Structural loads induced in a containership by abnormal wave conditions

J M a r Sci Technol (2006) 11:245-259 D O I 10.1007/S00773-006-0222-9

O R I G I N A L A R T I C L E

Structural loads induced in a containership by abnormal wave conditions

Received: December 13, 2005 / Accepted: M a y 16, 2006 © J A S N A O E 2006

Abstract A n analysis is presented of the vertical bending

moments induced in a containership by a set of abnor-mal waves measured at different locations and on differ-ent occasions. A systematic investigation was carried out by using a large set of wave traces that included abnor-mal waves. I n this way i t was possible to assess the influence of the height, length, and shape of the abnor-mal waves on the wave-induced structural loads. The probabihty distributions of the ship responses to the sea states that included the abnormal waves were also calcu-lated and were compared to the responses induced by the abnormal waves and to fitted distributions. Finally, the structural loads induced by the abnormal waves were compared with rule values and with long-term predictions.

Key words Abnormal waves • Global structural loads •

Time domain numerical model • Design loads

1 Introduction

The current definition o f wave-induced structural loads for the design of ship structures is i n many cases still based on empirical formulas f r o m classification societies and are sometimes complemented by direct calculations. However, as computers become faster and cheaper, there is a tendency to apply procedures based on direct calcu-lations to define the design wave loads. These procedures

N . Fonseca • C. Guedes Soares ( E l ) • R . Pascoal U n i t o f M a r i n e Engineering a n d T e c h n o l o g y , Technical U n i v e r s i t y o f L i s b o n , I n s t i t u t o Superior T é c n i c o , A v . Rovisco Pais, 1049-001 L i s b o n , P o r t u g a l

e-mail: guedess@mar.ist.utl.pt

rely on hydrodynamic models derived f r o m first prin-ciples, together with a proper stochastic characterization of the waves and of the ship responses. There are several advantages to using these more advanced methods: ulti-mately, the design wave loads will be more accurate and will be tailored f o r the specific ship characteristics; novel ship concepts can be assessed (empirical formulas are in principle vahd f o r existing ships only); and, besides glo-bal structural loads, i t is possible to obtain consistent hydrodynamic load distributions f o r finite element calculations.

For the linear case, the maximum wave-induced struc-tural loads during a long period of time can be efficiently calculated by applying linear potential flow hydrody-namic models i n the frequency domain, together with spectral analysis and a weighted summation of short-term Rayleigh distributions o f t h e maxima.' However, at least f o r ships w i t h small block coefficients, the wave-induced structural loads are highly nonlinear. I n these cases the linear procedure cannot be apphed and the design wave loads must be determined by nonlinear time domain numerical models together with appropriate ex-treme value distributions. Several approaches have been proposed and currently i t is not clear which is the best. A review of such methods has been presented by Guedes Soares et al.^

Although the methodologies referred to i n the previ-ous paragraph take into account the stochastic nature of the waves encountered by a ship during its life, condi-tions associated with encounters w i t h abnormal waves are not taken into account. This is because the prob-abilistic models describing the waves do not seem to consider abnormal waves. However, there are some accidents reports that describe encounters with waves much larger than those of the sea state in which they

246 J M a r Sci T e c l i n o l (2006) 11:245-259

occurred. I t is also believed by some authors that such abnormal waves may be responsible for the mysterious vanishing of some ships. For this reason, Faulkner and Buckley' suggested that the methods for determining the design loads should be revised to account for the effects of such abnormal waves on the ship structure.

Fonseca et al.'' proposed a method to calculate the structural loads induced by deterministic wave traces of abnormal waves, in which the ship responses are calculated by a numerical nonlinear time domain sea-keeping model.'''' The methodology was applied to a containership encountering a wave sequence that in-cludes an abnormal wave with a height of 26 m. This wave trace was measured in the N o r t h Sea during a severe storm. I n this case i t was found that the maxi-mum sagging moment i n the abnormal wave was be-tween the linear and the nonlinear long-term prediction for a ship operating in the N o r t h Sea over a period of 20 years.

The same methodology was applied by Clauss et al.' and Guedes Soares et al.** to investigate the structural wave loads induced by deterministic rogue waves on a Floating Production Storage and Offloading Platform (FPSO). Comparisons between experiments and simula-tions showed that the numerical model is able to repre-sent remarkably well the wave-induced midship bending moment in highly nonlinear waves, including the asym-metry o f t h e sagging and hogging peaks.

I n this article, the same procedure is applied to obtain the wave-induced structural loads on the above mentioned containership, but a systematic investigation is carried out by using a large set o f wave traces. These wave traces were measured on different occasions in different places and they include abnormal waves. I n this way i t is possible to assess the influence of the abnormal wave height, wavelength, and shape on the wave-induced structural loads. I t is also possible to produce some statistics regarding ship responses and structural loads induced by abnormal waves.

The time traces used herein originate f r o m the G u l f of Mexico during the passage o f hurricane Camille on A u -gust 17, 1969; f r o m the Draupner oil rig in the central N o r t h Sea when i t was struck by a storm f r o m December 31, 1994, to January 1, 1995 (a trace containing the wave that has become known as the "New Year Wave"); and f r o m N o r t h A l w y n in the northern N o r t h Sea f r o m a storm extending f r o m November 16 to 22, 1997. A l t o -gether, 20 wave traces were considered. These wave data have been analyzed by Guedes Soares et al.'''" to charac-terize the properties of the abnormal waves.

Abnormal, freak, or rogue waves can be generated by different mechanisms and there is not a clear criterion to identify such a wave. However, a general approach has

been to consider a wave to be abnormal when i t is larger than the usually adopted wave theories would predict. Therefore, the criterion would depend on the duration o f the record and whether a linear or second-order theory was used as the reference condition. While this criterion is stifl open to discussion, several authors have opted to choose as reference an amplification or abnormality i n -dex defined by the ratio between the maximum wave height and the significant wave height of the record, whether they are calculated using the down-crossing or up-crossing definition. When this amplification ratio is larger than 2, the wave is considered to be abnormal and this was the definition adopted by Guedes Soares et al.'''" Fonseca et a l . " presented an exploratory study of the adoption o f this approach with initial results of a few wave traces. The present article includes additional wave traces of abnormal waves, and this inclusion aflows a statistical analysis, a more in depth analysis of the condi-tions in which the maximum structural loads occur, an analysis o f abnormal wave encounters i n the water on deck effects, and comparisons of the maximum re-sponses due to the abnormal waves with probability dis-tributions o f the same responses.

2 Calculation method

2.1 Representation of the wave field

The calculation o f the structural loads induced by ab-normal waves using wave traces is based on a numerical time domain seakeeping model; however, the first step is the calculation of the exciting forces induced by the wave field that includes the wave trace. To do this, it is neces-sary to derive a representation in the time and space domains o f the incident wave field that is consistent w i t h the time history o f the wave elevation defined at a par-ticular point in space.

The time record is transformed f r o m the time domain to the frequency domain by a fast Fourier transform (FFT) algorithm. The frequency domain representation of the time signal is then used to simulate the original time history o f t h e wave elevation. Deep-water waves i n unidirectional seas and zero current are assumed. Fur-thermore, i t is assumed that the kinematics of the waves may be represented by the superposition of linear and harmonic wave components. W i t h the v/ave trace de-composed into harmonics and assuming linear superpo-sition, i t is possible to calculate the wave exciting forces on the ship. Details of the procedure may be f o u n d i n Fonseca et al.''

The apphcation of the linearity assumption and su-perposition principle to represent very large and steep

J M a r Sci Teclinol (2006) 11:245-259 ₂₄₇

waves may seem inadequate, but Slunyaev et al.'^ liave used vanous nonlinear wave theories to study the mechanisms of propagation of freak waves and con-cluded that linear theory was able to describe properly various properties of the wave field. Furthermore, one should keep in mind that the focus here is on the global responses of the ship and not on local responses. I t is known that nonlinear wave effects are important i f one needs to represent local flow effects, e.g., relative mo-tions at the bow, but the global responses are less sensi-tive to higher order effects of the wave elevation. The piocedure presented here was applied to the investiga-tion of vertical moinvestiga-tions and bending moments on a FPSO subjected to wave traces o f abnormal waves and the comparisons with experimental data showed remark-ably good agreement.*

The quality of the simulated wave elevation compared to the original time series depends on the time interval that is Fourier analyzed and the number of harmonics used i n the simulation. On the other hand, the computa-tional effort of the numerical seakeeping model increases with the number of harmonic components. For the present problem, the interests hes in the assessment of the ship responses to an abnormal wave that is inserted in the wave signal. The simulation of the ship responses must start before the large wave crest is encountered such that the transient effects induced by the wave field are correctly represented. The hydrodynamic transient effects are usually felt for a period of less than 1 min, thus i f one considers a period o f 2 min before the abnormal wave and l m i n after, it is assumed that all transient effects are taken into account. The number of harmonic components needed to represent correctly 3 min of the wave record is compatible with the numerical seakeeping model.

2.2 Numerical seakeeping model

Regarding the numerical time domain seakeeping model,'"'' the method assumes that the nonhnear contri-_ bution to the vertical bending moment is dominated by

hydrostatic and Froude-Krylov forces; these compo-nents thus depend on the instantaneous hull wetted sur-face. The exciting forces due to the incident waves are decomposed into the diffraction part and the Froude-K r y l o v part. The diffraction part, which is related to the scattering of the incident wave field due to the presence of the moving ship, is kept linear. Since this is a linear problem and the exciting waves are known a priori, i t can be solved in the frequency domain and the resulting transfer functions can be used to generate a time history of the diffraction heave force and pitch moment. The Froude-Krylov part is related to the incident wave

po-tential, and is evaluated by integration at each time step o f t h e associated pressure over the wetted surface o f t h e hull under the undisturbed wave profile.

The radiation forces are represented in the time domain by infinite-frequency added masses, radiation restoring coefficients, and convolution integrals of memory functions. The convolution integrals represent the effects o f the whole past history of the motion ac-counting for the memory effects due to the radiated waves. Both the radiation and diffraction coefficients in the frequency domain are calculated by a strip method.

The vertical forces associated with green water on the deck, a situation that occurs when the relative motion is larger than the free board, are calculated using the mo-mentum method." The mass of water on the deck is proportional to the height of water on the deck, which is given by the difference between the relative motion and the free board of the ship.

According to the classificarion of Committee V I . 1 of the International Ship and Offshore Structures Commit-tee,''' this numerical model is based on a "partially non-linear method." This means that the equations of motions and loads combine linear and nonlinear terms. The Committee has reviewed the methods available to calculate nonlinear ship motions and loads i n large am-plitude waves and concluded that, for practical applica-tions, the methods that are more appropriate are those based on approaches similar to the one described above. The fonnulation of the method for a ship advancing in regular waves was presented by Fonseca and Guedes Soares' and generalized for irregular seas by Fonseca and Guedes Scares.'^ I n Fonseca and Guedes Soares," a semiempirical model was added to the previous formula-tion to account for the viscous forces associated with large-amplitude vertical ship motions. Extensive com-parisons with experimental data have been presented by Fonseca and Guedes Soares."^"* The limits o f applica-tion o f the method were tested by comparisons with experimental data for two fast monohulls."'^"

F r o m these comparisons it was concluded that the time domain model is an improvement compared to the equivalent linear model. I t was able to quahtatively rep-resent all nonlinear characteristics detected i n the experi-mental data; however, several discrepancies show that some improvements can be introduced, especially re-garding forward speed effects.

3 Characteristics of the wave traces

The procedure presented in the previous section is used to calculate the responses of a containership to several wave traces that include abnormal waves. These wave

248 J M a r Sci Teclinol (2006) 11:245-259

traces were measured in different places and on different occasions, as described i n the following paragraphs. I n this work, following the general consensus, an abnormal wave is taken to exist whenever the ratio between its height, be it calculated by the up-crossing or the down-crossing definition, and the significant wave height of the wave record in which it is embedded (usually having a duration of about 20 min, during which stationarity is a good approximation) is larger than 2. Due to the design-oriented nature of the calculations, smah abnormal waves have been disregarded; only waves with heights larger than 10 m are considered.

During the passage o f hurricane Camille across the G u l f of Mexico on August 17, 1969, wave data were registered by a wave measurement system installed on a platform fixed in waters 100m deep. Original data, sensed by an inductive wave staff was recorded continu-ously on magnetic tape until the measurement system was damaged.^' The digitized time sequence used herein contained approximately 12 h of surface elevation at a 0.5-s sampling period. Earle identified the degree of nonstationarity o f the data. The significant wave height of the signal changes at approximately 0.8m/h. These data have been further analyzed by Guedes Soares et a\.'° and, therein, abnormal waves were identified. Some of the segments determined to contain abnormal waves with high crests have been used here, namely segments identified as 37 and 41, which correspond to the end of the elevation time trace, just before the wave staff broke. F r o m the Draupner jacket platform, a time trace that contained a very high wave crest was first reported by Haver and Karunakaran." The platform is positioned in the central N o r t h Sea and the water depth at the mea-surement site is 70 m. The sensor was laser based and the samples were 0.468 s apart. The high crested wave has become known as the New Year Wave because it oc-curred during a storm that lasted f r o m December 31, 1994, to January 1,1995. I t was registered in a time series lasting 20 min that started at 1520 hours of the second day.

N o r t h Alwyn has also provided some time series con-taining abnormal waves with high crests. The data were cohected by laser-based sensor equipment installed on a fixed jacket platform positioned in the northern N o r t h Sea. The data used were f r o m a storm that lasted f r o m November 16 to 22, 1997. From this stonn there exist 421 files, each corresponding to 20 min of data sampled at 5 Hz and with 2-niin gaps between the files, thus pro-viding an almost continuous throughput. F r o m this set of files, those identified as containing abnormal waves' have been used.

Table 1 presents the file numbers attributed to each wave trace, grouped according to the three different

Table 1. File numbers associated w i t h d i f f e r e n t data sets D a t a File n u m b e r

N o r t h 0 1 , 02, 03, 05, 06, 07, 08, 09, 1 1, 12 A l w y n 14, 15, 16, 17, 19, 20, 24

D r a u p n e r 04

Camille 3 7 , 4 1

Table 2. Characteristics o f the a b n o r m a l waves

i ^ . ( m ) 7 ; ( s ) AI, AI, O I N A 8.11 13.00 2.29 2.31 0 2 N A 9.01 11.96 2.04 2.03 0 3 N A 9.36 12.23 2.02 2.23 0 4 D R 11.92 13.24 2.23 2.19 0 5 N A 7.17 12.60 2.26 2.38 0 6 N A 8.16 12.90 2.08 2.12 0 7 N A 5.51 12.23 2.13 2.18 0 8 N A 6.50 13.11 2.21 2.20 0 9 N A 8.75 12.70 2.03 2.14 U N A 5.92 9.99 2.13 1.88 1 2 N A 7.85 12.70 2.12 1.94 1 4 N A 6.47 12.14 1.78 2.07 1 5 N A 7.84 13.43 2.27 1.90 1 6 N A 9.63 12.32 1.93 2.02 1 7 N A 7.41 12.90 2.19 1.87 1 9 N A 10.39 13.88 1.75 1.88 2 0 N A 8.29 13.11 1.98 2.18 2 4 N A 7.81 12.60 1.96 2.13 3 7 C A 10.65 13.47 2.32 2.22 41 C A 10.69 .13.52 2.10 1.79

H„ significant wave height; T^, peak wave period; AI„ and AIj

represent the a b n o r m a l i t y indexes calculated using zero u p -crossing and zero down--crossing; NA, N o r t h A l w y n ; DR, D r a u p n a r ; CA, hurricane Camille

Abnormal Waves

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F i g . 1. T i m e traces o f the a b n o r m a l wave events

sources. Altogether, 20 wave traces that include abnor-mal waves were analyzed and used f o r time domain simulations of the ship response. Table 2 presents the abnormality indexes of the waves, the significant wave heights, and the peak wave periods of the wave records that include the abnormal waves. The abnorinality i n -dexes were defined as the ratio between the abnormal wave height and the significant wave height f o r the record.

J M a r Sci Teclinol (2006) 11:245-259 249

Fig. 2. Wave properties o f data f r o m N o r t h A l w y n , D r a u p n e r , and hurricane Camille W a v e period W a v e height 12 10 8 6 4 350 300 250 200 1S0 100 EO 0102 030405 0607 080911 121415 1617 192024 3741 file number Estimated w a v e length 035 0.3 0.25 0.2 a. o " 0.15 0.1 005 0 0102 0304 0506 070809 11121415161719 2024 3741 file number Front slope 01 02 030405 0607 080911 1214151617 192024 3741 file number 0102 0304 0506 070809 11121415161719 2024 3741 file number

Figure 1 siiovvs tlie time histories of the wave traces around the abnormal wave events. This group of wave traces looks synnnetric around the large crests; however, the symmetry is not always found in each wave trace isolated, as can be seen in the graphs of section 4, which confirm the earlier findings of Guedes Soares and Pascoal.'"

Some parameters of the abnormal waves were calcu-lated to compare the characteristics of the different wave events, and the results are plotted in Fig. 2. The param-eters are: the wave period, wave height, estimated wave-length, and front slope. These values have been calculated as follows. The wave height was calculated using the up-crossing definition, i.e., the vertical differ-ence between the value of the trough immediately after the maximum crest and the maximum crest (in the time domain). The wave period was determined as twice the time i t took for the events used to estimate the height. The wavelength was estimated using linear deep water waves; the record F F T along with linear dispersion was used to calculate the wave profile f o r the time at which the abnormal crest occurred and the wavelength, was taken to be the horizontal distance between the preced-ing and subsequent trough with respect to the abnormal crest (provision was made to exclude small zero level crossings existing i n the calculated profile). The wave slope was calculated as the ratio between wave height and the horizontal distance between the trough and sub-sequent abnormal crest.

The graphs in Fig. 2 show some parameters f r o m a large number o f abnormal waves and it is clear that this group includes waves with a wide variety o f characteris-tics. For instance, the wavelengths vary f r o m around 140 to 300 m and the f r o n t slope of the abnormal waves vary f r o m around 0.1 to 0.27. For this reason it is expected that the ship responses to these wave conditions will give a good general picture of the global structural loads induced by abnormal wave conditions.

4 Vertical bending moments induced by the abnormal waves

This section presents the results f r o m the time domain simulation of the ship responses to the group o f wave traces presented in Sect. 3. The calculations are f o r the Iinernational Towing Tank Conference (ITTC) S-175 containership, which has the foUowing main particulars: length betv/een perpendiculars, 175m; beam, 25.4m; draft, 9.5 m; displacement, 24742 tonnes; and service speed, 22 knots. The ship advances i n long-crested head waves with a reduced speed of 13 knots, which is a little less than 60% of the service speed. Reports o f long-crested extreme waves have been compiled and classified as type I by B u c k l e y . H e r e a discussion will not be provided as to whether abnormal waves are part of the extreme wave population, but simply state that these numerical simulations aim at characterizing responses to

250 _{J M a r Sci T e c l i n o l (2006) 11:245-259}

these type I waves. Furthermore, according to Sellars and Setterstrom^'' (as referenced by Mandei"), 13 knots is a reahstic speed in a sea state with around 8-9 m o f significant wave height f o r a containership similar to the one in this study.

I n all cases, the ship is forced to pass through the location in space where the v/ave record was measured exactly at the time when the abnormal wave crest is generated (measured). The reference point in the ship is the centre of gravity. The simulated abnormal wave crest elevation is equal to the measured value, whereas the wave elevation around the abnormal wave crest is simi-lar to the actual wave record. I n fact, most of the simu-lated wave elevation time record that is felt by the ship is different f r o m that measured at a fixed point i n space. This is because the ship is moving with a constant for-ward speed. However the sea state that the ship encoun-ters has the same statistical characteristics as those of the measured time record.

The wave exciting forces are given by deterministic wave traces. To calculate the excitation forces, a summa-tion of harmonic wave components represent the wave traces. FFTs are used to transform the wave signal to the frequency domain, and i n all cases 1024 points were used. The major part of the wave signals were collected at a sampling rate of 5 Hz (North A l w y n records), thus 1024 points corresponds to 3 min of record and some seconds of zeroed wave elevation. The hurricane Camille wave records were collected at a sampling rate of 2 Hz and the Draupner records at roughly 2.13 Hz, and thus 1024 points correspond to a longer period of time f o r these locations. Between 80 and 90 harmonic compo-nents were used to reconstruct the original wave traces.

Figure 3 presents the type of simulation responses that were obtained by running the numerical time do-main model with the wave traces as input. This wave trace was measured during hurricane Camille and is des-ignated trace 41. The first graph presents the simulated wave elevation represented i n the reference system ad-vancing with the ship's speed. This is the wave elevation at the longitudinal position of the centre of gravity (2.5m aft of midship). A very large wave occurs at a time of 190s. The crest has an amphtude of around 10.5m and the next trough is similar with opposite sign, which results in a wave height of around 21.Om.

The second graph presents the relative motion at the bow (at the forward perpendicular), and the horizontal auxiliary lines represent the height of the deck and the bottom at the same position. The third graph shows the vertical downward force per unit length due to green water on the deck at the forward perpendicular. The last graph presents the midship vertical bending moment (hogging positive).

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F i g . 3. S i m u l a t i o n o f the ship response i n head waves f o r wave trace 41 ( f r o m hurricane Camille)./ii'c/p/j, f o r w a r d perpendicular;

VBM, V e r t i c a l Bending M o m e n t

This set of results shows that the ship's forward deck becomes submerged not when the bow encounters the large wave crest, but when the bow encounters the next crest, which is much smaller. This means that the ship is able to climb the large crest, but then the bow dives into the fohowing crest. Associated with this second event there is an estimated height o f water on the deck of around 10 m. This height of water on the deck produces a vertical downward force at the position of the forward perpendicular with a maximum value o f approximately 1800 k N / m . The midship vertical bending moment is also a maximum when the ship bow dives after the large crest. It is interesting to note that the forces resulting f r o m green water on the deck produce a hogging contribution to the midship vertical bending moment that limits the maximum value of the sagging moment.

Figure 4 presents graphs with simulations o f the wave elevation (dashed lines) and the midship vertical bending moment (continuous hne) around the abnormal waves. The figure gives a general overview of how the vertical bending moment behaves with respect to this type o f wave. I t is possible to conclude that the maximum hog-ging moment always occurs when the large wave crest passes through the midship. The magnitude of the maxi-mum sagging moment is around twice that f o r the hog-ging moment. I n most cases, the maxiinum saghog-ging moment occurs not when the ship bow encounters the large crest but when the bow dives into the following

J M a r Sci T e c i m o l (2006) 11:245-259 251

F i g . 4. Simulations o f the wave elevation a r o u n d the a b n o r m a l waves and the corresponding m i d s h i p vertical bending m o m e n t

File OINA File 02NA File 03NA 2 1.5 1 „ 0-5

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i

File 06NA File 07NA File 08NA

110 118 126 134 142 t[sl 120 130 140 150 t[sl 118 126 134 142 t[s] 120 130 140 150 160 t[sl 18 13.5 9 4.5 0 -4.5 -9 -13.5 -18 2 1.5 1

I

I

xio" File09NA File U N A File 12NA Flie 14NA

110 118 126 134 142 t[sl 128 136 144 152 t[s] 120 130 140 150 t[sl 126 132 138 144 150 t[sl 18 13.5 9 4.5 0 -4.5 -9 -13.5 18

File 15NA File 16NA File 17NA File 19NA

110 120 130 140 150 l[sj 142 154 166 178 t[s] 128 136 144 152 t[sl 118 126 134 142 150 t[s]

File 20NA File 24NA 2 1.5 „ 1

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File 37ca File 41ca

— —. V B M 4 -( - 1 1 1 1 1 1 1 1 i 1 1 1

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sate ffc

wave crest. The only cases where this does not happen occur because the wavelength after the large crest is long and the height of the foUowing wave relatively smaU, so the ship is able to foUow the wave instead of the bow diving into the free surface. The simulations seem to indicate also that there is a strong correlation between the abnormal wave height and the maximum moments (which is not surprising). This is confirmed i n the graphs given in the following figures.

Figures 5 to 7 present correlations between the maxi-mum hogging and sagging moments and, respectively,

252 J M a r Sci T e c h n o l (2006) 11:245-259

VBM vs Wave Height (Upcrossing)

• Hogging 1 1 1 1

A Sagging 1 1 1 A 37 ' 245l17,52Ë6 + 20öö2.625a'C , R' = 0.69

— £5950e-73£6 + 54g35.0a5G< , R' = 0.76 .

10 12

V B M v s wavelength/Lpp (Dow ncrossing)

Hccgirg Sagging . 305175.9393 + 2259l820a7x',R'=0.4a 69739Z9707 + 395951.1639:('. 022 - J * 1 2 _ >1^07

F i g . 5. C o r r e l a t i o n between the m a x i m u m sagging and hogging

m o m e n t s and the height o f the a b n o r m a l waves (77„) F i g . 7. C o r r e l a t i o n between the m a x i m u m sagging and h o g g i n g moments and the a b n o r m a l wavelength (A). L p p , length between perpendiculars V B M vs Slope (Downcrossing) 1 1 1 » u - ^ i . 3 7 1 ^ 4 1 , , • Hogging A Sagging 611070.629Ö-1713l2.125a(',R=^0.0073 103&ie0.2D93 + 8S5477.0952(',R'=0.029 -1 -1 -1 1 ! 1 _ 1 A 2 0 A 0 3 1 A 24 - _ _ ^ ' v i - - ' US ' ' ' A 24 _ _ ^ ' v i -' A 1 6 1 1 1 A 07 • 15 " • 0 4 ! ! • 09 1 0 5 ' ' 1 1 1 2 1 m 14 ' • O ? 2 0.14 0.16 0.

VBM vs wave heiglit (Upcrossing)

02 022 024 0.26 0.2B 0 3 A 14A 07 • B B 0.5 h - • - , . « „ , I ^° ^ 'f ^ ' • •„•, • B-• Hogging A Sagging - Q " Rules Hogging W Rules Sagging DnLTDp-ID"'* — | - n!inLTDp= 10"^ -pr--B B

-F i g . 6. C o r r e l a t i o n between the m a x i m u m sagging and hogging

moments and the a b n o r m a l wave steepness F i g . 8. C o m p a r i s o n o f the m a x i m u m sagging and h o g g i n g m o -ments and the r u l e values. LTD, l o n g - t e r m d i s t r i b u t i o n ; Im, linear; /;//';;, nonlinear

moment, wliereas the maximum sagging moment is very dependent of the wave that follows the large crests, as already analyzed in the previous paragraph.

Figure 6 shows that, on average, the maximum verti-cal bending moments are almost independent o f the f r o n t slope of the abnormal waves (the f r o n t wave slope is defined as the ratio between the wave height and the estimated wavelength). This result is somewhat surpris-ing, but it should be noted that the midship vertical bending moment is a global response that is more depen-dent on the wavelength and height o f t h e wave and o f the phasing of the wave with respect to the pitch motion. However, it is believed that the wave slope is important to local responses such as the loads resulting f r o m green water on the deck or bow slamming impacts, as demon-strated by Guedes Soares et al.'^ On the other hand, the present seakeeping model does not represent the tran-sient structural response to slamming impact loads, and

Springer

these effects may contribute to the maximum global ver-tical bending moments.

F r o m the three graphs shown i n Figs. 5-7, it is clear that while there is a strong correlation between the maxi-mum sagging and hogging moments and the height of the abnormal waves, there is also a weaker correlation with the f r o n t wave slope and wavelength (although these three variables are not independent).

J M a r Sci Technol (2006) 11:245-259 253

refers to the northern N o r t h Sea and thus this probabil-ity corresponds approximately to a period of 24 years. Regarding the linear long-term distribution, the maxi-mum wave-induced bending moment experienced over a long period of time is calculated by applying a linear strip theory i n the frequency domain, together with spec-tral analysis and a weighted summation of short-term Rayleigh distributions of the maxima.' However, the vertical bending moment f o r this containership is highly nonlinear, and the actual sagging peaks are largely un-derestimated by the hnear theory. To calculate correctly the long-term distribution of the nonlinear bending mo-ments, the numerical time domain model described i n Sect. 2.2 was applied to produce nonhnear transfer func-tions f o r a range of wave amphtudes. Then the response spectra were directly calculated with these functions f o r all sea states in the scatter diagram and finally, as i n the linear case, the weighted summation of the short-term Rayleigh distributions gives the long-term distribution. The corresponding results are included in the graph shown i n Fig. 8. Guedes Soares et al.'' have described the former method and have presented the linear and non-linear long-term distributions o f the midship vertical bending moment for a fast monohuU operating i n the N o r t h Sea (scatter diagram with the annual probabihties f r o m Hogben et al.^').

Figure 8 also shows that the maximum hogging mo-ments induced by the abnormal waves are always beUow the rule value, whereas a few o f the sagging results are larger than the value required by the rules. However, it is more meaningful to compare the maximum moments induced by the abnormal waves with the long-term dis-tribution values, since both type of results were com-puted using the same nonlinear numerical time domain model. Comparison shows that the maximum expected sagging moment during a period of 24 years in the northern N o r t h Sea is considerably larger than all o f the moments induced by the abnormal waves. The result seems to indicate that there are many "nonabnormal" wave conditions that can induce larger bending mo-ments than those induced by the abnormal waves.

Figure 9 shows the long-term distribution of the wave crests and of the wave heights f o r the northern N o r t h Sea. The results are based on Rayleigh short-term distri-butions; the circles represent the crest and the height o f the largest abnormal wave used f o r the calculations, i.e., the New Year Wave ( N Y W ) . The probability of occur-rence o f the crest (18.5m) is 1.58 x 10"', a smaller figure than the reference value of 10"l On the other hand, the probability of occurrence o f t h e height is 2.21 x 10"^, a much higher value than f o r the crest. The reason is that the N Y W is characterized by a very large crest with adjacent relatively small troughs.

Probability of Exceedance Max. Crest Probability of Exceedance Max. Heiglit

a [m] H [m]

F i g . 9. L o n g - t e r m scatter-based p r o b a b i l i t y d i s t r i b u t i o n s o f wave crests a n d wave heights f o r European area 8 (EA8) and the super-imposed D r a u p n e r m a x i m u m crest and height ( N e w Y e a r Wave,

ivrtro

The right graph shows that over a period of 24 years the ship will encounter many waves with heights larger than that of the N Y W . Waves with heights larger than the N Y W are not necessarily abnormal because they are contained in sea states of the scatter diagram higher than that i n which the abnormal wave was measured. Addi-tionally, as demonstrated i n Sect. 5, within a stationary sea state there may exist "nonabnormal" wave sequences that induce vertical bending moments larger than those induced by the abnormal wave.

5 Analysis of the probability distributions

The results f r o m the previous section seem to indicate that abnormal waves do not induce the largest global structural loads that a ship experiences during its opera-tional life. This conclusion is derived f r o m comparing the long-term distribution prediction w i t h the range of maximum moments induced by a large set o f wave records that include abnormal waves. The group of ab-normal waves used, besides representing waves w i t h ex-tremely large heights, also includes many waves with estimated wavelengths around the length of the ship (see Fig. 7). The maximum vertical bending moments occur for wavelengths slightly larger than the length of the ship between perpendiculars. From these reasons, it is be-lieved that the applied wave records are very much rep-resentative of the abnormal wave conditions that would induce the largest vertical bending moments on the ship. This section demonstrates that abnormal waves do not necessarily induce the largest ship responses and, i n particular, they do not induce the largest vertical bend-ing moments. The graphs shown i n Figs. 10 and 11 present probability distributions of the maximum wave elevations and of the ship responses to wave records lasting 20 min. These are the wave elevations represented

254 J M a r Sci Teclinol (2006) 11:245-259

F i g . 10. Exceedance p r o b a b i l i t y distributions o f the wave elevation and ship response f o r the sea state described by wave record 0 4 D R (the N e w Y e a r W a v e ) w i t h a significant wave height ( i ï j ) o f 11.92 m and a peak wave p e r i o d {Tp) o f 13.24s. T h e symbols m a r k e d w i t h a n

arrow correspond to the

a b n o r m a l wave crest and t r o u g h f o r the first graph, and to m a x i m u m and m i n i m u m responses induced b y the a b n o r m a l wave f o r the other graphs

File 04DR, Wave Maxima Comulative Exceedance Distributie

File 04DR, Heave Maxima Comulative Exceedance Distribution File 04DR, Pitch Maxima Comulative Exceedance Distribution

in a reference system advancing with the ship's forward speed and they inchide the abnormal waves o f the mea-sured records.

The probabilities of exceedance have been calculated for the inaxima and minima o f the simulated time records, named in the graphs respectively as crests and troughs f o r the wave, heave, pitch, and the relative mo-tion at the forward perpendicular. I n the case of the midship vertical bending moment, the inaxima and minima are represented as hogging and sagging,

respec-tively. Only one maximum and one minimum are consid-ered between consecutive up-crossing zeros o f the time histories. I n the graphs of the wave elevation, larger circles and arrows mark the peaks corresponding to the crest and to the trough of the abnormal wave. The same procedure is used in the graphs of the ship responses to mark the largest responses induced by the abnormal wave. I n this way, i t is possible to easily check i f there are ship responses larger than those induced by the abnor-mal wave.

J M a r Sci T e c l i n o l (2006) 11:245-259 255

F i g . 1 1 . Exceedance p r o b a b i l i t y d i s t r i b u t i o n s o f the wave elevation and ship response f o r the sea state described by wave record 1 6 N A w i t h f / i = 9.63ra and r / ) = 12.32s. Symbols marked w i t h an arrow correspond to the a b n o r m a l wave crest and t r o u g h f o r the first graph, and to m a x i n i u m and i i i i n i n i u m responses induced by the a b n o r m a l wave f o r the other graphs

File 16NA, Wave Maxima Comulative Exceedance Dislribution

File 16NA, Heave Maxima Comulative Exceedance Dislribution File 16NA, Pitch Maxima Comulative Exceedance Distribution

0,12 0.14 0.16

Finally, the Rayleigh distribution is fitted to the distri-bution of aU peaks, whereas WeibuU distridistri-butions are fitted to the maxima and to the minima. These distribu-tions are presented for qualitative assessment of the merit of these theoretical models to represent empirical distributions.

Figure 10 corresponds to the sea state of wave record 04DR, which is characterized by a significant wave height of 11.92m and a peak period o f 13.24s. This wave record is chosen because i t includes the well known New Year Wave, which has an extremely large height of

about 26m. The graphs i n Fig. 11 correspond to the sea state of wave record 16NA with a significant wave height of 9.63m and a peak period of 12.32s. I n this case, the abnormal wave is around 19.5m high. The latter is an interesting example because the wave record includes many "normal" waves that induce larger bending mo-ments than the abnormal v/ave.

The empirical distributions of the wave crests and troughs are very siiriilar, reflecting a global symmetry o f the wave elevation with respect to zero. The same obser-vation is approximately valid for the peaks of the pitch

256 _{J M a r Sci T e c l i n o l (2006) 11:245-259}

motion and of tlie relative motion at the bow. Regarding heave, the ship tends to move upwards more than it moves downwards. Finally, a large asymmetry is ob-served for the midship vertical bending moment, with the sagging peaks having values around twice those of the hogging peaks. This nonlinear behavior has been identified before and is related to the nonhnear hydro-static and Froude-Krylov forces associated with the nonvertical sides of the hull.

Regarding the theoretical distributions, the Rayleigh distribution correlates weh with the distribution of wave peaks, except for the abnormal wave crests. The Weibull distribution with a shape factor of less than 2 is more appropriate f o r representing the distributions of the ship responses, especially those responses characterized by larger nonhnear effects.

Focusing now on the aspects related to the abnormal waves, the first graph of Fig. 10 shows that the crest of the New Year Wave is located far f r o m the empirical and theoretical distributions. Observation of the other graphs of this figure leads to the conclusion that the abnormal wave induces the largest ship responses f o r the whole time record. However, while the abnormal wave crest is very far f r o m the empirical and theoretical distri-butions, this is not reflected in the ship responses, for which the largest peaks are much more i n line with the overall tendency of the distributions.

Regarding the probabihty distributions of Fig. 11, the wave crest of the abnormal wave is also located far f r o m the empirical distribution. I n this case, the abnormality index calculated using the zero down crossing is 2.02. However, looking at the distributions of the ship re-sponses, one observes that the responses induced by the abnormal wave fit quite well within the empirical distri-butions. The absolute motions induced by the abnormal wave are the largest o f the records; however, the relative motions at the bow and the vertical bending moment are not the largest. I n fact there are many wave sequences

that induce larger vertical bending moments than those induced by the abnormal wave. These results demon-strate that there aie wave sequences with heights of nearly half the abnormal wave that induce global bend-ing moments larger than those of the abnormal wave. The wave sequences discussed here are analyzed in more detail in Figs. 12 and 13.

Figure 12 presents the simulations of wave elevation and midship vertical bending moment corresponding to the probability distributions of Fig. 11. Auxihary hori-zontal lines represent, in the first graph, the crest and trough of the abnormal wave. I n the graph showing the vertical moment, the horizontal lines represent the maxi-mum sagging and hogging moments induced by the ab-normal wave. It is thus easy to identify the peaks that surpass those reference values. I n Fig. 13, the wave sequences that induced the largest vertical bending moments are shown i n more detail, together w i t h the corresponding relative motions at the forward perpen-dicular and the vertical bending moments. Analysis o f the graphs leads to the conclusion that the maximum sagging moments are very much related to the magni-tude o f t h e positive relative motions at the bow, meaning when the bow dives into a wave.

6 Conclusions

This article presents a systematic study of the struc-tural global loads induced by abnormal waves on a containership. A numerical nonlinear time domain seakeeping model was used to calculate the ship re-sponses to deterministic wave traces that included ab-normal waves. A large set of wave traces was used, and these data were measured on different occasions and at different places. I n this way i t was possible to assess the influence o f abnormal wave heights, wavelengths, and shapes on the wave-induced structural loads. The results

J M a r Sci Teclinol (2006) 11:245-259 ₂₅₇

F i g . 13. Simulations o f the wave elevation, the relative m o t i o n s at the f o r w a r d perpendicular, and midship vertical bending moments f o r the sea state described by wave record 16NA (Hs = 9.63 i n , Tp= 12.32 s). Shown are the selected wave groups that induced the largest vertical bending nioinents

Wave elevation (in)

20 30 40 t(s} 50 60 70

110 120 1 3 0 f f s J 140 150 160

Ret motion at the fwdpp (m)

40 t(s) 50 60 0 120 130 ( ( s J 140 150 160 240 250 260 ( f s j 270 280 370 3 8 0 ( ( s J 390 400 750 t(s}76Q 770 s s o f f s j g e o 9 7 0 Wave grotip thai includes Ihe abnormal wave

240 250 2 6 0 f f s j 270 260 VBM al midship (kNm) 40 t ( s ) 50 6.0x10' 3.0x10' 0.0x10°

/ \ j \ p

sbnoOTiaJ Visvs tnireman^ 1

1 3 0 ( f s J 1 4 0 150

t(s) 390 400 410

750 ( f s ; 760 770

950 ( (s) 960 970 Relal. motion response to the abiwnnnl wave

130 1140 i i s o f f s j n e o i i 7 0 1130 1140 1150ffsJ1160 1170 260 ƒ f s ; 270 370 3 8 0 f f s ; 390 400 6.0x10' 3.0x10^ 0.0x10° -3.0x10' -6,0x10' -9.0x10' •1.2x10*

" abnomial wave ma>imum ^ •

- -

--A-A /--A-A f

abnonnal I'.avs mViimum'' 1

740 750 ( ( ' s j 7 6 0 770

9 5 0 / f s ; 960 970

VBM response to Ihe abnonnal wave

r abnomial vi

1130 1140 1150(fs;i160 1170

258

J M a r Sci T e c l i n o l (2006) 11:245-259

showed that the maximum hogging bending moments always occurred when the abnormal wave crest passed through the midship position. The maximum hogging moments increase almost linearly with the height of the abnormal wave. The maximum sagging moments are around twice the maximum hogging moments. The larg-est sagging moments occurred, in most cases, after the large wave crest passed through the midship and the bow dived into the foUowing wave crest. I t was also observed that in some cases the downward forces due to green water on the bow deck tended to reduce the maximum sagging moment. When analyzing the results for the whole group of abnormal waves, on average, the maxi-mum sagging moments also increased w i t h the height o f the abnormal waves. However, compared to the hogging peaks, there was a larger dispersion relatively to the mean value. This is in part explained by the fact that the sagging moment depends on the height and shape o f t h e wave crest that the ship encounters immediately after the abnormal wave. I t was also observed that maximum bending moments seem to be independent of the f r o n t slope of the abnormal wave.

The sagging moments induced by the 20 wave records were compared with the long-term distribution of the nonhnear vertical bending moment for a ship operating in the northern N o r t h Sea. The reference value sponds to an exceedance probabihty o f lO"'', which corre-sponds approximately to a period of 24 years. The long-term distribution value was larger than all the maximum moments induced by the abnormal waves. Since both types of results were calculated by the same numerical seakeeping model, this result seems to indicate that there are "nonabnormal" wave conditions that in-duce larger bending moments than the abnormal waves. Acknowledgments. This w o r k was p e r f o r m e d w i t h i n the scope o f

the M A R S T R U C T project. N e t w o r k o f Excellence o n M a r i n e Structures, ( w w w . m a r . i s t . u t l . p t / m a r s t r u c t / ) , w h i c h is financed by the E U t h r o u g h the G R O W T H Programme under contract T N E 3 -CT-2003-506141. The w o r k builds u p o n methods and data ob-tained d u r i n g the research project titled R o g u e Waves—Forecast and I m p a c t on M a r i n e Structures ( M A X W A V E ) , p a r t i a l l y f u n d e d by the European C o m m i s s i o n under the p r o g r a m Energy, E n v i -r o n m e n t a n d Sustainable Development ( C o n t -r a c t n o . EVI<:3:2000-00544). W o r k by the t h i r d a u t h o r was financed b y grant S F R H / BD/10527/2002 o f the Portuguese F o u n d a t i o n f o r Science and Technology.

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