Long term prediction of non-linear vertical bending moments on a fast monohull

ELSEVIER

Applied Ocean

Research

Long term prediction of non-linear vertical bending moments

on a fast monohull

C. Guedes Scares *, N . Fonseca, R. Pascoal

Unit of Marine TecI,noIosy ané En.ineerin,, Tecknical University, of Lisbon. Instituto Superior Téaüco. Av. Ro.isco Pais. 1049-001 Lisboa, Portugal

Available online 25 October 2005

Abstract

An assessment is made of the maximum wave induced bending moment expected to occur during the operational lifetime of a fast monohull, bafed on o"r e^^^^^^^ calculations of the non-linear vertical bending moment at mid-ship. The ship is assttme to operate n, the Northern N::hSea:Uhanoperationanifeof25years.Asuccess^

results m the long-term distribution of the structural loads. Non-linear pseudo transfer functions are used in the pioceduie to calculate the v ' i c e " Ï he s i ttei-m responses. The amplitudes of these transfer functions are calculated by a non-linear time domain seakeeping program^ " s u u f a l e compared with'rules minimum required values and also with results from simpler calculation procedures such as adopting design sea states.

© 2005 Elsevier Ltd. All rights reserved.

1. Introduction

The definition of the minimum wave induced stractural loads for the design of ship structures is in most cases still based on empirical formulas from Classification Societies. However, presently there is a tendency to apply procedures based on direct calculations to define the design wave loads [1]. These procedures rely on hydrodynamic models derived from first principles, together with a proper stochastic characteriz-ation of the waves and of the ship responses. There are several advantages f r o m using these more advanced methods: eventually the design wave loads will be more accurate and tailored for the specific ship characteristics, novel ship concepts can be assessed (while empirical formulas are in principle valid only for existing ships), and, in addition to the global structural loads, it is possible to obtain consistent hydrodynamic load distributions for finite element calculations.

For the tinear case, the maximum wave induced structural loads during a long peiiod of time can be efficientiy calculated applying linear potential flow hydrodynamic models in the frequency domain, together with spectral analysis and a weighted summation of short term Rayleigh distributions of the maxima [2,3]. However, at least for ships

* CoiTesponding author. Tel.: -H351 218417607; fax: -^351 218474015.

E-mail address: guedessOmar.ist.utl.pt (C. Guedes Soai'es).

0141-1187/$ - see front matter © 2005 Elsevier Ltd. A l l rights reserved. doi:10.1016/j.apor.2005.08.002

with small block coefiicient, the wave induced süTictural loads are highly non-linear. The asymmetry of the vertical bending moment is the visible non-finear characteristic of this response and also the most important from the practical point of view. In fact the sagging peaks can be more than double the magnitude of the hogging peaks. In these cases the linear procedure cannot be applied and the design wave loads must be determined by time domain non-linear codes together with appropriate extreme value distributions.

Several approaches have been proposed and presenüy i t is not clear i f any one is better than the others. The next paragraphs present a biief review of such methods. Some of the methods require only one time domain simulation of the ship responses in a selected wave trace that is assumed to be the condition in which the extreme wave loads occur. These methods are indicated for utilization together with most advanced and computationally demanding seakeeping time domain codes. Other procedures require more extensive time domain simulations in a variety of conditions and these are appropriate to be used together with computationally simpler time domain codes.

Guedes Soares [4] and Guedes Soares and Schellin [5] generatized the linear long term predictions procedure to account for the non-linear asymmetry of the vertical bending moment, by using form functions that transform the amplitude of the linear transfer functions to that of the non-Unear transfer functions associated with sea states of different intensity.

Several other procedures are based on the assumption that the linear model is a good identifier of the conditions in which the extreme wave loads occur. Adegeest et al. [6] present

the design regular wave method, in which a regular wave is determined based on linear long-term calculations and then a non-linear simulation is mn f or that regular wave to determine the design stmctural loads.

The coefficients of contribution method also uses linear long term calculations to identify the sea states of the scatter diagram that contribute most to the probability of exceedance of the stractural loads during the ship's life. This way the non-linear simulation program is applied to a selected small number of sea states only. Sagli and Moan [7] studied the non-linear vertical bending moment on a containership in the Northern North Sea.

The critical wave episodes method is another possibility, which is adequate for the most advanced and time consuming hydrodynamic models. Torhaug et al. [8] identified the random incident wave sequences that result on the ship extreme response by performing linear time domain analysis for each relevant stationary sea state. The wave sequences resulting in the largest linear responses are then applied to the non-linear analysis. In the most likely response method [6,9], the most likely extreme response is first determined by linear frequency domain analysis. Then, using the theoiy of Gaussian processes near a maximum, the corresponding deterministic wave elevation is produced and applied in the non-linear simulation.

Another possibility is to use the contour Hne approach. With a long-term analysis of the wave climatology it is possible to define a set of sea states corresponding to a specific return period. These sea states include the worst environmental conditions and lie in a closed contour on the scatter diagram. The design extreme response is the most probable extreme value determined within all short term sea states of the contour Une.

Although the methods referred in the previous paragraphs take into account the stochastic nature of the waves encountered by the ship during its life, the conditions associated with the encounter of the ship with abnormal waves are not taken into account. This is because the probabilistic models describing the waves do not consider abnormal waves [10]. In order to calculate the wave loads induced by these conditions, Fonseca et al. [11] proposed a method to calculate the structural loads induced by determi-nistic wave traces of abnormal waves, where the ship responses are calculated by a non-linear time domain seakeeping code. The method was also applied to investigate the loads induced by abnormal waves on a FPSO [12] and also on a containership [13].

The approach initially proposed by Guedes Soares [4], and later used by Guedes Soai'es and Schellin [5], has been used herein, but now taldng direct advantage of a time domain seakeeping code. Non-linear transfer functions are calculated for a range of wave amplitudes and then the response spectra are directly calculated with these functions for all sea states in the scatter diagram. This procedure has also been used by Guedes Soares et al. [12] to calculate the design wave bending moment on a FPSO. The same procedure is applied here to calculate the maximum wave induced vertical bending moment on a fast monohull. An operational life of 25 years is

considered and the ship is assumed to operate in one of the most severe ocean areas, namely the Northern North Sea.

2. Long term prediction method

The approach for calculating long-term cumulative prob-ability distributions of sliip responses has been established for several years and is thought to be an accepted one. It was initially proposed by Fukuda [2] and by Guedes Soares and Moan [3] among others. In this approach it is necessary to have the probability density ftinction of a response, conditional on certain physical conditions being met.

The method used herein is both the conventional calculation based on transfer functions and the approach using so-called 'pseudo transfer functions'. The first con'e-sponds to determining the short term linear response spectram using transfer functions for specific situation, broadly named 'condition' in the equations to follow, which includes loading condition, headings, speed, and wave period, then using the result of Wiener-Khinchin theorem to calculate the variance of the response as the area of the response spectrum. Since the sea states and the different headings are considered independent realizations and the only statistics available are the variance and zero mean, theoretically there is a constraint to use the Rayleigh distribution for each condition and the result is the sum over all sea states and headings weighed by the coiTesponding probability of occurrence.

The pseudo transfer function procedure is very similar, but the linear transfer function is replaced by one detemined from a code accounting for non-linear effect for the situation in which the vessel is subject to a harmonic wave with a wave height equal to the sea state's significant wave height. The transfer function amplitude for sagging and hogging, which are now different, are then determined separately by dividing the corresponding peak response values by the wave amplitude. This last procedure carries therefore some inconsistency from a strict mathematical view of non-linear systems, namely because the response will be periodic but not harmonic, however, i t is thought to be adequate for practical purposes.

The response spectrum 5R(W) is obtained from the input wave spectram 5w(w) and the transfer function H(w), all considered to be one-sided:

(1) The narrowband response then has a probability of exceeding the level r.

e s ( ^ > ' - k ) = e x p

2a' (2)

where a is the variance of the process. The variance can be obtained from the spectram by integration:

290 C. Guedes Soares et al. /Applied Ocean Research 26 (2004) 288-297

which is applicable to both the input and the response spectrum.

Thus the probability of exceeding an amplitude r in a long-teim is given by Refs. [2,3]:

QLiR>'-) Qs(R> i-\a)f(a)dff (3)

where Qs is the short term Rayleigh distribution given by (2) and the distribution of response standard deviation is given by scatter diagram, which represents the statistics of the wave climate, encountered by the ship during its operational life.

The integral kernel is slightly changed to account the fact that a ship will be subjected to a different number of cycles in different sea states. The original kernel will be multiplied by the following condition dependent weighing factor:

^•''condition

-^21 condition

where the average response period, Z'2, is given by:

f2^

3. Non-linear transfer function results

As discussed i n Section 2, the long term prediction of the wave induced non-linear vertical bending moment is based on the calculation of non-linear transfer functions by a time domain non-linear code. This section presents the non-linear transfer function results for a fast monohull. It starts by describing the geometrical and physical characteristics of the ship, afterwards the time domain code is presented together with some typical results and a few comparisons with experiments and finally a set of non-linear transfer functions are presented and analyzed.

3.1. Fast monohull chafacteristics

The calculations are earned out for a fast monohull, which has a length of 134 m and a maximum speed of 27 Imots {F^ = 0.40). Due to the severity of some sea states, a reduced ship speed of 13.5 knots was considered in the calculations. Fig. 1 presents the hull bodylines and Table 1 the ship main particulars and physical properties.

Table 2 shows the inertial characteristics of the complete ship and of each of the four segments. These are the mass properties coiTesponding to a segmented model of the ship as it was tested in the El Pardo model basin, Madrid. The model was made up of four segments connected by a rigid bar, with cuts at mid-ship and l/4Lpp forward and aft of mid-ship. In the table, '/,„,' represents the longitudinal inertia with respect to the

Fig. 1. Fast monoliull bodylines.

centre of gravity of the whole ship, is the longitudinal position of the centre of gravity relative to mid-ship and Zg is the vertical position of the centre of gravity with respect to the baseline. The order of the segments numbering is from to stem bow.

3.2. Non-linear time domain code

The numerical model to calculate the ship motions and structural loads induced by large amplitude waves is a non-linear time domain seakeeping code which assumes that the non-linear contribution to the vertical ship motions and loads is dominated by hydrostatic and Froude-Kiilov forces, thus these components depend on the instantaneous hull wetted surface. The exciting forces due to the incident waves are decomposed into a diffraction part and the Froude-Kiilov part. The diffraction part, which is related to the scattering of the incident wave field due to the presence of the advancing ship, is kept Hnear. The Froude-Kiilov part is related to the incident wave potential and, at each time step, results from the integration of the associated pressure over the wetted surface of the hull under the undisturbed wave profile.

The radiation forces are represented i n the time domain by infinite frequency added masses, radiation restoring coefficients

Table 1

Fast monohull main particulars Monohull

Length overall i-oa (m) 133.7

Length between perp. Lpg (m) 122.0

Breadth overall B ( m ) 15.19 Depth D ( m ) 9.2 Draught r ( m ) 4.66 Displacement A (ton) 4329.9 Service (max) V (kn) 27 Long position of CG LCG - 1 . 3 4 7 Vertical position of CG VCG 4.374 Block coefficient CB 0.49

Table 2

Inertial properties of tlie model (translated to f u l l scale)

Description Segment 1 (stem) Segment 2 Segment 3 Segment 4 Ship

Weight W(ton) 888.6 1373.8 1214.1 852.9 4329.6

Long. Pos. centre of X„ (m) -43.483 - 1 4 . 8 2 4 13.747 42.697 - 1 . 3 6 3

gravity

Lat. pos. centre of J'g(m) 0.000 0.000 0.000 0.000 0.000

gravity

Vert. pos. centre of Z„ (m) 4.374

gravity

Long, inertia with lyy (ton m^) 1,634,378 376,532 410,800 1,701,199 4,122,909

respect to model centre of gravity

Pitch radius of _0.253

gyration

and convolution integrals of memory functions. The convolu-tion integrals represent the effects of the whole past history of the motion accounting for the memory effects due to the radiated waves. The vertical forces associated with the green water on deck are calculated using the momentum method.

The formulation was presented in Refs. [14,15]. Some comparisons with available experimental data are presented in Ref. [16]. In Ref. [17] the method is compared with other similar formulations.

In order to obtain a comprehensive set of experimental data necessary to validate the numerical model, an experimental programme was carried out at the Laboratory of Ship Dynamics of the El Pardo Model Basin, Madrid. The motions and loads induced by large amplitude regular and iiTegular waves were measured on a containership. The experimental data and compaiisons with the numerical model have been presented in Refs. [18-19]. Finally the hmits of application of the method regarding the Froude number were assessed by comparing numerical results with experimental data for a containership [20] a Fast Ferry [21], and for the Fast Monohull that is being investigated here, by Fonseca et al. [22].

The general conclusions from all these comparisons are that the linear time domain method is able to capture all non-linear effects detected in the experimental data, the predictions

Heave Amplitudes (m/m) - Fn = 0.20

1.2 i- ;

are improved compared to the linear predictions, however there are some discrepancies mostiy related to the high speed effects that are simplistically represented by the numerical model.

As examples, the graphs in Figs. 2 and 3 show comparisons between calculated results and experimental data for the motions and stractural loads on the fast monohull being investigated here, advancing with a Froude number of 0.20 in head regular waves. This Froude number corresponds to the speed considered here for the long-term distribution calcu-lations. The experimental program and analysis of the data is presented in [23] and comprehensive comparisons with the numerical model are given by Fonseca et al. [22].

Fig. 2 presents the amptitudes of the transfer function of heave and pitch as function of the wavelength normalized by the ship length. The solid symbols stand for experimental data and the lines for numerical results. Different regular wave amplitudes were considered for each wave frequency in order to investigate the related non-linear effects, thus, in the graphs, different symbols coirespond to different wave steepness. The effects of non-linearity on the amplitudes of the vertical motions are obviously small. The agreement between the numerical results and the experiments is good and there is only a small over-estimation of the experiments for normalized wavelengths around 1.1.

Pitch Amplitudes (°/m) - Fn = 0.2

202 C. Guedes Soares et al. /Applied Ocean Researcli 26 (2004) 288-297

100000

V B M at midship - Max. and M i n . - Fn = 0.2

ia 50000 (kNm/m) O -50000 -100000 \ -150000 9 exp. (Lw/Hw=80) - B — num. (Lw/Hw=80)

Fig. 3. Sagging (negative) and hogging (positive) peaks of the vertical bending moment at mid-ship for the fast monohull in head regular waves.

In Fig. 3 one observes the sagging (negative) and hogging (positive) peaks of the vertical bending moment at mid-ship when the ship is advancing in head regular waves with a steepness given by the ratio wavelength to wave height of 80. The correlation between calculations and experiments is very good, especially for the sagging peaks. This small set of comparisons show that the non-linear time domain code is able to represent very well the vertical ship responses for the Froude number of 0.20.

3.3. Non-linear pseudo transfer functions

The transfer functions of the vertical bending moment at mid-ship are determined from time domain simulations using the non-linear code. The ship advances in incident harmonic waves. Fig. 4 shows one example of simulations for a harmonic wave with a length of 127.6 m, which is approximately the length of the ship, and it advances in head waves with a Froude number of 0.20. Three different wave amplitudes are considered, namely 1.0, 3.0 and 5.0 m. The graphs present results of heave, pitch, relative motions at the bow (3 m forward of the fwdpp), vertical forces on the deck at the bow (3 m forward of the fwdpp) due to green water, and the vertical bending moment at mid-ship. Heave, pitch and the bending moment are normalized by the wave amplitudes, while the other responses are not.

It is clear that the vertical motions are non-tinear with respect to the wave amplitude, and i n particular the bow downward pitch motion, when divided by the wave amplitude, reduces significantly with the wave amplitude for the larger wave (one should not that the results in fig. 2 are for wave amplitudes smaller than 5.0m and therefore nonlinear effects ai-e not detected there). This same effect results on a reduction of the vertical bending moment sagging peaks, normalized by the wave amplitude, for the largest wave.

The two largest wave amplitudes result on green water on deck and on corresponding downward vertical forces on the

deck. This is observed on the fourth graph where the maximum forces per unit length reach a value close to 370 kN/m, which corresponds to an average pressure over the deck at the forward perpendicular of around 80 kN/m^. These downward forces on the deck induce a hogging global vertical bending moment on the ship structure, which opposes the sagging moment, thus tending to reduce the maximum sagging moments that the ship would otherwise experience. This is one of the reasons why the sagging peaks divided by the wave ampUtude reduce with the wave amplitude.

While the usual definition for the amptitude of the transfer function is given as the amplitude of the first harmonic of the time record divided by the wave amplitude, a different definition is used here. In the present work, the transfer function amplitude for a particular wave frequency and amplitude is defined by the absolute value of the sagging peak divided by the wave amplitude. This value may in some cases be quite different from the normalized first harmonic amplitude and additionally it is dependent of the wave amplitude. For these reasons the result is called 'pseudo transfer function'.

In order to obtain complete transfer functions, simulations are canied out for around 20 wavelengths covering the whole frequency range of interest, while the wave amplitude is kept constant. Similar calculations are repeated for several wave amplitudes, since the transfer functions vary with the wave amplitude, and also for each heading of the ship with respect to the waves.

Fig. 5 presents the pseudo transfer functions for the fast monohull advancing with a Froude number of 0.20 in head, bow and beam waves. In head and bow waves 5 different wave amptitudes (i^a) are considered and the Ca^O limit stand for linear results corresponding to infinitesimal wave amplitude. The graphs show that, from small to medium wave amplitudes, the transfer functions increase with the wave amplitude. As the wave amplitude increases further, the transfer function starts to decrease due to the reasons described earlier.

4. Calculation of design wave bending moment

As described in Section 2, one obtains the long-term probabilities of exceedance by using the pseudo transfer functions within a spectral formulation and combining the results with the wave climatology of the ocean areas where the ship is to operate.

This section presents the long-term sagging probabilities of exceedance of the vertical bending moment at mid-ship. Assuming a probability level associated to the operational life of the ship, one obtains the most probable maximum wave induced bending moment induced on the ship stmcture during its lifetime. Calculations include tinear and non-linear long-term values, which are compared to the mie requirements and to other simpler methods of estimating the design wave bending moment. Linear and non-linear long-term values are calculated respectively ti'om linear transfer functions and pseudo transfer functions.

. wave amp. = l.Om wave amp. = 3.0m wave amp. = 5.0m 1.2-1 160 164 168 172 176 time (s) 160 164 168 172 176 time (s) 160 164 . 168 172 176 time (s) =• - 1 0 0 0 0 0 - 1 ^ ^ , 1 , ^ , 1 , . 160 164 168 172 176 180 time (s)

294 C. Guedes Soares et al./Applied Ocean Research 26 (2004) 288-297

1

l.OE+05

8.0E+04

6.0E+04

Vertical Bending Moments at midship for 120°

f a - Om ^ f a = 1 m - a - C a = 3 m ^ f a = 5 m ^ f a = 7 ra iJ> 4.0E+04 2.0E+04 4 O.OE+00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 fflg (rad/s) l.OE+05 i.OE+04 6.0E+04 >J 4.0E+04 2.0E+04 4

Vertical Bending Moments at midship for 90°

O.OE+00

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 (Bo (rad/s)

Fig. 5. Pseudo transfer functions of the vertical bending moment at mid-ship for different heading and wave amplitudes (F,, = 0.20).

The long-term calculations were performed for long crested seas and headings from head to beam waves. Altogether seven equally probable headings were used. A t this stage no quartering or following waves were considered. Linear calculations show that the difference between considering all headings between head and following waves and only between head and beam waves is very small (Fig. 6). The scatter diagram with the probabihties of occurrence of the sea states corresponds to the Northern North Sea ocean area and it includes annual probabilities [24].

Fig. 6 presents the long-term probability of exceedance of the sagging moment at mid-ship together with the rules minimum requirements. The difference in considering all headings or just the five headings in linear calculations is seen to be of marginal importance with respect to the estimated long-tenn response values. In the graph, the line at the exceedance probability of 10~^ coiTesponds approximately to a period of 24 years, thus the point of intersection of this line with the long-term distribution lines defines the expected maximum wave bending moment during 24 years of operation in the Northern North Sea.

X 10' Wave Induced Bending Moment 1 1 -linear (12 head) linear (5 head) —B— nonlinear- (5 head) 1 1 -linear (12 head) linear (5 head) —B— nonlinear- (5 head) -1 ó i

T R U . es sagging niinimum valUe''*V. " \ T R U es hogging lirinimum value

i i

10^'° 10^** 10-'5 lO-"* 10-2 10" Exceedance Probability

The long-term values calculated from linear transfer functions are roughly twice the minimum requirement value and the non-linear are roughly thiice. This is in fact a very large difference, which needs to be analyzed. Earlier investigations for other vessels [12,13], have identified that the long term predictions for European area 8 [24] results in maximum bending moments larger than the minimum rule requirements. It happened systematically for vessels that do not operate with restricted conditions.

For instance Guedes Soares [25,26] has presented results that are 43% higher for a containership and 24% and 26% for a large and small tanker, respectively. The non-tinear effects were not considered in these cases, meaning the asymmetry of the vertical bending moments was neglected. Guedes Soares et al. [12] presented results for an FPSO hypothetically moored in European Ai'ea 8 and there would be an estimated long-term sagging bending moment of 45% (non-tinear prediction) and 30% (linear prediction) greater value than the rules. In Ref. [27] dme domain calculations with the S175 in storm conditions complement the long-term predictions in saying that minimum mle values are not adequate for such situations. Parunov et al. [28] present design wave bending moments for two tankers calculated applying the lACS procedure for long-term prediction. Large overestimations of the rule vertical bending moments were obtained. The authors state that: '...rule vertical bending moment should be substantially increased in order to have safe prediction for navigation i n the North Atiantic'. It is then clear the overestimation of rule values is serious for vessels that travel in um-estiicted condition and that may at one time cross the North Atlantic waters.

With the objective of verifying i f the very lai'ge value obtained from the non-linear long-teim distribution is realistic or um-eaUstic when the ship operates in this ocean area, a set of additional calculations with alternative methods were performed. The objective is to obtain indicative results to compare with the rale requirements and with the long-term distiibution values.

One possibility is to assume that the maximum wave bending moment occurs during a specifically chosen storm, or a design storm. In this case a 10-year return period storm was selected from the statistics of the North Sea ocean area. The significant wave height is 11.5 m and the zero crossing wave periods is 9.5 s, which coiTesponds to the seastate that maximizes vertical bending moment spectral energy. The estimated most probable bending moment for a storm duration of 3 h is 5.1 X10^ kNm. The calculation is done applying a short term Rayleigh distribution, and the variance of the response is based on the pseudo transfer function of the sagging peaks con-esponding to a wave amplitude of 5.75 m. The resulting bending moment is between the tinear and the non-linear long-teim distributions. For 100-year return the value would be 5.3X10^ kNm.

This simple method has two drawbacks: the selected stoiTn may have a wave height and period that are not the most severe one in terms of wave induced moment during the life of the vessel, and, probably and most important, the duration of the storm may be inadequate and insufficient to represent the most severe wave conditions that the ship will encounter during the

24 years period. This method seems to result in underestimated maximum bending moments but still much larger than the nunimum Rule values.

One possibility to overcome these drawbacks is to identify the conditions that most contribute to the long-term distri-bution, at the long-term value for the desired exceedance probability. This can be done by applying the coefficient of contribution method presented, e.g. by Sagli [7]. The coefficient, which represents short-term response probability of exceedance for the given value of r with respect to the long-teim value, is defined by:

CoC()-) = wQ,iR> ,-\a)f(cr)

QdR>r) (6)

The larger the coefficient, the larger is the contribution of the condition to the long-term value, meaning to the maximum expected bending moment during the specified period of time.

The use of the coefficient is meant to reduce the calculation time especially in case of non-linear time domain calculations and is normally calculated using the tinear transfer functions, which provides a good estimate. After having the coefficient it is possible to perform non-linear simulations in iiTegular waves only for those sea states with the coefficient above some threshold.

The coefficient of contribution method is used here to identify the sea state that most contributes to the long-teim distribution, and then a short-term calculation is earned out for this sea state only. Fig. 7 presents the contour results of (6) calculated when r is the value attained by the non-linear long-term distribution at the 10^^ probability of exceedance.

Using the method of coefficients of contribution, it is found that a sea state with H^= 10.5 m and 7; = 9.5 s i n the scatter diagram, has the largest contribution to the vertical bending moment in 24 years of service. This sea state occurs with 8 X

Coefficient of Contribntion [%] for p(R>r) = 10

Fig. 7. Coefficient of contribution contour lines for the 10-8 probability of exceedance.

296 C. Guedes Soares et al./Applied Ocean Research 26 (2004) 288-297

Table 3

Mid-ship design wave bending moment from different methods

Method Wave bending

moment (10^ kNm)

(a) Rules—hogging 1.5

(b) Rules—sagging 1.9

(c) Linear long tenn distribution, 24 years 4.1 (d) Non-linear long temi distiibution, 24 6.3

years—sagging

(c) Non-linear, 10 years design storm, 3 h 5.1 duration—sagging

(f) Non-linear, 100 years design storm, 3 h 5.3 duration—sagging

(g) Non-lineai', coefficient of contribution 6.1 method—sagging

10~ probability in the region, thus roughly 5.84 days are spent in the sea state, the mean response period in this sea state is 6.4 s which means a probability of the most expected bending moment of 1.27X10"^ and a corresponding value of 6.1 X 10^ kNm. Using this sea state alone one obtains a bending moment, which is thus 97% of the complete long-term distribution.

Table 3 summarizes all results presented i n this section. Linear long-term distribution results are around two times the iTile required sagging value. This is a larger difference than that obtained by Guedes Soai'es [25] for a containership. When considering the asymmetry of the vertical bending moment, then the non-linear long-term distribution value is around tlnee dmes the rule value. In order to confirm the former result, short term selected sea states (e.g. design storms) were considered and the con-esponding results, although smaller than the f u l l long-term distribution results as expected, they are close to the value in (d). Given that this analysis have resulted in values higher than other obtained for fuller ships at a lower speed, study was made of the impact of some of the assumptions adopted. One was the speed that was assumed for the vessel for calculations which was 13.5 knots conesponding to a Froude number of 0.2, which represents half of the maximum speed of the vessel. The long-term calculations were repeated for the same vessel at zero speed so as to have a lower bound value and the result was 3.92X10^ kNm, as compared with the value of 4.12X 10^ kNm obtained for the speed of 13 knots with the North Sea wave climate. So it is clear that even without speed the value is much larger that the minimum values prescribed by classification sociedes.

Another decision that has a major impact on the results is about the weather conditions chosen for calculations. In this case a scatter diagram representing the typical wave climate for North Sea was chosen. However i f a different scatter diagram would have been chosen as for example the one for the Meditenanean that was adopted in Schellin and Lucas [29] then a value of 1.79 X10^ Id^Im would have been obtained for the long term value at 13 knots and 1.74X 10^ IcNm for zero speed. Comparing these values with the 1.9 l<Nm prescribed by the rules would mean that this ship would be within the minimum values prescribed by classificadon societies minimum

requii-ements, which was also the conclusion of Schellin and Lucas [29], who did an independent study with different calculation tools and applied to a different fast ship.

5. Conclusions

This work has shown that when using direct calculadon methods the long-term wave induced loads that are predicted are much higher than the minimum wave induced loads specified i n the rules of classification societies. This conclusion is the same as various authors have akeady reported in the past for ships of fuller forms but in this case large numerical differences were obtained.

Various different procedures were adopted to obtain altemadve predicdons of the most probable long-term value of the wave induced bending moments and it was effectively demonstrated that any calculation for the present vessel operating in the North Sea leads to loads that are largely exceeding the rule minimum values.

It was also shown that the choice of the design wave climate is a critical step that very much governs the values of the predicted long-term values of wave-induced loads suggesting that more attendon needs to be devoted to the specificadon of what is the probabilisdc description to be adopted as the design weather conditions.

Acknowledgements

The presented work was performed within research project BE-4406, Advanced Methods to Predict Wave Induced Loads for High Speed Ships (WAVELOADS). This project was partially funded by the Commission of the European Commumty under the BRITE/EURAM program; contract BRPR-CT97-0580. The third author has been financed by the Portuguese Foundation for Science and Technology (Fundajao para a Ciência e Tecnologia) through grant SERHyBD/10527/2002.

References

[1] Guedes Soares C, Dogliani M , Osteigaard C, Parmentier G, Pedersen PT. Reliability based ship structural design. Trans SNAME 1996;104:357-89. [2] Fukuda J. Theoretical determination of design wave bending moments.

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