 8. For the time-domain approaches, the standard deviations of fatigue damage areshown in Fig. 8 by a bar.

It is observed that fatigue damage predicted by the FEAapproach exhibits good agreement with the measurement resultfor all of the sea states. The FEA approach yielded overall accuratefatigue damage estimates on the windward side (starboard in thisstudy), whereas the fatigue damage on the leeward side (port) wasoverestimated to some extent. One possible explanation for thisresult may be that the hydrodynamic code overestimates waveloads on the leeward side. This topic should be investigatedfurther.

In general, all of the approaches making use beam theory,except for the strip theory-based code, predicted fatigue damagereasonably well. This implies fatigue damage can be calculatedusing beam theory with the VBM as the only sectional forceconsidered. However, this approach is limited to the deck regionof container vessels. Reasonable results are expected from thisapproach, indicating that a FE analysis is not always needed. Thisapproach simplifies the fatigue assessment procedure but intro-duces some uncertainty in the calculated fatigue damage byneglecting other sectional forces.

The results of the current investigation demonstrate that thetime-domain linear approach “Panel linear” distinguished itselffrom the nonlinear approach with respect to fatigue damage;however, the differences were smaller than expected. This resultindicates that nonlinear effects played a minor role in the fatigueassessment of the deck level for the case study vessel. Althoughthe maximum hogging/sagging ratio determined from a nonlinearanalysis may differ significantly from that determined from alinear analysis, the stress ranges calculated using these twoalgorithms are similar. Furthermore, for relatively small waveheights, the linear algorithm yielded results almost identical orslightly higher than those obtained with the nonlinear algorithm.However, as the wave height increases, the difference betweenthese two algorithms increases, and the nonlinear approach tendsto predict greater fatigue damage than the linear approach. This

Table 3CoV, skewness, and kurtosis of the WF stress histories amidships under individualsea states. (DLS: deck longitudinal starboard; DLP: deck longitudinal portside).

Sea state 1 Sea state 2 Sea state 3

DLS DLP DLS DLP DLS DLP

CoV 0.206 0.204 0.147 0.159 0.120 0.121Skewness �0.021 0.033 0.132 0.189 0.179 0.200Kurtosis 3.260 3.096 2.996 2.809 2.970 2.835

Table 4Major characteristics of the various approaches.

Approach Wave loads Stress calculation Fatigue calculation

1 Strip Beam theory NBA2 Panel FD Beam theory NBA3 Panel linear Beam theory RFC4 Panel nl Beam theory RFC5 Panel nl FEA RFC

Table 5Twenty-minute accumulated fatigue damage for the measured and calculated data. (DLS: deck longitudinal starboard; DLP: deck longitudinal portside; std: standarddeviation).

Accumulated fatigue damage (1�10�6) in 20 min

Sea state Measured Strip (Appr. 1) Panel FD (Appr. 2) Panel linear (Appr. 3) Panel nl (Appr. 4) Panel FEA (Appr. 5)

DLS DLP DLS/DLP DLS/DLP DLS/DLP DLS/DLP DLS DLP

Mean Std Mean Std

1 72.5 5.83 80.5 4.76 110.6 68.3 85.2 110 69.3 95.82 29.8 4.04 43.0 4.43 63.8 50.4 29.1 33.0 34.8 54.33 15.9 1.62 18.7 1.47 16.9 19.7 19.4 19.1 18.2 26.3

Z. Li et al. / Ocean Engineering 82 (2014) 65–74 71

increasingly large difference can be explained by the asymmetrybetween sagging and hogging moments amidships, whichbecomes more prominent at greater wave heights. This asymme-try can only be captured by the nonlinear approach.

Slight differences were observed in the results from the “PanelFD” and “Panel linear” approaches. These slight differences indi-cate that for the wave-frequency load applied on the investigatedship deck structures, the NBA predicts fatigue damage reasonablyaccurately in comparison to the rainflow counting method.The results also indicate that the strip method yields reasonablefatigue damage estimates for relatively small waves. However, asthe wave height increases, the strip theory code tends becomeincreasingly inaccurate for fatigue damage prediction compared toother approaches.

Finally, the computational effort required by the above-mentioned approaches for fatigue assessment was compared. Ananalysis using the spectral method takes less than 1 min, whereas apanel method-based time-domain hydrodynamic simulation takesseveral hours on a standard PC for a 20 min sea state, depending onthe number of component waves, the panel resolution, and the sizeof the time step. Experience indicates that a nonlinear analysis takesapproximately one third longer than the corresponding linear one.The FE analysis approach requires the most computation resources:an FE analysis of an entire ship hull may take at least several hours tocomplete. However, given that fatigue failure can occur at manylocations along a ship hull, it is necessary to carry out a fatiguescreening of an entire ship structure rather than analysing only thedeck position in the midship section. In view of this result, the FEanalysis approach is recommended, although considerable computa-tional effort is required.

6. Various measures of wave environments

Ships are designed to be operated in specific trade regionscorresponding to specific wave climates. The variation in waveenvironments considered in ship design, i.e., the distribution of seastates encountered, is often described by a wave scatter diagram.In classification society rules, the scatter diagram can provide theoccurrence frequency of a pair parameter (Hs, Tz), where Tz denotesthe mean zero up-crossing wave period. This occurrence frequencymay be one of the most convenient sources (and most often usedin engineering practice) used in ship fatigue design. One of thelargest uncertainties in ship fatigue design is because the distribu-tion of Hs used in the classification rules may differ greatly fromthe real values.

Due to the importance of the encountered wave environmentsin ship design, various instruments in addition to onboard wave

radars, such as satellites and buoys, have been used to obtainreliable wave measurements. In this study, three sources of thedistribution of Hs were used to quantify the difference betweenthe encountered and assumed wave environments. The area of theNorth Atlantic was considered. The first source was the measure-ments taken on the 2800 TEU container ship over a six-monthperiod along its route between Europe and North America. Thesecond source was a spatio-temporal wave model for Hs built upbased on satellite measurements over an area of the North Atlanticthat extends between 421 and 621 in latitude and �481 to �121 inlongitude (Baxevani et al., 2009). The third source is commonlyaccepted as the North Atlantic climate among classification socie-ties, which is specified in classification guidelines for exampleprovided by DNV (2010b). A comparison is illustrated in Fig. 9.

As shown in Fig. 9, the observed data include more moderatesea states (Hs from 1 to 5 m) than the satellite model and DNV rule,whereas the probabilities of large Hs are almost identical for theDNV rule and satellite model but greater than in the observeddata. The results also agree well with those reported by Olsen et al.(2006). If fatigue damage caused by the observed wave environ-ments is assumed to be a unit, the damage computed using Hs

from the DNV rule is 1.5, compared to the damage of 1.6 from thesatellite model. These results indicate that at least 50% of theuncertainty in fatigue life prediction is attributable to differencesin the assumed distributions of Hs.

However, the difference between the predicted fatigue andthat calculated from onboard measurements could be due to

Fig. 8. Fatigue damage accumulated over 20 min at (left) DLS and (right) DLP under various sea states; values are normalised with respect to the measured fatigue damage.

Fig. 9. Probability density functions for Hs from different sources. Scaled fatiguedamage, D, with respect to observed Hs (D¼1.0); using DNV Hs (D¼1.5), usingsatellite Hs (D¼1.6).

Z. Li et al. / Ocean Engineering 82 (2014) 65–7472

the routing plan system installed in the monitored ship. With thehelp of such a system, a ship can easily avoid some large stormareas. Furthermore, the difference could also be due simply tostatistical error, given that the first estimate is based on only6 months of measurements, whereas the second and thirdestimates are based on data collected over many years.

7. Influences of the wave spectrum and wave spreading

In practice, idealised wave spectra are used instead of the wavespectra derived for particular times and places. Examples of theseidealised wave spectra are the P–M spectrum, which describesfully developed seas, and the JONSWAP spectrum, which describesdeveloping seas. The JONSWAP spectrum is formulated as amodification of the P–M spectrum shown in Eq. (5) for a devel-oping sea state in a fetch-limited situation:

SJONSWAP ¼ SP�MðωÞγexp � 1=2ð Þ ω�ωp=sωpð Þ2� �

ð6Þwhere SP–M(ω) is the P–M spectrum, ωp is the peak angularfrequency, γ is the non-dimensional shape parameter, and s isthe spectral width parameter, the value of which depends on thewave frequency ω compared to the peak frequency ωp.

s¼sa ¼ 0:07; ωoωp

sb ¼ 0:09; ω4ωp

(ð7Þ

A fatigue model developed by Mao et al. (2010) has been foundto be convenient for analysing fatigue damage accumulationduring a ship's voyage. There is a proportional relationshipbetween the standard deviation of the response and the significantwave height encountered. For a stationary sea state, a factor C thatdepends on the wave period Tz, ship speed U, and heading angle θ,as well as the ship's structural details, can be used to describethis relation. In fatigue analysis, hydrodynamic loadings are oftencomputed using linear theory. The waves encountered are mod-elled by a power spectral density function S(ω|Hs,Tz). In this case,the factor C will not depend on the value of Hs (assumed to beunity) and can be computed as follows:

CðU;θ; TzÞ ¼ 4KMu

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 1

0H2sðωjU;θÞSðωjHs ¼ 1; TzÞdω

sð8Þ

where Hs(ω|U,θ) is the bending moment transfer functionobtained from a linear hydrodynamic analysis, Mu is the sectionmodulusof the cross section at the fatigue location of interest (e.g., inthe deck), and K is the value of the stress concentration factor.The fatigue damage for a sea state of length T can be computed asshown in Eq. (9). More details could refer to Mao et al. (2012,2010).

D� 0:47α

Tf zðU;θÞ½CðU;θÞHs�3 ð9Þ

In this study, a sea state with Hs¼5 m and Tz¼10 s was chosento investigate the difference in fatigue damage for different wavedescriptions. The 2800 TEU container ship was used as thenumerical model. For a given sea state, the ship was assumed tobe operated at 10 and 5 m/s, and the corresponding headingangles were 1801 (head sea) and 801, respectively. WAVESHIPwas employed for computation of the transfer functions. Fourdifferent wave models were used to generate the wave elevation:the P–M spectrum, with and without energy spreading, and theJONSWAP spectrum, with and without energy spreading, with apeak enhancement factor γ¼5. The sea condition was assumed tolast for 30 min. The fatigue damage amounts computed usingEq. (9) are listed in Table 6. Because the ocean surface wasgenerated from different wave models, the differences in thefatigue results can differ by as much as a factor of 10 for closebeam sea operations. Head sea operation is known to be moreimportant in ship fatigue design because more fatigue damage canbe accumulated in this situation. In this case, it has also beenshown that different approaches to wave modelling can result indifferences of more than a factor of two in fatigue damageestimates.

8. Conclusions

Various direct calculation approaches for ship structural fatigueassessment were investigated using two container vessels as casestudies. A real wave environment and real operating conditionswere employed as inputs to the direct calculations, and the fatiguedamage amounts obtained from full-scale measurements wereused to verify the direct calculation results. The following conclu-sions can be drawn based on the results for the case study of the4400 TEU vessel:

� In general, most of the investigated direct calculationapproaches predict rather similar fatigue damage results. TheFE analysis approach yields reasonable and conservative fatiguedamage results and is thus recommended.

� Alternatively, fatigue damage can be calculated using beamtheory with the VBM as the only sectional force considered.However, this approach is limited to the deck region ofcontainer vessels. Reasonable results are expected from thisapproach, indicating that a FE analysis is not always needed.This approach simplifies the fatigue assessment procedure butintroduces some uncertainty in the calculated fatigue damage.

� In general, nonlinear effects played a minor role in the fatigueassessment of the deck structures of the case study vessel. Forrelatively small waves, the linear algorithm yielded fatiguedamage estimates that are almost identical to or slightly higherthan those obtained with the nonlinear algorithm, but asthe wave height increases, the difference between thesetwo algorithms increases, and the nonlinear approach tendsto predict greater fatigue damage than the linear approach.

� For wave-frequency loads applied to the investigated ship deckstructures, the NBA was found to predict reasonably accuratefatigue damage compared to the rainflow counting method.

� The strip method was found to yield reasonable fatigue damageestimates for relatively small waves. However, as the waveheight increases, the strip theory code tends to produceincreasingly inaccurate fatigue damage estimates compared toother approaches.

Different measures of the wave environment can lead tosignificant differences in the predicted fatigue damage. The resultsof the fatigue analysis conducted using the 2800 TEU vessel as acase study demonstrate that the wave spectrum description used

Table 6Thirty-minute accumulated fatigue damage in the 2800TEU container ship com-puted by linear strip theory and Eq. (9), with various wave models and a sea statewith Hs¼5 m, Tz¼10 s (Unit: 10�6).

Wave spectrum models U¼10 m/s U¼5 m/sHeading angle (1) Heading angle (1)

180 80 180 80

P–M no spreading 47.1 0.4 25.2 0.4P–M cos2 spreading 30.8 5.6 19.9 4.6JONSWAP γ¼5; no spreading 81.2 0.5 40.7 0.4JONSWAP γ¼5; cos2 spreading 49.0 7.8 30.9 6.4

Z. Li et al. / Ocean Engineering 82 (2014) 65–74 73

and the assumption made concerning wave spreading are impor-tant sources of uncertainty in ship fatigue life prediction.

Acknowledgements

The authors acknowledge the support of VINNOVA (the Swed-ish Governmental Agency for Innovation Systems) and the Light-house Maritime Competence Centre (www.lighthouse.nu).

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