Stochastic models of load effects for the primary ship structure

Structural Safety, 8 (1990) 353-368

353

Elsevier

STOCHASTIC MODELS OF LOAD EFFECTS FOR THE PRIMARY SHIP STRUCTURE * C. Guedes Soares Naval Architecture and Marine Engineering, Technical University of Lisbon (Instituto Superior T~cnico), Av, Rovisco Pais, 1096 Lisbon, Portugal (Accepted December 1989)

Key words: floating structures; stochastic processes; load combinations; extreme loads; permanent loads; passage problems; renewal processes.

ABSTRACT The available stochastic models for the characterization of the main load effects acting on ship structures are presented and discussed. A description is provided of the still water and the wave-induced load components, both as concerns fatigue loading and extreme situations. Load combination is also adressed as well as the derivation of characteristic values appropriate for design. While fatigue loading will be the basis of linear structural analysis, the extreme load conditions referred here will often require non-linear structural analysis.

1. INTRODUCTION The design and analysis of ship structures is normally conducted at various levels of detail to which are associated different sets of load effects. The most important design consideration is given to the global behaviour of the ship structure, which is provided by its primary components, i.e. the longitudinally continuous material that bends together in a beam like fashion. This is made up of the different plate elements found in the decks, sides, bottom and by the longitudinal stiffeners of these elements which have longitudinal continuity. To complete the classification of the structural elements, there are also secondary structures, which are components such as transverse frames or the stiffened panels of the deck, side, bottom or bulkhead which do not have longitudinal continuity. Tertiary structures, which is the lower level of detail, are for example the plate elements between stiffeners. * Paper presented at the Euromech 250 Colloquium on Nonfinear Structural Systems under Random Conditions, Como, Italy, June 19-23, 1989. 0167-4730/90/$03.50 © 1990 - Elsevier Science Publishers B.V.

354 In the linear analysis of secondary and tertiary structures one superposes on the response of the primary structure the behaviour due to the localized loadings. In general while the primary structure bends, the deck and bottom will be subjected to compression and tension in their own plane. Local loads are out of the plane and will induce additional bending of these elements. Because of its implications, the behaviour of the primary structure is the main concern in design and safety assessments of ship structures. Local failures have very often only economic consequences, which are thus serviceability problems of less concern [1,2]. This consideration of the various levels of ship response is important in that it influences the load effects to be accounted for. Although in local structures several loads are important, when addressing the primary structure there are only two main load components, the still water and the wave-induced one. A ship floating in still water is subjected to her own weight (the light ship weight) and to the weight of her cargo (the deadweight). This weight is balanced by the buoyancy generated on the immersed part of the hull. Thus the ship will adjust her draft and trim until global equilibrium is achieved. The different transverse sections along the ship hull will be loaded by the difference between the weight and buoyancy. When the ship is subjected to the action of a passing wave, the distribution of buoyancy along her length changes, inducing motions. The static loads due to changes of buoyancy and the inertia forces associated with the resulting ship motions make up the wave-induced load effects. The two main components of the load effects on the primary ship structure separate the effect of the cargo distribution (still-water load effect) and of the wave excitation and associated inertia forces (wave-induced load effect). Other load effects that exist are the thermal and the vibratory load effects, but these have a more localized effect and are not generally accounted for in the preliminary design of the main structural elements. Thus they will not be addressed in the present work. The still-water load effects are mainly vertical shear forces and bending moments although in some special cases torsional moments may also be present. The wave-induced components are vertical and horizontal shear forces and bending moments as well as torsional moments. For design purposes it is necessary to describe not only each load component individually but also their combined value. In doing so, due account must be given to the fact that they have different characteristic durations. The load models described here are required to assess the structural capacity in relation to fatigue strength and to ultimate collapse strength. While in the first case one uses linear structural analysis, in the latter situation different types of non-linear structural formulations [3,4] are required. The ultimate collapse load of the primary structure of the ship hull involves the overall bending of the structure as a box girder. Collapse will be due to the large deformation yielding of one of the deck or bottom under tension while the other flange of the box girder will fail under compressive loads. In general this latter mode of failure will govern the overall strength and this mode of failure involves large deflection, elasto-plastic behaviour of plate elements which is only possible to analyse with non-linear programs as described in [4]. 2. STILL-WATER LOAD EFFECTS The still-water loads depend basically on the amount and distribution of the cargo carried onboard because the light ship weight is almost constant during the ship's lifetime. The changes

355 of cargo usually occur in port. There is a gradual consumption of fuel during the voyages, inducing a small change in the level of the still-water load effects but its value is not significant. Thus, interest is focused on the description of the intensity and duration of the load effects between consecutive arrivals in ports. Once the distribution of the cargo is known, there is no difficulty in obtaining the distribution of the load effects. They result from integrating the difference between cargo and buoyancy twice along the length of the ship. 2.1. Distribution of the load effects at a random point in time

There is no theoretical model available to predict how the cargo will be distributed along a ship. Thus the approach taken by various authors has been to conduct a statistical analysis of operational data and to construct theoretical models from those results. The first type of models to be proposed did not account for the voyage duration and they represented the intensity of the load effects in successive voyages as a random variable [5-7]. Formal non-parametric statistical tests have shown that the data from a sequence of departures could be considered as random and independent outcomes of a variable. Furthermore a normal distribution was shown to be an adequate model to describe those load effects [7]. Recently, ships that are more than 150 m in length are forced to have on board a load distribution equipment which gives indication to the ship master about the distribution of shear forces and bending moments along the length of the ship. Furthermore, Classification Societies prescribe the maximum values of the still-water load effects at each transverse section which should not be exceeded during operation. Use of this equipment has decreased the number of times that the still-water load effects exceed the allowed values, acting almost as a truncation of the probability distribution at those levels. In fact, tests of fit have indicated that sometimes a truncated distribution provided a slightly better fit to the data than the normal distribution [7]. The availability of the load distribution equipments on board and the respective records providing information of load effects at different points along the length of the ship, made it possible to collect data and to develop models that describe the longitudinal distribution of the load effects [8], while previous works [5-7] have considered only the midship section. Some types of ships have distinct operating modes, often in full load, partial load and in ballast, during which the load effects change according to different patterns. This has been taken into account in [8], in which the load effects in various transverse sections along the length of the ship, usually between ten and fifteen, were modelled as normal distributions conditional on the load condition of the ship. Thus at each section j the marginal probability density function of the load effect X is given by:

x(x) = f

c(c)

x(xlc) dc

(1)

where C is the load condition in which the ship may be at a random point in time. This model is based on the assumption that the load effects in neighbouring transverse stations can be modelled as independent and identically distributed r a n d o m variables. The correlation coefficient between load effects at the different longitudinal sections have been calculated in [8], indicating, as expected, that it decreases with increasing distance between sections. A model of the linear dependence of the correlation coefficient on the distance between

356 stations was shown to be appropriate to describe the results. The coefficients of the regression equation vary slightly with the type of the ship but a typical expression is: O(y) = - 1 . 2 y + 0.9

(2)

where y is the longitudinal distance between transverse stations along the ship length L, normalized by L and p is the correlation coefficient. Thus for distances of 0.1L one has a correlation coefficients of 0.78 while for y = 0.4L the coefficient drops to 0.42. These results indicate that having data from one transverse section in the midship region already provides some information about a significant part of that region. The analysis in [8] has shown that there exist significant differences in the probabilistic description of the load effects in different ship types. The statistical analysis that was performed allowed the separation of the variability of the load effects in successive voyages of the same ship, from the variability between different ships of the same type, and between different ship types. The statistical model used to analyse the data represents the intensity of the load effect X at voyage i on ship j of type k by: Xij k = m o + m~ + m j + ei

(3)

where c i represents the normally distributed random variation of the load effect from its mean value in successive voyages of one ship and the other terms represent the characteristics of the specific ship. The mean value of all ships is m 0 while the mean value for the k t h ship type is (m 0 + ink) and the mean value of the j t h ship is (m 0 + rn k + m j). The variable e has a zero mean value. The variance of the observations Xij~ is equal to the sum of the variance of all effects. The variance of c represents the random nature of the process of cargo distribution in one ship in successive voyages. In the k th ship type the variance of the load effect is (o2 + o~) where o/ is the variance of the mean values of ship types. A more refined analysis showed that the parameters defining the probability distribution vary not only with the load conditions and with ship type but also with ship size. Thus a linear regression model was used to represent the mean value or the standard deviation, genericaly designated by Y, as a function of the ship length L, the deadweight W and the type of ship: Y = A o + A , W + A 2 L + Y'~ B i D ~

(4)

i=1

where A~ and B i are regression coefficients and D i is a d u m m y variable that is unity for ships of type i and zero otherwise. Thus the coefficient B~ indicates the difference between class i and the reference case. 2.2. Distribution of the maximum load effect

For design purposes one is often not so much interested in knowing the distribution of the load effect at a random point in time but instead one looks for the maximum value experienced during a specific time duration. With a random variable model in which the probability distribution is given by F x ( x ), one can determine the probability distribution of the maximum value in n outcomes Fm,n(x) as:

Fm,n(X)= [Fx(x)] n

(5)

357 which can be approximated by different expressions such as for example:

Fm,n(x)=n[1-Fx(x)] for high values of Fx(x) [9].

(6)

If one is interested in including in the model the duration of the loads, that is of the voyages, a stochastic process idealization can be used. Since the still-water loads are characterized by pulses of constant intensity that change intensity at specific points in time, they can be modeled by a renewal pulse process [10]. Thus each load occurrence is described by its intensity and duration which are themselves random variables following different probabilistic laws. A complete description of the load process must account for the fact that voyages alternate with periods in port, each of which have different time durations. This can be accomodated in an alternating renewal process as proposed in [10]. If the two different types of pulse duration in the alternating process are denoted by D 1 and D 2, the process D = D 1 + D 2 is an ordinary renewal process. For appropriately large values of the time t, the expected number of occurrences N of any pulse type is given by [11]: 0 2 _ #2 E [ N ( / ) ] = t. + _ 2_. 2 + O(1)

(7)

where . and a are the mean value and standard deviation of D and the terms of order of magnitude of one, O(1), are neglected. The moments of the process D are related with the ones of the individual processes by:

, =,1 +,2.

o2 = ot +

(8)

where .1, .2, 01 and o2 are the mean value and standard deviation of processes 1 and 2 respectively. The asymptotic expression for large values of t is E[N(t)] =

t/.

(9)

If the loads occur according to a Poisson process, the durations of D are exponentially distributed, which implies t h a t . and o are equal and the asymptotic expression is also the exact one. In an alternating renewal process, the probability p that the first pulse is acting at an arbitrary point in time is given by [10]: P = "1/("1 -F "2)

(10)

The rate of occurrence of each of the pulse processes is given by: = 1/. = 1/(.1 + .2)

(11)

and the mean duration of the load pulses 1 and 2 is .a and .2 respectively. This is a special feature of the alternating pulse processes, which have an arrival rate different from the inverse of the pulse duration. Renewal processes are special cases of counting processes which are completely described by the number of points occurring in a time interval t or by the interarrival time h. In stationary conditions these parameters depend o n . as can be observed in the previous two equations. For short periods the variance o 2 of the pulse durations is also required as indicated by eqn. (7). Typical values of these parameters are given in Table 1 for different ship types [10].

358 TABLE 1 Mean value (#) and standard deviation (0") of the data on duration of voyage and periods in port for different ship types Ship

Voyages duration (days)

Time in port (days)

type

N

/11

o1

N

/t 2

0"2

Container ship Bulk carrier Ore/oil carriers Small tankers Large tankers

72 105 114 501 169

3.7 15.7 27.9 12.5 23.5

3.5 5.3 10.3 11.0 11.3

22 107 111 541 197

1.0 11.7 3.1 5.1 4.9

0.8 8.8 2.4 7.0 6.3

The probability of the maximum of the process can be approximated by an upper bound on the probability of upcrossing a sufficiently high level of the process. This bound is established by noting that the probability Q(a, T) that the process X(t) exceeds level a in the period of time from 0 to t is given by: Q(a, t)=P[X(O)>a] + P [ X ( 0 ) < a ] - P [ N ( a , T)>~I] (12) where N(a, T) is the number of upcrossings of level a during time T. It can be shown that the following bound is obtained: Q(a, T) <~Q(a, 0) + [ 1 - O ( a , 0 ) ] E [ N ( a , T)] (13a) where the mean number of level upcrossings during time T is given by E [ N ( a , T)] = t,(a)T (13b) and p(a) is the mean upcrossing rate of level a. This bound can be specialized to the present pulse process to read [10]: Q(a, T) <~[1 - Fx(a)](1 + XT) (14) where Fx(x ) is the probability distribution of pulse intensity. This probability of exceeding level a is related to the probability of occurrence of the maximum of the process because for large values of a the probability of more than one upcrossing is negligible. When applying this bound to the vertical bending moments one must notice that exceedances can occur with two types of moments, which induce compression (sagging condition) or tension (hogging condition) in the deck. Typical values of the yearly probability of exceedance of the reference design value are indicated in Table 2. TABLE 2 Annual probabilities of exceedance Q(-, 1) of the design value of bending moments in hogging (100) or in sagging ( - 100) in individual ships of different type [10] Ship type

Load condition

Bending moment /~x °x

Bulk carrier

Ballast Loaded

13.5 - 7.9

33.7 27.6

Small tanker

Ballast Loaded

29.7 - 15.7

Large tanker

Ballast Loaded

12.8 - 32.6

~ year -1

Q (100, 1)

Q ( - 100, 1)

Q ( + 100, 1)

Qa (_+ 100, 1)

6.7 6.7

0.0393 0.0003

0.0031 0.0031

0.0424 0.0034

0.0458

21.7 18.7

10.4 10.4

0.0068 0.0000

0.0000 0.0000

0.0068 0.0000

0.0068

25.6 22.6

6.5 6.5

0.0023 0.0000

0.0000 0.0105

0.0023 0.0105

0.0128

359 2.3. Influence of human control on the maximum load effects

The stochastic model just described provides adequate predictions for most ships in many different time spans. However in the estimation of lifetime extremes of some ships it may occur that the model predictions result well above the limit value allowed by Classification Societies. In these cases that model is not appropriate and one must include also the effect of using the load distribution equipment on board. To study this problem the decision process of the ship master has been reproduced by a Monte-Carlo simulation [12], which yielded results very similar to the ones that had been observed in the analysis of operational data in [8]. In short, the ship master will load the ship with the cargo that he has to transport. If the load distribution equipment indicates that the cargo condition will imply an exceedance of the allowed value he will probably redistribute the cargo to result in a lower value. The probability of the cargo being redistributed will increase with the difference between the allowed limit and the intensity of the initial distribution, i.e., if the exceedance is only one or two percent the ship master may accept his initial guess but if it results in a 40% exceedance he will most certainly redistribute the cargo. The ship master's decision has been idealized in probabilistic terms and the resulting probability density function has indicated a clear decrease in the probability of occurrence of values larger than the design value. This effect is similar to a truncation but it is not a real one because some cases do exceed the allowed values. Thus, instead of adopting a truncated normal to represent the still-water load effect intensity, a truncation factor T R has been proposed in [12], so that the probability of exceeding the allowed limit, which has been normalised to +_100, is given by:

P[X>IOOI=TR[1-Fi(IO0)],

0~
(15)

where Fi(x) is the initial probability distribution of the load effect intensity. The modified probability density function fm(X) is related to the initial Gaussian distribution fi(x) by: fm(X) =

TFfi(x),

X < 100

(16)

where the new factor T F is given by: 1 - TR[1 - Fi(lO0)]

TF =

F~(100)

(17)

The factor T R is a measure of the efficiency of the control existing on board. Whenever there is no control, T R and T F are equal to one and the initial normal distribution is not changed. For a perfect control T R is zero and one has exactly a truncated distribution in which the maximum allowed value is never exceeded. In this case T F becomes equal to 1/Fi(100 ) which is the typical case considered in textbooks. The values of X exceeding the limit of 100, which will be denoted by X e ( = X - 100), will be described by a shifted exponential distribution given by

&(x)=(1/~'e)exp(-x/oo),

Xe>O

(lS)

where o~ is the scale parameter which is given by:

Oe= /0 Xe/m(Xo) dxo = Rfo Xo/i(Xe) axe

(19)

360 TABLE 3 Results of the calculations of distribution of maximum still-water bending moments in ship classes, indicating the probability of exceeding the design value Q(100) in two reference periods as well as the characteristic moment X~ for two return periods and for different values of the truncation parameter TR [12] Data Set

TR

Q(100) 1 year

Tankers sagging

0.25 0.50 1.00

Tankers hogging Container ships hogging Container ships sagging

20 years

Xc 1 year

20 years

0.073 0.141 0.262

0.781 0.952 0.998

101 105 124

122 165 196

0.25 0.50 1.00

0.017 0.034 0.066

0.317 0.497 0.747

90 92 98

112 144 172

0.25 0.50 1.00

0.456 0.952 0.998

0.999 1.00 1.00

115 128 144

130 156 171

1.00

0.000

0.1300

1

26

Taking eqn. (15) in consideration, the upper tail of the distribution is given by:

fro(X)= TR[a- 2Fi(ao0)]exp -

(X-100) oe

'

X > 100

(20)

With this description of the modified distribution it is possible to make predictions of the probability of exceedance by using eqn. (14) where now Fx(a ) is the modified distribution just described. The value of T R can be adjusted so that the model predictions agree with the observed data. In two specific data sets studied in [12] T R was found to be about 0.5 for a group of tankers and 0.25 for a containership. The probabilities of exceedance and the resulting characteristic values are indicated in Table 3. Significant differences can be observed from the comparison with the predictions without this effect, i.e., with T R equal to one.

3. WAVE-INDUCED LOAD EFFECTS The effect of the waves on floating structures is to change the longitudinal distribution of the buoyancy forces which results in changes of hydrostatic forces as well as in motions of the structures with the corresponding inertial forces. In general, motions and the induced forces are linearly proportional to the wave height. Thus they have the same general properties of the wave elevation process which is often modeled in two different time scales, the short-term and the long-term.

3.1. Short-term description The elevation of the sea surface is well represented by a non-stationary stochastic process which for appropriately short periods of time can be considered to be stationary. During these

361 periods, called sea states, the wave elevation is idealised as an ergodic zero mean Gaussian process. This implies that in the frequency domain the process is completely described by a power spectrum. The area under the spectrum is related to the variance of the process which is all the required information to completely describe a zero mean Gaussian process. In a narrow-band Gaussian process, the amplitudes follow a Rayleigh distribution. LonguetHiggins [13] showed that this model was adequate to describe sea states even when the narrow-band assumptions was not satisfied. In these cases the probability Qs(x) of the wave amplitude exceeding the value x in a sea state is given by Qs (x) = exp(

- x 2/2

R)

(21)

where R is the variance of the process, that is the zeroth moment of the spectrum (m 0 = R). In a broad-band process the Rice distribution is the appropriate description but Longuet-Higgins has shown that even in this case the Rayleigh distribution can be used if the parameter R is adjusted accordingly [14]. A sea state is characterised by its significant wave height H s which is related to the variance of the process by: Hs = 4 m~0

(22)

and by an average period which can be for example the mean zero upcrossing period T~, which is related to the zeroth and the second moment of the spectrum by: Tz = 2¢rm0~-~2

(23)

The response of a linear system to a stochastic excitation can be conveniently determined in the frequency domain in which case the spectrum of the response variable SR(t~ ) is obtained from the spectrum of the excitation S(to) by the application of a response amplitude operator which is the square of the transfer function H(to): SR(o~ ) = S(to)H2(to)

(24)

In reality the transfer function H depends not only on the encounter frequency to but also on the relative heading a between ship and waves, on ship speed V and even on its load condition C. Thus the previous expression should read: Sa(to , Hs, T~, or, V, C) = Sn(to, H~, T~)HE(to, a, V, C)

(25)

where S n is the spectrum of sea surface elevation. The variance of the response is the zeroth moment of the spectrum: R = f0°°SR(to) dto

(26)

In general, theoretical formulations of sea spectra are used to represent S n. Fully developed sea states are represented by the Pierson-Moskowitz spectrum [15], developing sea states by the JONSWAP spectrum [16] and the double peaked spectra of combined wave systems can be represented by a four-parameter model [17]. This latter model reduces to the JONSWAP spectrum when one of the peaks tends to zero, and the JONSWAP model tends to the Pierson-Moskowitz spectrum when the peak enhancement factor becomes equal to one. When predictions are wanted and one does not know which sea state will occur, Bayesian predictions can be made, using all the information from the theoretical models. The probability

362 of occurrence of each type of spectrum depends on the governing parameters of the sea state H~ and TZ. Thus, the expected value of the response variance to a given sea state can be obtained by weighting appropriately the variances to each type of spectrum by its probability of occurrence. This study has been reported in [12], which also allows assessments to be made about the uncertainty involved in those predictions. It was found that the common procedure of using always the Pierson-Moskowitz spectrum can lead to variance predictions that range from 0.7 to 1.3 of the correct expected value, depending on the combination of the values of the various variables in eqn. (24). It has also been shown [18] that whenever one is not interested in the response in one specific sea state but instead one looks for the mean response in a series of sea states, those effects tend to cancel out and very often the use of only the Pierson-Moskowitz spectrum in response calculations can give good results. An exception would be the fatigue load predictions for large ships, which would be underpredicted by that procedure. The transfer functions that are required to be used in eqn. (24) are usually determined by the use of strip theory methods, as for example [19]. These methods are based on certain assumptions that make their results have a different degree of accuracy depending on the governing parameters a, V and C (eqn. (24)). This has been clearly shown in [20] which includes systematic comparisons of the predictions of a linear response model [19] with measured responses of several ship models. Based on those results a probabilistic model error can be postulated which corrects the transfer function predictions. This allows improvements to be incorporated in the short-term prediction models, as reported in [21]. 3.2. Wave-induced loads in extreme sea states

Particularly important situations in short-term predictions of wave-induced load effects are the extreme sea states, which are of particular interest as reference design situations. As happened with the still-water load effects, an accurate prediction cannot be achieved in this situation if one does not account for the actions taken by the ship master. Specific measures are taken in heavy weather to reduce speed and change the ship course so as to aleviate the effects of the weather. This will change the probability distribution of V and a in those sea states and consequently the expected responses. A generalised belief has existed that those voluntary actions by the ship master would reduce the levels of wave-induced stresses on the structure. However, it was shown in [22] that the opposite happens in many situations. The ship response that is most easily detected on board is the transverse oscillation or the rolling of the ship, which is largest when the relative direction of the ship and waves differ by 90 degrees, i.e. the waves are comming from a transverse direction (beam seas). To avoid the possibility of capsizing as a consequence of large amplitude rolling, ship masters change the course of their ship so as to decrease the relative angle between ship and waves. However, although this decreases the transverse oscillations, it will induce significant increases of the wave-induced vertical bending moments which are the most important primary load effects. The effect of these changes has been studied in [22] with a Monte-Carlo simulation procedure. Based on interviews with ship masters, probabilistic based decision rules were established as to when and how voluntary course changes would be performed in heavy weather. A probabilistic distribution of relative ship headings was established from the analysis of operational data. This

363

distribution was used to sample initial values of the heading of the ship relative to the waves and then the probabilistic decision rules were applied to change the initial heading. The statistics of the results allowed a probability mass function to be derived for the headings in heavy-weather sea states. In short, the joint probability density function of the response variables was represented as a joint distribution of the sea state parameters H s and Tz and two probability distributions of heading and of velocity conditional on those parameters:

f(h s, tz, v, a)=fn~,r~(hs, tz)f~(alH~, Tz)fv(v[Hs, Tz, a)

(27)

The conditional distributions are derived from the results of a Monte-Carlo procedure. The specific feature of this formulation is the use of conditional distributions which were normaly used as marginal distributions i.e., a and v would be independent of H~ and T~. "Short-term responses can then be determined from the application of eqn. (24) with the appropriate transfer function. It was shown that the response variance can be underpredicted by as much as 30% if this effect is not taken into account [22].

3.3. Long-term description The long-term probabilistic description of wave-induced load effects have the same basic formulation as the ones of wave height since the first is obtained as a linear transformation of the latter. The long-term models of wave heights are obtained by combining the short-term probability of exceedance as given by eqn. (20), with the long-term distribution of the parameters that define the short term situation. The probability of exceedance QL(X) of the value x in the long term is given by: OL(X) =

fn(r)as(xlr)fR(r) dr

(28)

n(r) is a weighting factor that accounts for the relative duration of each sea state, and is the probability density function of the variance R as appropriate for the long-term time frame of reference. In the case of wave heights, R is substituted by H s as given by eqn. (21) and fR(r) becomes the joint distribution of wave parameters fns, r~ (h, t) that is appropriate to the ocean area and the time frame under consideration. An example of these different types of models applied to measured wave data from the Portuguese coast can be found in [23]. The time frame to which the data refers is important in that the long-term reference values which are predicted from the models, i.e., the 100-years design wave height, can vary by as much as 50% depending on the length of the data base even though it is from the same ocean area, as was shown in the predictions made for the North Sea in [24]. The other source of uncertainty is the long-term extrapolation model adopted. Alternatives to the model of eqn. (28) are formulations based on the occurrence of storms or short-term maxima. Significant differences can occur in the predictions of the 100-years wave height by these different methods even when they use the same data base, as was shown in [25]. One possible way to deal with the difference is to adopt a Bayesian formulation that combines the predictions of the different long-term models according to a subjective evaluation of their adequacy. This was proposed in [26] and applied to long-term predictions of wave height in a North Sea weather station. where

fR(r)

364

To apply the long-term formulation of eqn. (28) to ship responses it is necessary that the probability distribution of the response variance be related to all the variables on which it depends:

fR(r) = f ( h s , tz, v, t~, c ) = f ( h s ,

tz, u, a]C)fc(C )

(29)

where fc(c) is the probability density function of the cargo condition and the conditional distribution is given by eqn. (27). The long-term formulation can be improved by incorporating model uncertainty factors that correct for the effects described in the two previous sections. An interesting aspect that is worth mentioning is the uncertainty or the sensitivity of the long-term predictions to the input wave data either as concerns the predictions of extreme values [27] or of the fatigue load distribution [28]. Long-term reference values of the response can vary by as much as 50% whenever one uses different sources of wave statistics from the same ocean area. This is the same order of magnitude of the differences resulting from using data from different ocean areas as reported in the same data source. The implication that the uncertainty of a data source is of the same order of magnitude than the differences in results from various ocean areas points to the importance of improving the quality of the existing wave data sets.

4. COMBINATION OF LOAD EFFECTS It is often required to know the maximum of the combined value of still-water and wave-induced load effects. There is some correlation between the two load processes in that significant changes of deadweight may imply some changes in the wave-induced load effects. However this effect is not pronounced and has been neglected in the present study. The still-water loads have been modelled both as a random variable and as a stochastic process. Thus two formulations are possible to study the load combination process.

4.1. Ferry Borges-Castanheta model Ferry Borges and Castanheta proposed a representation of a stochastic process by a sequence of rectangular pulses of fixed duration [29]. The pulse amplitudes are represented by a random variable, and the probability distribution of the maximum in n repetitions is given by eqn. (5). Whenever one has two processes Xl(t) and X2(t) such that during each occurrence of Xl(t) there are m occurrences of process X2(t ), the distribution of the maximum combined value is [30]

fmax,T(X)=

(f_ fx,

n

Fx2
t30/

where z is a d u m m y variable and during the time period T there are n repetitions of X], the longest process. To apply this model to the combination of still-water and wave-induced vertical bending moments, as was reported in [30], it is required that the response to the waves be considered a sequence of independent and identically distributed pulses which correspond in fact to the wave cycles. Denoting the average duration of voyages by "q and the mean period of the waves by ~'2 implies that n = T/'q and m = "q/'r 2 in eqn. (30).

365 The Ferry Borges-Castanheta model has been shown to be very accurate for long-duration loads and to be a good model for code purposes [31,32].

4.2. Upcrossing rate solution In this formulation both loads are modelled as stochastic processes and the aim is to determine the rate of upcrossing of different load levels by the combined process. The still-water loads are represented by an alternating renewal pulse process as described in Section 2.2 and the wave-induced loads are modelled as a continuous process, exponentially distributed. Denoting again these processes by Xl(t ) and X2(t ), the upcrossing rate of the combined process is [30]: =

(31)

where Vx(X) is the upcrossing rate of level x by the process X(t). This expression is exact when the two processes are rectangular pulse processes. However, this expression is an upper bound to the true upcrossing rate for the sum of any two processes [33]. Furthermore, eqn. (31) is exact whenever the two processes satisfy the condition that at all times: P[3~'/(/)>0and Xj(/)<0] =0,

i, j = 1 , 2

(32)

where the dot indicates the time derivative. This condition insures that the increase'in process i due to its positive slope will not be cancelled by a decrease of the process j as a consequence of its negative slope. The combination of a rectangular pulse process with any other process satisfies these conditions because the probability of simultaneous changes of the processes is zero. Thus eqn. (31) provides an exact solution to the combined still-water and wave-induced components. Having determined the upcrossing rate of the process, estimates of the probability of exceeding given levels of loads can be obtained by the bound formulation of eqn. (13), as was reported in [30].

4.3. Load combination factors The design values of the load effects that are used in the Rules of Classification Societies are in general the ones that correspond to the 10 -8 probability level. It is convenient for code utilization to prescribe the combined load effect in terms of the same values. Thus, the c o m m o n approach is to multiply those values by a load combination factor smaller than unity. In the present case two formulations have been considered, yielding the combined m o m e n t M c as [30]: Mc = q~(Ms + Mw)

or

M c = M s + ~bMw

(33)

where the load factors q~ and q~ affect the sum of the characteristic values or only the wave-induced moment. Both of the two previously mentioned load combination formulations have been used to predict M~ and to derive the load combination factors. Some results of [30] are summarized in Table 4, as a function of the truncation factor (eqn. (15)) of the still-water load probability distribution. An alternative formulation that can be appropriate for code purposes is to adopt

366 TABLE 4 Load combination factors predicted by the Ferry Borges-Castanheta model and by the upcrossing rate solution [30] TR

Ferry Borges-Castanheta

Upcrossing solution

Tankers sagging

0.10 0.50

Ms+Mw 219 227

~ 0.97 0.96

~ 0.94 0.93

,~ 0.97 0.95

0.94 0.91

Container ships hogging

0.10 0.50

227 250

1.00 0.95

1.00 0.90

1.00 0.94

1.00 0.88

the Turkstra rule [34] and to obtain the maximum of the combined load effects as the maximum of each process when the other is at its expected value: Mc = max{(Ms + Mw); (~ts + Mw)}

(34)

where the bar denotes mean values. One must be aware that the Turkstra rule may underestimate the maximum combined values because it may occur when none of the loads are experiencing their maximum value in the reference period. Despite this drawback, this method is well suited for code formulations and has been used in [35] to derive partial safety factors for code requirements of primary ship structures.

5. CONCLUDING REMARKS An overview has been presented of recent formulations of primary load effects for ship structures. The major advances of recent formulations are based on the stochastic model of the still-water load effects, which improves the existing random variable formulations. This model allows the prediction of the probability of exceedance of this process and of its combination with the wave-induced load effects. Another significant aspect of the recent contributions is the incorporation of the effect of human actions in the occurrence of the extremes, both of the still-water and of the wave-induced load effects. Although there might be a degree of uncertainty in predicting human actions in those extreme situations, the estimates of the expected load effects are certainly improved. Finally, the combination of the load effects has been presented in a way that is especially appropriate for design codes i.e., in the form of load combination factors. The load models described here can be used to determine the probability of occurrence of a load effect at a random point in time or to derive extreme values appropriate for design. In the first case linear analysis is often adequate and fatigue strength is the major concern. Extreme loads are generaly associated with ultimate carrying capacity which requires a non-linear analysis for an adequate prediction.

ACKNOWLEDGEMENT This work has been partially funded by INIC, the National Institute for Scientific Research through CEMUL, the Center for Mechanics and Materials of the Technical University of Lisbon.

367

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