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Combination of primary load effects in ship structures
Probabilistic Engineering Mechanics 7 (1992) 103-111
Combination of primary load effects in ship structures C. Guedes Soares Department of Naval Architecture and Marine Engineering, Technical University of Lisbon ( Instituto Superior Tdcnico), Av. Rovisco Pais, 1096 Lisboa, Portugal (Received January 1991; accepted September 1991) The models currently used for representing the still-water and the wave-induced load effects in ship structures are discussed. Models based both on random variables and on stochastic processes are considered. Load combination solutions are obtained based on the Ferry Borges-Castanheta model for random variables and also on an upcrossing formulation for stochastic processes. Load combination factors appropriate for code purposes are suggested and their values are derived from comparison with the load combination results.
1 INTRODUCTION
scale of the order of the gross dimensions of the structure. The separation of the load components is not always unambiguous but in the case of ships the established procedure is to consider the still-water and the wave-induced components. 2 In the analysis, design and reliability evaluation of the ship structure one often considers different levels of detail in the structural behaviour. The main concern from a global point of view is the primary response in which the ship bends like a beam which is supported along its length by the vertical forces of the buoyancy.3 In addition to this global mode of deformation, ship components that are limited by stiff boundaries also deflect on a secondary and tertiary behaviour. Examples would be the behaviour of a deck panel between the ship sides and two transverse watertight bulkheads and the behaviour of a plate element in the deck between two longitudinal and two transverse stiffeners.4 This work is concerned with the primary ship structure, in which case the two most important load effects are the still-water and the wave-induced components. The first one depends basically on the longitudinal distribution of the cargo and results from the difference between the cargo and the buoyancy forces at the various transverse sections along the length of the ship. The second component results from the changes of buoyancy due to the presence of a wave and from the inertia forces associated with the wave-induced ship motions. The two load effects have been studied extensively in the past but not so many studies exist about their combined value. This results from the fact that for a long time the still-water load effects were considered as a
In structural design and in safety assessments it is often required to know the maximum value of the combination of various load effects. The main difficulty results from the fact that the load processes depend on variables that have different time variation. This implies that in general the maxima of all load effects will not occur simultaneously and thus the maximum of the combined process is, in general, smaller than the sum of the maxima of the individual loads. The choice of the probabilistic model for a particular type of load depends on the available information and on the application in mind. The most advanced models give information about the variation of the load intensity in time and space. However, the less sophisticated ones give the possibility of modelling most loads in a common framework, which is especially interesting to the study of load combinations. Some loads can even be treated as random variables. They experience a random value at a given point in time but, conditioned on that value, their intensity at any other point in space is quantified deterministically. In general, the loading of a structure can be represented as a sum of several components. Often there exists a gross, structure-free representation of the loading environment which combines with other factors that result from the interaction between the load and the structure, l These factors have a time scale of the order of the fundamental period of the structure and a spatial Probabilistic Engineering Mechanics 0266-8920/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 103
104
C. Guedes Soares
deterministic quantity based on the argument that knowIo edge about the cargo distribution would allow an accurate assessment o f the longitudinal distribution of the load effects. However, at the design stage, it is not possible to know how the ship will be loaded in successive tr/ps along her lifetime. Indeed, analysis of operational data has shown that the variability of the cargo embarked in successive voyages will make the load effects experience values that change in a r a n d o m manner as was demonstrated by non-parametric statistical tests. 5 Thus proposals have been put forward to model the still-water load effects probabilistically. The first models that have been proposed represented the intensity of the vertical bending m o m e n t in the rrfidship region as a normally distributed r a n d o m variableo 6'7 More recently stochastic models have been proposed to describe also their time variability. The stillwater load effects were represented as an alternating renewal pulse process. 8'9 The stochastic nature o f the wave-induced load effects had been recognised much earlier. The wave elevation during short-periods has been modelled as a stationary Gaussian process with zero mean, which becomes fully described by its variance. The wave elevation considered in a tong time framework is a non-stationary process which is studied by modelling the wave elevation as a sequence o f discrete short periods of stationarity which are characterised by parameters such as the significant wave height and the average period. The nonstationarity of the process is studied by modelling the long-term variability o f these sea state parameters} TM The wave-induced vertical bending moments are a linear response to the input process of sea surface elevatiom Thus they are also a stochastic process that can be described in the same fashion, both in the short term. ~'~ and in the long term} 4-I7 The deviation from iinearity that may occur for high sea states was analysed in R o t t3, where it was concluded that it was not large enough to discard the hypothesis o f linearity in the applications concerned with long-term, formulations. This work will study the maximum combined stillwater and wave-induced vertical bending moment using different types o f probabilistic models. A load combination rule is also proposed for design purposes and different values o f toad combination factors are derived, to be incorporated in codes o f ship structural design.
2 STILL-WATER LOAD EFFECTS The intensity of the stilLwater load effects in successive voyages can be modelled as a random variable which has been shown to be normally distributed, for each ship and each load condition. 7 This description can be generalised to account for the time dependency of the process. An alternating renewal process has been proposed in which the voyages and the time spent by the ships in port are
modelled as two pulse processes} These processes aiwaya occur in an alternating sequence so that if the duration, of each of them is D1 and Dz the process that has a duration D = D, + D a [s an ordinary renewal process Simple distribution-free results exist for this type of process. For exampte, for appropriately large ~,aiues e f the ume t the mean nurnbe~ of occurrences ~7(0 of each pulse type is given by (;2 _ /z E[N(0 ]
=
/~
+
2~ 2
w h e r e #: a n d ~ a r e t h e m e a n
o(f ~ ' ,.
(l;
vak~e a n d t h e s t a n d a r d
deviation of D and the terms with an order o f magnkude less than one, o(1), are neglected. The statistical moments of the process with duration D are related with the ones of the individuai processes by u -
#~ + #2
(2)
where tq, ~2, ffl and 0 2 are the mean value and stan.o dard deviation of the durations o f processes I and 2. respectively. The mean rate of occurrence o f each of the rm!se processes ~s given by 2 •
#
~'b #1 + #2
and the mean duration o f each pulse ts ~ and #:. This ~s a special feature o f the alternating pulse process, which has an arrival rate different from the iwverse o f the pu[se duration. Renewal processes are completely described by the probability distribution o f either the number of eomts occurring in a time [ntervai or of its mterarnva~ time between events. Thus, information about mean voyage durations, mean time in port and about the mean and standard deviation o f toad effect intensity ts a require°, merit fbr a complete description of the load process. Statements about Q(~, T), the probability of this process exceeding the level ~ during a period of T can be made using a bound formulation that starts from the equality
(5) "where N(~,T) is the number of upcrossmgs of' ~eve~ ~: during time T. Noting that the probability of more than one upcrossing is smaller than the mean number o ~ uncros~mgs~ .~ ~_~.
which is easily seen
j=2
by
making explicit these expressions
r=~'
Combination o f Primary Load Effects in Ship Structures For a stationary process satisfying the condition P [N(~,T) = j] ~ P [N(a,T) = l]
(7)
j=2
a good approximation for the mean number of level upcrossings is given by 18 E[N(a,T) = v+ T
(8)
where v+ is the mean upcrossing rate of level ~. It is defined as the limit as At tends to zero of the probability that the process is below a at a time t and over a at t + At v~+ = 2[1 -- Q(a,t)Q(~,t)
(9)
where 2 is the arrival rate in the case of point processes or the mean zero upcrossing rate for continuous processes. It expresses the probability that there is a change in the process in the period between t and t + At. Substituting the inequality (6a) and eqn (8) in eqn (5) leads to a useful bound 18'19
Q(a,r) <. Q(a,O) + [1 - Q(~,0)lv + T
(10)
In order to exceed high levels the term in square brackets must be very close to unity and can thus be omitted. In this case the term in square brackets also disappears from eqn (9), and eqn (10) becomes
Q(~,T) <~ [1 -- Fx(a)](1 + 2T)
(l l)
if one notes that the probability of the pulse exceeding the level a at any time depends on the distribution function of the process intensity
Q(a, t) =
1 - Fx(~)
(12)
The bound of eqn (11) can be applied to the still-water load effects in ships knowing the distribution Fx(x), which was determined in Ref. 7 for different ship types and the mean rate of occurrence, 2, which was reported in Ref. 8 for various ship types. 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 happen that the model predictions result in values well above the design value. 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 shipmaster has been reproduced by a Monte-Carlo simulation, 9 which yielded results very close to the ones that had been observed in the analysis of operational data in Ref. 7. In short, the shipmaster will load the ship with the cargo that he has to transport. If the load distribution equipment indicates that the cargo condition results in exceeding the allowed value he will probably redistribute the cargo to result in lower values of the load effects. The shipmaster's decision has been idealised 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
105
effect is similar to a truncation but it is not a real one because in some cases the loads do exceed the allowed values. Thus, instead of adopting a truncated normal to represent the still-water load effect intensity, a truncation factor TR has been proposed in Ref. 9, so that the probability of exceeding the allowed limit, which has been normalised to _ 100, is given by
P [ X > 100] = TR[1 -- F~(100)],
0 <~ TR <<. 1"0 (13)
where F~(x) is the initial probability distribution of the load effect intensity. The modified probability density function fro(X) is related to the initial Gaussian distribution f ( x ) by
fro(x) = Tel(x),
x < 100
(14)
where the new factor TF is given by
TF =
1 -- TR[1 -- Fi(100)]
(15)
E(100) The factor TR is a measure of the efficiency of the control existing on board. Whenever there is no control, TR and Tr are equal to one and the initial normal distribution is not changed. For a perfect control TR is zero and there is exactly a truncated distribution in which the maximum allowed value is never exceeded. In this case TF becomes equal to 1/F~(100), which is the typical case considered in textbooks. The values of X exceeding the limit of 100, which will be denoted by Xe(= X - 100), are described by a shifted exponential distribution given by
fx~(Xe) = (1/ffe)eXp(--Xe/ffe),
Xe > 0
(16)
where ae is the scale parameter which is given by
(Te
=
TR ~oo (X -- lO0)f(x)dx
(17)
To be more precise, this distribution could also be defined as a conditional density of Xe given that X > 100, i.e. fXelX>'O0, which is the implication of the condition X~ > 0 in eqn (16). However, to simplify the notation we stick to the one of eqn (16). Taking eqn (13) into consideration, the upper tail of the distribution is given by fm(X) = x > 100
TR[1 --o'eFi(100)] exp -- (X --~r~100~,./ (18)
which complements eqn (14). With this description of the modified distribution it is possible to make predictions of the probability of exceedance by using eqn (11) where now Fx(~) is the modified distribution just described. The value of TR can be adjusted so that the model predictions agree with the observed data. In two specific data sets studied in Ref. 9, TR was found to be about 0.5 for a group of tankers and 0.25 for a container ship.
C. Guedes Soares
106 3 WAVE-INDUCED LOAD EFFECTS
A zero-mean stationary Gaussian process can be completely described by a power spectrum, which has been used to describe the wave elevation in sea states. Theoro etical models exist to represent the average spectrum in fully developed sea states, a° in developing seas 2~ and in combined wave systems? 2 The linear response of a ship to that random excitation is also a stochastic process described by a respor~se spectrum S~(co) given by
&(co) = S.(o~)H~(co)
(i9)
where Su(m) is the wave elevation spectrum and H(m) is a transfer function depending on the encounter frequency ~o and on the relative heading between the ship and the direction of the wave system. The variance of any o f those spectra is given by R =
f ~ S(co)dco
(20)
In a narrow-band Gaussian process the amplitudes are Rayleigh distributed, so that the probability Qs of exceeding the amplitude x is given by: 23
Qs(xlR) = exp
- ~
(21)
The narrow-hand assumption does not always hold for the free-surface elevation but it is often a good model for the responses because the transfer function acts as a sort o f filter which reduces the frequency bandwidth o f the response. It should be noticed that Qs is fully described by R which is obtained from the knowledge of the spectrum. To obtain Q~(x), the probability of exceeding the load efl?ct x at a random point in time, one must weigh Qs(x) by the probability of occurrence of a given value of R and integrate over alt possible outcomes: 14q7
QAx) = fo Qs(xlr)f~(r)dr
= f(h s, ~z, O, v, c)dh, dtzdOdvdc
(23)
where the joint distribution o f the five variables is usually represented by the product o f several conditional distributions.~ 6,i 7 The tong-term distribution QL(x) calculated according to expressions (21)-(23) is usually fitted by a theoretical distribution. Experience has shown that one o f the best choices is the Weibull distribution, which is given by F)c(x) =
1 - exp -
Ship No. 1 Ship No. 2 Ship No 3 Ship No. 4 Ship No~ 5 Shio No. 6 Ship No 7 N, QUEEN HAVKONG BERGE COM. TAIMIR
1°08 I-!4 0°99 0-84 0-9~ 1.04 i-03 0-54 0.75 0.76 0'99
WOLV. STATE CAL. BEAR UNIV. ~RELAND £SSO MALAYS!A R. G FOLLIS FOTINI L ID, MARU BOSTON ESSO BONN TOYAMA Average
~,05 0-99 0.85 ~9,88 ~'63 0,97~ 0% ~.{~5 9.75 ~;~.% 085~
wave-induced vertical bending moments reported in Ref~ 24, 25 and the values that resulted for the exponent .a are. indicated in Table i. ~nspection of this table indicates that the shape pa> ameter A is often very ciose to unity aitho~.gh ~ the average somewhat lower. However additional resuI-.:~°s~ which were obtained from calculated transfer %ncuo~s resulted in exponents closer ~o unity. Th~s suggests tbe adoption of the simpler exponential distributions to reno resem the long-term distribution of waveoiv~duced bending moments, which ~as been used in Ref. 3 for example_ The adoption o f the ~ni~ exponen~ for eq~ ~2~) although being acceetable for the whole populatio~ of ships, cm3 be serious in error when applied to one spec~.fic ship. This is the general problem of defining reqmrements for codes that must be applied to classes of ships m structures and quantifying the load effec~ for one specific case. At any rate the formulation presented is valid for any value of the exponent arid thus can accommodate both cases. 4 PROBABILIST[C COMBINATION OF LOAD EFFECTS
(22)
where Qs(xtr)is given by eqn (21). The probability density function of the variance of a stationary period, fa(r) will depend on several variables such as the wave climate represented by /Is and Tz, the ship heading 0, speed v and loading condition c, i.e. f~(r)dr
r u n e 1o Exponent ef the Weiba~ DNtrib~ge~ t}~at was ~r~e~ te measured data of wave-indtuee~boning meme~t~ (Data from Refs 24, 25)
(24)
This distribution has been. fitted to data sets o f measured
The general objective in the design and operation of structural systems is to ensu~re that they wi!l be able ~-:o maintain their serviceability 'under ordinary loading conditions and that they will have a "verysmall probability of failure under extreme loads throughout their !ifetime. Classical structural reliability theory models the u n t o > tainties in s t r u c t ~ a l strength and in the loadi~g as random variables, not accounting explicitly for their time variability; However, even. this type of approa&~ ea;~ be used to quantify the time-odependent reliability as suggested in Refo 3. While the strength variables have a relatively narrow range of variation with time, the opposite occurs with the load effects. The most unfavourable conditions will often occur when there is a simultaneous occurrence of different types of loads at high levels of intensity° To take the time domain load fluctuation i~to co>~ sideration, one needs to model the loads as ray,dora
Combination of Primary Load Effects in Ship Structures processes. Failure will be the first crossing of a vector process out of a safety domain. However, because this is a complex problem, different simplifications have been proposed to transform the random process into a manageable random variable formulation. An especially important case is the linear system, in which the load effects S(t) are linear functions of the load intensity. In this case, the total load effect Si(t) is the sum of the various components si(t)
=
cuI j(t)
(25)
j=l
where Cij are coefficients transforming the load Lj into the load effect S~. The lifetime probability of failure in relation to the ith load effect is
Pr = P[R, <~ Sin,]
(26)
where Sm~ is the maximum value of the load effect S; during the lifetime T. This formulation substitutes the load effect process S~(t) by the random variable Sm~ allowing one to use the classical structural reliability analysis. The same approach can be applied to the reliability formulation for the combination of several load effects. This is why there is interest in predicting the maximum lifetime combined load effect. Examples of processes that are adequate for modelling several load effects and that allow an analytical solution to be obtained are the Gaussian process and the renewal pulse process, which can model, respectively, the linear responses to the excitation of the wave free surface elevation and the still-water loads. Mano and Kawabe 27developed one of the few published formulations of the probability distribution of the maximum combined still-water and wave-induced bending moments, considering the first one as a normally distributed variable. They formulated the probability distribution of the combined load effect x as
Fx(x)
=
[1 - Qw(x - y)] .f~(y)dy
(27)
where Qw is the exceedance probability of the wave load effect, fs is the probability density of the still-water load and n is the number of wave load cycles in the reference period. That formulation considers that both loads have the same time variation, i.e. in n wave cycles there will be n independent outcomes of the still-water loads. This implies that the lifetime characteristic value of still-water loads is obtained from the probability distribution of the maximum in 108 outcomes, because this is the reference value used for the number of wave cycles in a ship's lifetime. In fact, results presented in Ref. 26 indicate that the expected number of cycles is generally smaller than 108, with a value that depends on the type of the ship. However, the number of load conditions that a ship
107
experiences in her lifetime is only of the order of 103104. 7,8
Srding ~5represented the distribution of the combined load by a similar expression, without the exponent n. This is the probability distribution of the combined load at a random point in time. It does not model the repetitive nature of both load processes, and thus is not applicable to a duration larger than one wave cycle. The two formulations just referred to do not model the different time dependency of the two load processes. In reality, for each still-water load condition there may be as many as 104 wave cycles. These formulations also assume that the still-water and the wave-induced load effects are statistically independent. Mano & Kawabe 27 calculated the wave-induced bending moments in a ship's hull with five different displacements and obtained a negative correlation between the two load effects. However, all the five cases considered were from the same load condition, i.e. they had the same basic distribution of the load, with one peak at each of the quarter points along the ship's length. The changes of still-water bending moments were achieved by small load variations about these points. In addition, the changes were always symmetric, implying that there was no change of trim nor of displacement. However, it is exactly the changes of trim and displacement and the irregularities of the cargo distribution that will decrease the type of correlation that they obtained. Changing the deadweight will modify the underwater volume of the hull, leading to changes in the waveinduced loads. If it was not for the variations of trim, a strong correlation would be expected between the displacement and the wave-induced loads. However, the trim has the nature of a normally distributed random variable, as concluded in Ref. 5. It is expected that the variations of trim for a given value of displacement will diminish the correlation between wave-induced load effects and deadweight so that their correlation with the still-water load effects will be smaller than the value of - 0 - 5 0 mentioned above. Support for this viewpoint can also be found in the study of Lotveit et al.,28 who conducted a series of experiments with models of the same hull with different distribution of weights. They found that different distributions of weights produced different wave-induced bending moments, as noticed also by others. 29'3° However, they concluded that the variations of the wave-induced bending moments could not be described by the values of the still-water bending moment. In conclusions, it appears that the assumption of independence between still-water and wave-induced load effects may be adequate for practical purposes, and it has been adopted in the formulations described hereafter.
4.1 Ferry Borges-Castanheta model The model proposed by Ferry Borges and Castanheta 3t is
Co Guedes Soares
108
one o f the simplest load models, which assumes that the loads change intensity after prescribed deterministic, equal time intervMs, during which they remain c o n s t a n t The intensity o f the loads in the different elementary time intervals are outcomes o f identically distributed and mutually independent r a n d o m variables. The probabilistic distribution Freer o f the maximum value during n repetitions o f the toad i or equivalently during the time T = m:, where ~cis the pulse duration, is given by G,~(x)
:
[fAx)] ~
(2~)
where F~(x) is the probability distribution function o f the load intensity. The application of this formulation to model the stillwater load effects requires few assumptions since a rec~ tanguiar putse renewal has been shown in Section 2 to be a good representation of that load. Instead o f having pulses with different duration, the Ferry Borges-Castanheta model would require all pulses to be o f equal duration, which should be the average voyage duration. These durations range from 2 days for container ships to 28 days for large tankers} The approximation involved in the Ferry BorgesCastanheta model would not be a good one for periods corresponding to a few voyages, but it would gradually become better as the number o f voyages increases, and would be a good model for the maximum lifetime load. In this time span a ship will perform an integer number o f voyages, which fits well into the assumption o f the Ferry Borges-Castanheta model. To adopt this model to represent also the maximum wave-induced load effect requires that the time variation o f the process be substituted by a sequence of pulses, each representing the amplitude of a cycle of the response. The assumption of independence between successive cycles of response is not exact because waves separated by a few cycles are correlated. Watson 3~ studied this type of problem, defining a r a n d o m variable to be m-depeno dent when the variables in a sequence separated by more than m I outcomes are statistically independent. Watson has shown that the limiting distribution for the maximum o f m-dependent variables is the same as the limiting distribution of independent random variables. The correlation between wave heights is restricted to a small number of cycle% especially when considering the time scale o f long-term variability. Thus the assumption of independence seems reasonable for predictions of lifetime maxima. Another way of looking at the problem is to use a resuk o f Cramer 3~ who showed that the excursions of a high levet by a Gaussian process are asymptotically Poisson distributed° This implies that the exceedances o f high levels are independent° It is precisely the high levels of the wave-induced loads that are o f interest for the
predictions o f extreme values m which zase indepeno.. deuce can be assumed. Finally, because in a ship's ~ifetime there are many different toad conditions and daring each voyage there occur very many wave cycles, it is no major restn ction to consider that their number is an integer° Therefore, the main assumptions required by the Ferry BorgesCastanheta model are satisfied° Consider now two load process X~(0 and X2U) of this type with durations ~:i, and h , si~ch that ~, > 'c~ ~" oo~e is interested in the process ~<(,t) =
x,{t) + &(t)
(29.
the probability distribution of rise maximum of me combined processes during time T can be detennm.ed exactly provided that the two processes are in phase, and that khe ratios T 2 / Z [ =: m and T/z = ;~ are integers. ]l~ this case one has GA
X)
:
L (z)[G--<~ --
z)]mdz!
<~0)
This model could be extended to consider more than ~wo loads, using the same arguments° To apply ~.t ~o the distribution of the maximum combined stiHowater and wave-induced load one must identi~J the first one with Y and the second one with t l F r o m the preceding &s~. cuss,~on ~t is easily seen that the requiremem:s about t~:e ratio of durations and their being in phase are not very restrictive. A series of calculations has been performed for u~e application of this model to the cases of tankers and container ships. The distribution of stilt-water load effects was described by a normal distribution anc dif< ferent degrees o f truncation v,mre assumed, to mode~ th3 effect of h a m a n control. 9 F o r tankers the mean value and standard deviation of the distribution were in sagging, respective!y, !2.4% and 21.3% o~ U~e nomma~ value ,o'~ still-water vertical bending moment prescribed by the rules of Classification Societies, while for container ships ~hey were 71.5% and 16-3% b_ hogging, in accord~ncc with Ref. 7_ The mean voyage durations are assumed tc be 3.7 days for container ships and 23.5 days for m~ kers tn this description, the mean vMues e f the load effects are very small which imply very large coefficients o? variation. This results from the fact that tankers operate roughly 50% of the time loaded and 50% in bMlast b" the former situation they usually experience hogging stresses_ which are assumed to be posmve~ and in ballast they experience sagging stresses which are m e d e t b d as negat~_ve. Thus their average is very ~mait which results i;~. coefficients of variation around 1-7 However° if e~e considers only loaded or baitast conditions this va~.e decreases ) o r e i.? to around 0.5. The wave-induced ioads were represented by an exponential distribution with the parameters derived from tl:~e design wave-induced bending momen~ given in the m~e~ of Classification Societies. F o r exampteo in the case of th~
Combination o f Primary Load Effects in Ship Structures
109
Table 2. Results of the load eombination calculations using the Ferry Borges-Castanheta model, in terms of total load (Ms + Mw) or of the corresponding load combination factors (eqns (30), (31))
TR
1 year
20 years
30 years
Ms + Mw
q$
~k
Ms + Mw
q$
~b
Ms + Mw
¢
~k
Tankers sagging
0.10 0-50 0.90
219 227 247
0-97 0-96 0.94
0-94 0.93 0.88
259 311 337
0.95 0-94 0.91
0.91 0-90 0.78
263 322 354
0.94 0.90 0-90
0.90 0.78 0.78
Container ships hogging
0.10 0-50 0-90
227 250 282
1.00 0.95 0.84
1-00 0-90 0-63
256 302 342
0.96 0.90 0.79
0.92 0.79 0-51
260 310 353
0.95 0.83 0.78
0.91 0-64 0.48
Det norske Veritas rules of 1989, this requirement corresponds to the characteristic moment at the probability level of 10 -8 divided by 1.7. This information is enough to estimate the parameter of the exponential distribution. The mean wave period over all sea states was adopted as the duration of the wave load process. This is an approximation of a degree of accuracy that can be compared with other load modelling assumptions. It reflects the positive correlation that exists between wave height and period. While for low sea states one can have periods around 5 to 6s, for high significant wave heights they can be as large as 16s. This effect can be taken into account in the numerical integration just be making m in eqn (30) a function of (x - z), i.e. m will decrease with increasing (x - z). Some effect is expected on the characteristic values, although not a large one. The probability distribution of the combined load effect was obtained by numerical integration of eqn (23). Table 2 presents the characteristic values of the total load (Ms + Mw) for the probability levels corresponding to return periods of 1, 20 and 30 years. The load effects have been normalised by the Rule value of the still-water vertical bending moment, which corresponds to 100. Accounting for the differences in the definitions of nominal values of Ms and Mw, the nominal value of (Ms + Mw) would be 214, which should be compared with the ones indicated in Table 2. 4.2 Upcrossing rate solution The upcrossing rate for a scalar stationary process has been established by Rice as v+(e) =
=0 2fx, x,(ct, 2)d2
I~ v+(~'t)" dt
fi.~(y,9)
=
x-= - 7-_
(32)
which is valid for any non-stationary process and becomes v T for the stationary case.
i"1 = - ~
xfxd2(y-
I'XI
x')~:I
x,~-
~71)d2~-dx
(33)
Substituting now this expression on the upcrossing rate expression results in ......
tl ....
+2=--¢1
Xlfxi'£i(X'J$1)
× fx2.x2(~ - x, 22)d22d21dx
x fx~.s~2(~ - x, -t2)d-tzd2~ dx
(34)
The integrals can only be calculated in special cases. However they can be used as a basis to establish an upper bound on the upcrossing rate 34
+ f<~-oo v[ (o~ - x ) ' f x , ( x ) d x
(35)
It can be shown that this upper bound for any stationary stochastic process is in fact the exact result when both load components are renewal rectangular pulse processes, and the peak values are mutually independent and are also independent of pulse durations] 9 In fact, this equation is exact when the two processes satisfy the following conditions at all times t P[L(t) > 0 a n d L ( t )
(31)
wherefx;(X,2) is the joint probability density function of X(t) and J?(t) and v+ (e) is the mean upcrossing rate of level c~. Having the upcrossing rate, one can determine the mean number of upcrossings of level e in time T as E[N(~,T) =
Consider now the process Y(t) that is a linear combination of the independent stochastic processes X~(t) and X2(t) as indicated in eqn (29). This joint densityfi.~(y,D can be determined from processes X~ and X2:
< 0] =
0,
i, j, = 1,2 (36)
This condition ensures that the increase in process i due to its positive slope will not be cancelled by a decrease of process j as a consequence of its negative slope. Rectangular pulse processes have a non-zero slope only at two time instants: the beginning and ending of the pulse. Thus, the combination of a rectangular pulse process with any other process satisfies the above equation because the probability of simultaneous changes in both processes is zero. This means that eqn (9) provides the
t t0
C. Guedes Soares
Tab|e 3o Results of the load eomN~ation calculations using the nperossing rate sol~tion, i~ t e r ~ of torn ~oad (Ms 4- ?~2~,) er ef ff-~ corresponding load combination factors (eqns (30), (31))
Te
1 year
20 years
3~ years
Tankers sagging
0.I0 0-50 @90
219 227 247
0,97 0.95 0,93
0-94 0.91 0.86
259 3!~ 337
0-9t 0-89 0-85
0-84 0-76 0-61
263 322 354
0-9i 0~85 ~}.8~
0°84 0,~8 f) 5~
Container ships hogging
0-10 0-50 0.90
227 250 282
1,0{} 0"94 0.82
1.00 0-88 0.60
256 302 342
@95 0.84 ~72
0.90
260 310 353
0-93 ~)'83 0.7~
~-88 0°64 :3-3~
exact upcrossing rate of the combined still-water and wave-induced processes. The degree o f approximation o f this upper bour~d can be assessed by referring to eqn (11). The approximation is good when the probability o f occurrence of more than one upcrossing in the reference period is much smaller than the probability of one upcrossing (eqn (7)). This condition is often satisfied when determining the lifetime maximum o f the processes. This formulation has been used to determine the probability distribution of the combined still-water and wave° induced bending moments for tankers and container ships. The truncated distribution of the still-water loads was used, as given by eqns (7)-(i 1). The long-term distrio bution of wave-induced loads is given by eqns (t5) and (17) and the upcrossing rates are determined by eqn (35). The results o f the calculations are shown in Table 3 for the truncation levels of 0.1 and 0.5, which are more realistic to expect than 0.9. The results are very close to the ones obtained by the Ferry Borges-Castanheta model, which confirms the equivalence of the f o r m > lotions, as has been pointed out by Turkstra & Madsen, ~4 when studying other load models. 4.3 Load combinat~o~ ru~es
Defining Ms and Mw as the characteristic values o f the stiil-water and wave-induced load effects, the combined load effect L can be expressed by either of the following expressions L =
~ , (Ms + Mw)
(37)
5
e~s + ~ o M w
(38)
=
where 4~ and ff are factors that represent the load combination effect. The factor ~b has the character o f a bias while ~ is a load reduction factor. Both o f these systems are found in design codes. Examples o f the first one are in the National Building Code o f C a n a d a and the code o f the American Concrete Institute. The second format is adopted by the European Committee o f Pro-Stressed Concrete and by the CEBCIB-ECCS-FIP-IABSE-IASS-RILEM Joint Committee on Structural Safety) s For exampte, the 1989 ship rules of Dot norske Veritas adopt atso the tast format since the
0.67 0°43
nominai value of wave-induced bending moments ~ divided by to7, before being added to the nominal value of stilbwater bending moment. ToNe 3 indicates the sum of the characteristic vaiue o£ each individual load effect (/ffs - M~w) ar~ the values of 4) anc~ ~p derived from the characteristic value of ti~e combined b a d . Anatysis of the results indicates that the b a d combination factors depend on ~.l-~e probabiff~j level, as was previously observed by Turkstra g< Madsen. 34 Higher probability !evels lead ;o sinai!or values of the coefficients. The bias ~b had values between 0-95 and @80 whiie tf~e load reduction factor ~ varied between 0.90 and ©o5~ The variation depended not o n b / o n the probabiiity levei bm also on the degree o f truncation. For a m~id. ~runcat|on effect of the stilt-water load effects ~he ~oad co_,m.o bination coefficients diff?r significantly from oa_e However, for a strong truncation (TR = 0.~) :hey become d o s e to unity. This ~s because when there is a~ heavy truncation effect the stiltowater load effects a~'e ciose -~o their characteristic values so ores. tha~ I? becomes likely for the maximum waveqnduced ~oaC effect ~o occur when the still-wamr is a~ a high level A finai comment on the load combination fhci'~ors obtained is that they refer on!y to two ioads which m addition are of the sustained ~ype, i.e. during opera,son the structure is permanently subiected to both of ~hern~ despite the fact that high wave loads seldom occur. ~£fi~/e or six toads were considered as m the ease o f civil engineering smxctures, and if some were of ~raasient nature and short duration as are earthquake load~, smaller ic~ad combination Nctors could be expected. It is worth noting ehat ~:he eff%ct of' truncate.on tr @~;~c of the load models has not been treated previously in ar.~., of the toad combination studies regarding civil engmcer.ing load models] 93~-~4
5 C O N C L U D I N G RE}/~ARKS
This work has addressed the assessmen~ o f ti~e ~ong4.erm value of the combined stilt-water and wave-induced load effects ix snip structures. Solutions have been obtained
Combination of Primary Load Effects in Ship Structures by modelling both load effects as a sequence o f independent and identically distributed random variables and as two stochastic processes. In this latter case upcrossing rates of the combined process were determined and the probability distribution o f maxima was based on a bound formulation. L o a d combination factors have been introduced to express the combined load effect as a function of the characteristic values o f the individual processes. This approach, which is aimed at adoption in design codes, has been compared with the predictions of the combined load effects so as to determine the corresponding values of those factors. REFERENCES
1. Cornell, C.A. & Larrabee, R.D., Representation of Loads for Code Purposes. Proc. 2nd Int. Conf. on Structural Safety and Reliability (ICOSSAR 77), Munich, 1977. 2. Guedes Stares, C., Stochastic Models of Load Effects for the Primary Ship Structure. Structural Safety, 8 (1990) 353-68. 3. Guedes Stares, C. & Ivanov, L.D., Time Dependent Reliability of the Primary Ship Structure. Reliability Engineering and System Safety, 26 (1989) 59-71. 4. Guedes Stares, C., Uncertainty Modelling in Plate Buckling. Structural Safety, 5 (1988) 17-34. 5. Guedes Stares, C. & Moan, T., Statistical Analysis of Still-Water Bending Moments and Shear Forces in Tankers, Ore and Bulk Carriers. Norwegian Maritime Research, 10(3) (1982) 33-47. 6. Mano, H., Kawabe, H., Iwakawa, K. & Mitsumune, N., Statistical Character of the Demand on Longitudinal Strength (Second Report) - - Long Term Distribution of Still-Water Bending Moment. J. Soc. Naval Architects of Japan, 142 (1977) 255-63 (in Japanese). 7. Guedes Stares, C. & Moan, T., Statistical Analysis of Still-Water Load Effects on Ship Structures. Transactions Society of Naval Architects and Marine Engineers, 96 (1988) 129-56. 8. Guedes Stares, C., Stochastic Modelling of Maximum Still-Water Load Effects in Ship Structures. Journal of Ship Research, 34(3) (1990) 199-205. 9. Guedes Stares, C., Influence of Human Control on the Probability Distribution of Maximum Still Water Load Effects in Ships. Marine Structures, 3 (1990) 319-39. 10. Haver, S., Wave Climate off Northern Norway. Applied Ocean Research, 7 (1985) 85-92. 11. Guedes Stares, C., Lopes, L.C. & Costa, M.D.S., Wave Climate Modelling for Engineering Purposes. Computer Modelling in Ocean Engineering, ed. B.A. Schrefler & O.C. Zienkiewicz. A.A. Balkema Pub., Rotterdam 1988, pp. 169-75. 12. Guedes Stares, C., Effect of Spectral Shape Uncertainty in the Short Term Wave Induced Ship Responses. Applied Ocean Research, 12 (1990) 54-69. 13. Guedes Stares, C., Effect of Transfer Function Uncertainty on Short-Term Ship Response. Ocean Engineering, 18 (1991) 329-62. 14. Fukuda, J., Theoretical Determination of Design Wave Bending Moments. Japan Shipbuilding and Marine Engineering, 2(3) (1967) 13-22. 15. S6ding, H., Calculation of Stresses on Ships in a Seaway. Schiff und Hafen, 23 (1971) 752-62. 16. Guedes Stares, C. & Viana, P.C., Sensitivity of the Response of Marine Structures to Wave Climatology.
111
Computer Modelling in Ocean Engineering, ed. B.A. Schrefler & O.C. Zienkiewicz. A.A. Balkema Pub., Rotterdam 1988, pp. 487-92. 17. Guedes Stares, C. & Trov~o, M.F.S., Influence of Wave Climate Modelling on the Long-Term Prediction of Wave Induced Responses of Ship Structures. Dynamics of Marine Vehicles and Structures in Waves, ed. W.G. Price, P. Temarel & A.J. Keane. Elsevier Applied Science, Amsterdam, 1991, pp. 1-10. 18. Leadbetter, M.R., Extreme Value Theory and Stochastic Processes. Proc. 1st Int. Conf. on Structural Safety and Reliability (ICOSSAR), ed. A.M. Freudenthal. Pergamon Press, 1972, pp. 71-89. 19. Larrabee, R.D. & Cornell, C.A., Combination of Various Load Processes. J. Structural Div. (ASCE), 107 (1981) 223-39. 20. Pierson, W.J. & Moskowitz, L., A Proposed Spectral Form for Fully Developed Wind Seas Based on the Similarity Theory of S.A. Kitaigorodskii. J. Geophysical Research, 69 (1964) 5181-90. 21. Hasselmann, K., et al., Measurements of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP). Deutsche Hydrographische Zeitschrift, A8(12). 22. Guedes Stares, C., Representation of Double-Peaked Wave Spectra. Ocean Engineering, 11 (1984) 185-207. 23. Longuet-Higgins, M.S., The Statistical Distribution of the Height of Sea Waves. J. Marine Research, 11 (1952) 245-66. 24. Nordenstr6m, N., Further Analysis of Full-Scale Measurements of Midship Bending Moments. Report, Division of Ship Design, Chalmers University of Technology, Gothenburg, 1965. 25. Lindemann, K., An Evaluation of the Need to Measure Stresses in Practical Hull Surveillance Systems. Report 79-0603, Det norske Veritas, 1979. 26. Guedes Stares, C. & Moan, T., Model Uncertainty in Long Term Distributions of Wave Induced Bending Moments for Fatigue Design of Ship Structures. Marine Structures, 4 (1991) 295-315. 27. Mano, H. & Kawabe, H., Statistical Character of the Demand on Longitudinal Strength (Third Report) - Long-Term Distribution of Combined Bending Moment. J. Soc. Naval Architects of Japan, 144 (1978) 240-8 (in Japanese). 28. Lotveit, M., Mfirer, C., Vedeler, B. & Christensen, H., Wave Loads on a T2-Tanker. The Influence of Variations of Weight Distribution with Constant Mean Moment of Inertia on Shearing Forces and Bending Moments in Regular Waves. European Shipbuilding, 10 (1961) 2-32. 29. Vossers, G., Swaan, W.A. & Rijken, H., Vertical and Lateral Bending Moment Measurements on Series 60 Models. Int. Shipbuilding Progress, 8 (1961) 302-20. 30. Swaan, W.A. & Joosen, W.P.A., The Influence of Weight Distribution on Wave Bending Moments. Int. Shipbuilding Progress, 12 (1965) 389--407. 31. Ferry Borges, J. & Castanheta, M., Structural Safety. Laborat6rio Nacional de Engenharia Civil, Lisboa, 1971. 32. Watson, G.S., Extreme Values in Samples from m-dependent Stationary Stochastic Processes. Ann. Math. Statistics, 25 (1954) 798-800. 33. Cramer, H., On the Intersections Between the Trajectories of a Normal Stochastic Process and a High Level. Arkiv. Mat., 6 (1966) 337-49. 34. Turkstra, C.J. & Madsen, H.O., Load Combinations in Codified Structural Design. J. Structural Div. (ASCE), 106 (1980) 2527-43. 35. Ferry Borges, J., et aL, (Eds) General Principles on Quality Assurance. Int. Assoc. Bridge Struct. Eng., Zurich, 1981.
Combination of primary load effects in ship structures C. Guedes Soares Department of Naval Architecture and Marine Engineering, Technical University of Lisbon ( Instituto Superior Tdcnico), Av. Rovisco Pais, 1096 Lisboa, Portugal (Received January 1991; accepted September 1991) The models currently used for representing the still-water and the wave-induced load effects in ship structures are discussed. Models based both on random variables and on stochastic processes are considered. Load combination solutions are obtained based on the Ferry Borges-Castanheta model for random variables and also on an upcrossing formulation for stochastic processes. Load combination factors appropriate for code purposes are suggested and their values are derived from comparison with the load combination results.
1 INTRODUCTION
scale of the order of the gross dimensions of the structure. The separation of the load components is not always unambiguous but in the case of ships the established procedure is to consider the still-water and the wave-induced components. 2 In the analysis, design and reliability evaluation of the ship structure one often considers different levels of detail in the structural behaviour. The main concern from a global point of view is the primary response in which the ship bends like a beam which is supported along its length by the vertical forces of the buoyancy.3 In addition to this global mode of deformation, ship components that are limited by stiff boundaries also deflect on a secondary and tertiary behaviour. Examples would be the behaviour of a deck panel between the ship sides and two transverse watertight bulkheads and the behaviour of a plate element in the deck between two longitudinal and two transverse stiffeners.4 This work is concerned with the primary ship structure, in which case the two most important load effects are the still-water and the wave-induced components. The first one depends basically on the longitudinal distribution of the cargo and results from the difference between the cargo and the buoyancy forces at the various transverse sections along the length of the ship. The second component results from the changes of buoyancy due to the presence of a wave and from the inertia forces associated with the wave-induced ship motions. The two load effects have been studied extensively in the past but not so many studies exist about their combined value. This results from the fact that for a long time the still-water load effects were considered as a
In structural design and in safety assessments it is often required to know the maximum value of the combination of various load effects. The main difficulty results from the fact that the load processes depend on variables that have different time variation. This implies that in general the maxima of all load effects will not occur simultaneously and thus the maximum of the combined process is, in general, smaller than the sum of the maxima of the individual loads. The choice of the probabilistic model for a particular type of load depends on the available information and on the application in mind. The most advanced models give information about the variation of the load intensity in time and space. However, the less sophisticated ones give the possibility of modelling most loads in a common framework, which is especially interesting to the study of load combinations. Some loads can even be treated as random variables. They experience a random value at a given point in time but, conditioned on that value, their intensity at any other point in space is quantified deterministically. In general, the loading of a structure can be represented as a sum of several components. Often there exists a gross, structure-free representation of the loading environment which combines with other factors that result from the interaction between the load and the structure, l These factors have a time scale of the order of the fundamental period of the structure and a spatial Probabilistic Engineering Mechanics 0266-8920/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 103
104
C. Guedes Soares
deterministic quantity based on the argument that knowIo edge about the cargo distribution would allow an accurate assessment o f the longitudinal distribution of the load effects. However, at the design stage, it is not possible to know how the ship will be loaded in successive tr/ps along her lifetime. Indeed, analysis of operational data has shown that the variability of the cargo embarked in successive voyages will make the load effects experience values that change in a r a n d o m manner as was demonstrated by non-parametric statistical tests. 5 Thus proposals have been put forward to model the still-water load effects probabilistically. The first models that have been proposed represented the intensity of the vertical bending m o m e n t in the rrfidship region as a normally distributed r a n d o m variableo 6'7 More recently stochastic models have been proposed to describe also their time variability. The stillwater load effects were represented as an alternating renewal pulse process. 8'9 The stochastic nature o f the wave-induced load effects had been recognised much earlier. The wave elevation during short-periods has been modelled as a stationary Gaussian process with zero mean, which becomes fully described by its variance. The wave elevation considered in a tong time framework is a non-stationary process which is studied by modelling the wave elevation as a sequence o f discrete short periods of stationarity which are characterised by parameters such as the significant wave height and the average period. The nonstationarity of the process is studied by modelling the long-term variability o f these sea state parameters} TM The wave-induced vertical bending moments are a linear response to the input process of sea surface elevatiom Thus they are also a stochastic process that can be described in the same fashion, both in the short term. ~'~ and in the long term} 4-I7 The deviation from iinearity that may occur for high sea states was analysed in R o t t3, where it was concluded that it was not large enough to discard the hypothesis o f linearity in the applications concerned with long-term, formulations. This work will study the maximum combined stillwater and wave-induced vertical bending moment using different types o f probabilistic models. A load combination rule is also proposed for design purposes and different values o f toad combination factors are derived, to be incorporated in codes o f ship structural design.
2 STILL-WATER LOAD EFFECTS The intensity of the stilLwater load effects in successive voyages can be modelled as a random variable which has been shown to be normally distributed, for each ship and each load condition. 7 This description can be generalised to account for the time dependency of the process. An alternating renewal process has been proposed in which the voyages and the time spent by the ships in port are
modelled as two pulse processes} These processes aiwaya occur in an alternating sequence so that if the duration, of each of them is D1 and Dz the process that has a duration D = D, + D a [s an ordinary renewal process Simple distribution-free results exist for this type of process. For exampte, for appropriately large ~,aiues e f the ume t the mean nurnbe~ of occurrences ~7(0 of each pulse type is given by (;2 _ /z E[N(0 ]
=
/~
+
2~ 2
w h e r e #: a n d ~ a r e t h e m e a n
o(f ~ ' ,.
(l;
vak~e a n d t h e s t a n d a r d
deviation of D and the terms with an order o f magnkude less than one, o(1), are neglected. The statistical moments of the process with duration D are related with the ones of the individuai processes by u -
#~ + #2
(2)
where tq, ~2, ffl and 0 2 are the mean value and stan.o dard deviation of the durations o f processes I and 2. respectively. The mean rate of occurrence o f each of the rm!se processes ~s given by 2 •
#
~'b #1 + #2
and the mean duration o f each pulse ts ~ and #:. This ~s a special feature o f the alternating pulse process, which has an arrival rate different from the iwverse o f the pu[se duration. Renewal processes are completely described by the probability distribution o f either the number of eomts occurring in a time [ntervai or of its mterarnva~ time between events. Thus, information about mean voyage durations, mean time in port and about the mean and standard deviation o f toad effect intensity ts a require°, merit fbr a complete description of the load process. Statements about Q(~, T), the probability of this process exceeding the level ~ during a period of T can be made using a bound formulation that starts from the equality
(5) "where N(~,T) is the number of upcrossmgs of' ~eve~ ~: during time T. Noting that the probability of more than one upcrossing is smaller than the mean number o ~ uncros~mgs~ .~ ~_~.
which is easily seen
j=2
by
making explicit these expressions
r=~'
Combination o f Primary Load Effects in Ship Structures For a stationary process satisfying the condition P [N(~,T) = j] ~ P [N(a,T) = l]
(7)
j=2
a good approximation for the mean number of level upcrossings is given by 18 E[N(a,T) = v+ T
(8)
where v+ is the mean upcrossing rate of level ~. It is defined as the limit as At tends to zero of the probability that the process is below a at a time t and over a at t + At v~+ = 2[1 -- Q(a,t)Q(~,t)
(9)
where 2 is the arrival rate in the case of point processes or the mean zero upcrossing rate for continuous processes. It expresses the probability that there is a change in the process in the period between t and t + At. Substituting the inequality (6a) and eqn (8) in eqn (5) leads to a useful bound 18'19
Q(a,r) <. Q(a,O) + [1 - Q(~,0)lv + T
(10)
In order to exceed high levels the term in square brackets must be very close to unity and can thus be omitted. In this case the term in square brackets also disappears from eqn (9), and eqn (10) becomes
Q(~,T) <~ [1 -- Fx(a)](1 + 2T)
(l l)
if one notes that the probability of the pulse exceeding the level a at any time depends on the distribution function of the process intensity
Q(a, t) =
1 - Fx(~)
(12)
The bound of eqn (11) can be applied to the still-water load effects in ships knowing the distribution Fx(x), which was determined in Ref. 7 for different ship types and the mean rate of occurrence, 2, which was reported in Ref. 8 for various ship types. 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 happen that the model predictions result in values well above the design value. 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 shipmaster has been reproduced by a Monte-Carlo simulation, 9 which yielded results very close to the ones that had been observed in the analysis of operational data in Ref. 7. In short, the shipmaster will load the ship with the cargo that he has to transport. If the load distribution equipment indicates that the cargo condition results in exceeding the allowed value he will probably redistribute the cargo to result in lower values of the load effects. The shipmaster's decision has been idealised 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
105
effect is similar to a truncation but it is not a real one because in some cases the loads do exceed the allowed values. Thus, instead of adopting a truncated normal to represent the still-water load effect intensity, a truncation factor TR has been proposed in Ref. 9, so that the probability of exceeding the allowed limit, which has been normalised to _ 100, is given by
P [ X > 100] = TR[1 -- F~(100)],
0 <~ TR <<. 1"0 (13)
where F~(x) is the initial probability distribution of the load effect intensity. The modified probability density function fro(X) is related to the initial Gaussian distribution f ( x ) by
fro(x) = Tel(x),
x < 100
(14)
where the new factor TF is given by
TF =
1 -- TR[1 -- Fi(100)]
(15)
E(100) The factor TR is a measure of the efficiency of the control existing on board. Whenever there is no control, TR and Tr are equal to one and the initial normal distribution is not changed. For a perfect control TR is zero and there is exactly a truncated distribution in which the maximum allowed value is never exceeded. In this case TF becomes equal to 1/F~(100), which is the typical case considered in textbooks. The values of X exceeding the limit of 100, which will be denoted by Xe(= X - 100), are described by a shifted exponential distribution given by
fx~(Xe) = (1/ffe)eXp(--Xe/ffe),
Xe > 0
(16)
where ae is the scale parameter which is given by
(Te
=
TR ~oo (X -- lO0)f(x)dx
(17)
To be more precise, this distribution could also be defined as a conditional density of Xe given that X > 100, i.e. fXelX>'O0, which is the implication of the condition X~ > 0 in eqn (16). However, to simplify the notation we stick to the one of eqn (16). Taking eqn (13) into consideration, the upper tail of the distribution is given by fm(X) = x > 100
TR[1 --o'eFi(100)] exp -- (X --~r~100~,./ (18)
which complements eqn (14). With this description of the modified distribution it is possible to make predictions of the probability of exceedance by using eqn (11) where now Fx(~) is the modified distribution just described. The value of TR can be adjusted so that the model predictions agree with the observed data. In two specific data sets studied in Ref. 9, TR was found to be about 0.5 for a group of tankers and 0.25 for a container ship.
C. Guedes Soares
106 3 WAVE-INDUCED LOAD EFFECTS
A zero-mean stationary Gaussian process can be completely described by a power spectrum, which has been used to describe the wave elevation in sea states. Theoro etical models exist to represent the average spectrum in fully developed sea states, a° in developing seas 2~ and in combined wave systems? 2 The linear response of a ship to that random excitation is also a stochastic process described by a respor~se spectrum S~(co) given by
&(co) = S.(o~)H~(co)
(i9)
where Su(m) is the wave elevation spectrum and H(m) is a transfer function depending on the encounter frequency ~o and on the relative heading between the ship and the direction of the wave system. The variance of any o f those spectra is given by R =
f ~ S(co)dco
(20)
In a narrow-band Gaussian process the amplitudes are Rayleigh distributed, so that the probability Qs of exceeding the amplitude x is given by: 23
Qs(xlR) = exp
- ~
(21)
The narrow-hand assumption does not always hold for the free-surface elevation but it is often a good model for the responses because the transfer function acts as a sort o f filter which reduces the frequency bandwidth o f the response. It should be noticed that Qs is fully described by R which is obtained from the knowledge of the spectrum. To obtain Q~(x), the probability of exceeding the load efl?ct x at a random point in time, one must weigh Qs(x) by the probability of occurrence of a given value of R and integrate over alt possible outcomes: 14q7
QAx) = fo Qs(xlr)f~(r)dr
= f(h s, ~z, O, v, c)dh, dtzdOdvdc
(23)
where the joint distribution o f the five variables is usually represented by the product o f several conditional distributions.~ 6,i 7 The tong-term distribution QL(x) calculated according to expressions (21)-(23) is usually fitted by a theoretical distribution. Experience has shown that one o f the best choices is the Weibull distribution, which is given by F)c(x) =
1 - exp -
Ship No. 1 Ship No. 2 Ship No 3 Ship No. 4 Ship No~ 5 Shio No. 6 Ship No 7 N, QUEEN HAVKONG BERGE COM. TAIMIR
1°08 I-!4 0°99 0-84 0-9~ 1.04 i-03 0-54 0.75 0.76 0'99
WOLV. STATE CAL. BEAR UNIV. ~RELAND £SSO MALAYS!A R. G FOLLIS FOTINI L ID, MARU BOSTON ESSO BONN TOYAMA Average
~,05 0-99 0.85 ~9,88 ~'63 0,97~ 0% ~.{~5 9.75 ~;~.% 085~
wave-induced vertical bending moments reported in Ref~ 24, 25 and the values that resulted for the exponent .a are. indicated in Table i. ~nspection of this table indicates that the shape pa> ameter A is often very ciose to unity aitho~.gh ~ the average somewhat lower. However additional resuI-.:~°s~ which were obtained from calculated transfer %ncuo~s resulted in exponents closer ~o unity. Th~s suggests tbe adoption of the simpler exponential distributions to reno resem the long-term distribution of waveoiv~duced bending moments, which ~as been used in Ref. 3 for example_ The adoption o f the ~ni~ exponen~ for eq~ ~2~) although being acceetable for the whole populatio~ of ships, cm3 be serious in error when applied to one spec~.fic ship. This is the general problem of defining reqmrements for codes that must be applied to classes of ships m structures and quantifying the load effec~ for one specific case. At any rate the formulation presented is valid for any value of the exponent arid thus can accommodate both cases. 4 PROBABILIST[C COMBINATION OF LOAD EFFECTS
(22)
where Qs(xtr)is given by eqn (21). The probability density function of the variance of a stationary period, fa(r) will depend on several variables such as the wave climate represented by /Is and Tz, the ship heading 0, speed v and loading condition c, i.e. f~(r)dr
r u n e 1o Exponent ef the Weiba~ DNtrib~ge~ t}~at was ~r~e~ te measured data of wave-indtuee~boning meme~t~ (Data from Refs 24, 25)
(24)
This distribution has been. fitted to data sets o f measured
The general objective in the design and operation of structural systems is to ensu~re that they wi!l be able ~-:o maintain their serviceability 'under ordinary loading conditions and that they will have a "verysmall probability of failure under extreme loads throughout their !ifetime. Classical structural reliability theory models the u n t o > tainties in s t r u c t ~ a l strength and in the loadi~g as random variables, not accounting explicitly for their time variability; However, even. this type of approa&~ ea;~ be used to quantify the time-odependent reliability as suggested in Refo 3. While the strength variables have a relatively narrow range of variation with time, the opposite occurs with the load effects. The most unfavourable conditions will often occur when there is a simultaneous occurrence of different types of loads at high levels of intensity° To take the time domain load fluctuation i~to co>~ sideration, one needs to model the loads as ray,dora
Combination of Primary Load Effects in Ship Structures processes. Failure will be the first crossing of a vector process out of a safety domain. However, because this is a complex problem, different simplifications have been proposed to transform the random process into a manageable random variable formulation. An especially important case is the linear system, in which the load effects S(t) are linear functions of the load intensity. In this case, the total load effect Si(t) is the sum of the various components si(t)
=
cuI j(t)
(25)
j=l
where Cij are coefficients transforming the load Lj into the load effect S~. The lifetime probability of failure in relation to the ith load effect is
Pr = P[R, <~ Sin,]
(26)
where Sm~ is the maximum value of the load effect S; during the lifetime T. This formulation substitutes the load effect process S~(t) by the random variable Sm~ allowing one to use the classical structural reliability analysis. The same approach can be applied to the reliability formulation for the combination of several load effects. This is why there is interest in predicting the maximum lifetime combined load effect. Examples of processes that are adequate for modelling several load effects and that allow an analytical solution to be obtained are the Gaussian process and the renewal pulse process, which can model, respectively, the linear responses to the excitation of the wave free surface elevation and the still-water loads. Mano and Kawabe 27developed one of the few published formulations of the probability distribution of the maximum combined still-water and wave-induced bending moments, considering the first one as a normally distributed variable. They formulated the probability distribution of the combined load effect x as
Fx(x)
=
[1 - Qw(x - y)] .f~(y)dy
(27)
where Qw is the exceedance probability of the wave load effect, fs is the probability density of the still-water load and n is the number of wave load cycles in the reference period. That formulation considers that both loads have the same time variation, i.e. in n wave cycles there will be n independent outcomes of the still-water loads. This implies that the lifetime characteristic value of still-water loads is obtained from the probability distribution of the maximum in 108 outcomes, because this is the reference value used for the number of wave cycles in a ship's lifetime. In fact, results presented in Ref. 26 indicate that the expected number of cycles is generally smaller than 108, with a value that depends on the type of the ship. However, the number of load conditions that a ship
107
experiences in her lifetime is only of the order of 103104. 7,8
Srding ~5represented the distribution of the combined load by a similar expression, without the exponent n. This is the probability distribution of the combined load at a random point in time. It does not model the repetitive nature of both load processes, and thus is not applicable to a duration larger than one wave cycle. The two formulations just referred to do not model the different time dependency of the two load processes. In reality, for each still-water load condition there may be as many as 104 wave cycles. These formulations also assume that the still-water and the wave-induced load effects are statistically independent. Mano & Kawabe 27 calculated the wave-induced bending moments in a ship's hull with five different displacements and obtained a negative correlation between the two load effects. However, all the five cases considered were from the same load condition, i.e. they had the same basic distribution of the load, with one peak at each of the quarter points along the ship's length. The changes of still-water bending moments were achieved by small load variations about these points. In addition, the changes were always symmetric, implying that there was no change of trim nor of displacement. However, it is exactly the changes of trim and displacement and the irregularities of the cargo distribution that will decrease the type of correlation that they obtained. Changing the deadweight will modify the underwater volume of the hull, leading to changes in the waveinduced loads. If it was not for the variations of trim, a strong correlation would be expected between the displacement and the wave-induced loads. However, the trim has the nature of a normally distributed random variable, as concluded in Ref. 5. It is expected that the variations of trim for a given value of displacement will diminish the correlation between wave-induced load effects and deadweight so that their correlation with the still-water load effects will be smaller than the value of - 0 - 5 0 mentioned above. Support for this viewpoint can also be found in the study of Lotveit et al.,28 who conducted a series of experiments with models of the same hull with different distribution of weights. They found that different distributions of weights produced different wave-induced bending moments, as noticed also by others. 29'3° However, they concluded that the variations of the wave-induced bending moments could not be described by the values of the still-water bending moment. In conclusions, it appears that the assumption of independence between still-water and wave-induced load effects may be adequate for practical purposes, and it has been adopted in the formulations described hereafter.
4.1 Ferry Borges-Castanheta model The model proposed by Ferry Borges and Castanheta 3t is
Co Guedes Soares
108
one o f the simplest load models, which assumes that the loads change intensity after prescribed deterministic, equal time intervMs, during which they remain c o n s t a n t The intensity o f the loads in the different elementary time intervals are outcomes o f identically distributed and mutually independent r a n d o m variables. The probabilistic distribution Freer o f the maximum value during n repetitions o f the toad i or equivalently during the time T = m:, where ~cis the pulse duration, is given by G,~(x)
:
[fAx)] ~
(2~)
where F~(x) is the probability distribution function o f the load intensity. The application of this formulation to model the stillwater load effects requires few assumptions since a rec~ tanguiar putse renewal has been shown in Section 2 to be a good representation of that load. Instead o f having pulses with different duration, the Ferry Borges-Castanheta model would require all pulses to be o f equal duration, which should be the average voyage duration. These durations range from 2 days for container ships to 28 days for large tankers} The approximation involved in the Ferry BorgesCastanheta model would not be a good one for periods corresponding to a few voyages, but it would gradually become better as the number o f voyages increases, and would be a good model for the maximum lifetime load. In this time span a ship will perform an integer number o f voyages, which fits well into the assumption o f the Ferry Borges-Castanheta model. To adopt this model to represent also the maximum wave-induced load effect requires that the time variation o f the process be substituted by a sequence of pulses, each representing the amplitude of a cycle of the response. The assumption of independence between successive cycles of response is not exact because waves separated by a few cycles are correlated. Watson 3~ studied this type of problem, defining a r a n d o m variable to be m-depeno dent when the variables in a sequence separated by more than m I outcomes are statistically independent. Watson has shown that the limiting distribution for the maximum o f m-dependent variables is the same as the limiting distribution of independent random variables. The correlation between wave heights is restricted to a small number of cycle% especially when considering the time scale o f long-term variability. Thus the assumption of independence seems reasonable for predictions of lifetime maxima. Another way of looking at the problem is to use a resuk o f Cramer 3~ who showed that the excursions of a high levet by a Gaussian process are asymptotically Poisson distributed° This implies that the exceedances o f high levels are independent° It is precisely the high levels of the wave-induced loads that are o f interest for the
predictions o f extreme values m which zase indepeno.. deuce can be assumed. Finally, because in a ship's ~ifetime there are many different toad conditions and daring each voyage there occur very many wave cycles, it is no major restn ction to consider that their number is an integer° Therefore, the main assumptions required by the Ferry BorgesCastanheta model are satisfied° Consider now two load process X~(0 and X2U) of this type with durations ~:i, and h , si~ch that ~, > 'c~ ~" oo~e is interested in the process ~<(,t) =
x,{t) + &(t)
(29.
the probability distribution of rise maximum of me combined processes during time T can be detennm.ed exactly provided that the two processes are in phase, and that khe ratios T 2 / Z [ =: m and T/z = ;~ are integers. ]l~ this case one has GA
X)
:
L (z)[G--<~ --
z)]mdz!
<~0)
This model could be extended to consider more than ~wo loads, using the same arguments° To apply ~.t ~o the distribution of the maximum combined stiHowater and wave-induced load one must identi~J the first one with Y and the second one with t l F r o m the preceding &s~. cuss,~on ~t is easily seen that the requiremem:s about t~:e ratio of durations and their being in phase are not very restrictive. A series of calculations has been performed for u~e application of this model to the cases of tankers and container ships. The distribution of stilt-water load effects was described by a normal distribution anc dif< ferent degrees o f truncation v,mre assumed, to mode~ th3 effect of h a m a n control. 9 F o r tankers the mean value and standard deviation of the distribution were in sagging, respective!y, !2.4% and 21.3% o~ U~e nomma~ value ,o'~ still-water vertical bending moment prescribed by the rules of Classification Societies, while for container ships ~hey were 71.5% and 16-3% b_ hogging, in accord~ncc with Ref. 7_ The mean voyage durations are assumed tc be 3.7 days for container ships and 23.5 days for m~ kers tn this description, the mean vMues e f the load effects are very small which imply very large coefficients o? variation. This results from the fact that tankers operate roughly 50% of the time loaded and 50% in bMlast b" the former situation they usually experience hogging stresses_ which are assumed to be posmve~ and in ballast they experience sagging stresses which are m e d e t b d as negat~_ve. Thus their average is very ~mait which results i;~. coefficients of variation around 1-7 However° if e~e considers only loaded or baitast conditions this va~.e decreases ) o r e i.? to around 0.5. The wave-induced ioads were represented by an exponential distribution with the parameters derived from tl:~e design wave-induced bending momen~ given in the m~e~ of Classification Societies. F o r exampteo in the case of th~
Combination o f Primary Load Effects in Ship Structures
109
Table 2. Results of the load eombination calculations using the Ferry Borges-Castanheta model, in terms of total load (Ms + Mw) or of the corresponding load combination factors (eqns (30), (31))
TR
1 year
20 years
30 years
Ms + Mw
q$
~k
Ms + Mw
q$
~b
Ms + Mw
¢
~k
Tankers sagging
0.10 0-50 0.90
219 227 247
0-97 0-96 0.94
0-94 0.93 0.88
259 311 337
0.95 0-94 0.91
0.91 0-90 0.78
263 322 354
0.94 0.90 0-90
0.90 0.78 0.78
Container ships hogging
0.10 0-50 0-90
227 250 282
1.00 0.95 0.84
1-00 0-90 0-63
256 302 342
0.96 0.90 0.79
0.92 0.79 0-51
260 310 353
0.95 0.83 0.78
0.91 0-64 0.48
Det norske Veritas rules of 1989, this requirement corresponds to the characteristic moment at the probability level of 10 -8 divided by 1.7. This information is enough to estimate the parameter of the exponential distribution. The mean wave period over all sea states was adopted as the duration of the wave load process. This is an approximation of a degree of accuracy that can be compared with other load modelling assumptions. It reflects the positive correlation that exists between wave height and period. While for low sea states one can have periods around 5 to 6s, for high significant wave heights they can be as large as 16s. This effect can be taken into account in the numerical integration just be making m in eqn (30) a function of (x - z), i.e. m will decrease with increasing (x - z). Some effect is expected on the characteristic values, although not a large one. The probability distribution of the combined load effect was obtained by numerical integration of eqn (23). Table 2 presents the characteristic values of the total load (Ms + Mw) for the probability levels corresponding to return periods of 1, 20 and 30 years. The load effects have been normalised by the Rule value of the still-water vertical bending moment, which corresponds to 100. Accounting for the differences in the definitions of nominal values of Ms and Mw, the nominal value of (Ms + Mw) would be 214, which should be compared with the ones indicated in Table 2. 4.2 Upcrossing rate solution The upcrossing rate for a scalar stationary process has been established by Rice as v+(e) =
=0 2fx, x,(ct, 2)d2
I~ v+(~'t)" dt
fi.~(y,9)
=
x-= - 7-_
(32)
which is valid for any non-stationary process and becomes v T for the stationary case.
i"1 = - ~
xfxd2(y-
I'XI
x')~:I
x,~-
~71)d2~-dx
(33)
Substituting now this expression on the upcrossing rate expression results in ......
tl ....
+2=--¢1
Xlfxi'£i(X'J$1)
× fx2.x2(~ - x, 22)d22d21dx
x fx~.s~2(~ - x, -t2)d-tzd2~ dx
(34)
The integrals can only be calculated in special cases. However they can be used as a basis to establish an upper bound on the upcrossing rate 34
+ f<~-oo v[ (o~ - x ) ' f x , ( x ) d x
(35)
It can be shown that this upper bound for any stationary stochastic process is in fact the exact result when both load components are renewal rectangular pulse processes, and the peak values are mutually independent and are also independent of pulse durations] 9 In fact, this equation is exact when the two processes satisfy the following conditions at all times t P[L(t) > 0 a n d L ( t )
(31)
wherefx;(X,2) is the joint probability density function of X(t) and J?(t) and v+ (e) is the mean upcrossing rate of level c~. Having the upcrossing rate, one can determine the mean number of upcrossings of level e in time T as E[N(~,T) =
Consider now the process Y(t) that is a linear combination of the independent stochastic processes X~(t) and X2(t) as indicated in eqn (29). This joint densityfi.~(y,D can be determined from processes X~ and X2:
< 0] =
0,
i, j, = 1,2 (36)
This condition ensures that the increase in process i due to its positive slope will not be cancelled by a decrease of process j as a consequence of its negative slope. Rectangular pulse processes have a non-zero slope only at two time instants: the beginning and ending of the pulse. Thus, the combination of a rectangular pulse process with any other process satisfies the above equation because the probability of simultaneous changes in both processes is zero. This means that eqn (9) provides the
t t0
C. Guedes Soares
Tab|e 3o Results of the load eomN~ation calculations using the nperossing rate sol~tion, i~ t e r ~ of torn ~oad (Ms 4- ?~2~,) er ef ff-~ corresponding load combination factors (eqns (30), (31))
Te
1 year
20 years
3~ years
Tankers sagging
0.I0 0-50 @90
219 227 247
0,97 0.95 0,93
0-94 0.91 0.86
259 3!~ 337
0-9t 0-89 0-85
0-84 0-76 0-61
263 322 354
0-9i 0~85 ~}.8~
0°84 0,~8 f) 5~
Container ships hogging
0-10 0-50 0.90
227 250 282
1,0{} 0"94 0.82
1.00 0-88 0.60
256 302 342
@95 0.84 ~72
0.90
260 310 353
0-93 ~)'83 0.7~
~-88 0°64 :3-3~
exact upcrossing rate of the combined still-water and wave-induced processes. The degree o f approximation o f this upper bour~d can be assessed by referring to eqn (11). The approximation is good when the probability o f occurrence of more than one upcrossing in the reference period is much smaller than the probability of one upcrossing (eqn (7)). This condition is often satisfied when determining the lifetime maximum o f the processes. This formulation has been used to determine the probability distribution of the combined still-water and wave° induced bending moments for tankers and container ships. The truncated distribution of the still-water loads was used, as given by eqns (7)-(i 1). The long-term distrio bution of wave-induced loads is given by eqns (t5) and (17) and the upcrossing rates are determined by eqn (35). The results o f the calculations are shown in Table 3 for the truncation levels of 0.1 and 0.5, which are more realistic to expect than 0.9. The results are very close to the ones obtained by the Ferry Borges-Castanheta model, which confirms the equivalence of the f o r m > lotions, as has been pointed out by Turkstra & Madsen, ~4 when studying other load models. 4.3 Load combinat~o~ ru~es
Defining Ms and Mw as the characteristic values o f the stiil-water and wave-induced load effects, the combined load effect L can be expressed by either of the following expressions L =
~ , (Ms + Mw)
(37)
5
e~s + ~ o M w
(38)
=
where 4~ and ff are factors that represent the load combination effect. The factor ~b has the character o f a bias while ~ is a load reduction factor. Both o f these systems are found in design codes. Examples o f the first one are in the National Building Code o f C a n a d a and the code o f the American Concrete Institute. The second format is adopted by the European Committee o f Pro-Stressed Concrete and by the CEBCIB-ECCS-FIP-IABSE-IASS-RILEM Joint Committee on Structural Safety) s For exampte, the 1989 ship rules of Dot norske Veritas adopt atso the tast format since the
0.67 0°43
nominai value of wave-induced bending moments ~ divided by to7, before being added to the nominal value of stilbwater bending moment. ToNe 3 indicates the sum of the characteristic vaiue o£ each individual load effect (/ffs - M~w) ar~ the values of 4) anc~ ~p derived from the characteristic value of ti~e combined b a d . Anatysis of the results indicates that the b a d combination factors depend on ~.l-~e probabiff~j level, as was previously observed by Turkstra g< Madsen. 34 Higher probability !evels lead ;o sinai!or values of the coefficients. The bias ~b had values between 0-95 and @80 whiie tf~e load reduction factor ~ varied between 0.90 and ©o5~ The variation depended not o n b / o n the probabiiity levei bm also on the degree o f truncation. For a m~id. ~runcat|on effect of the stilt-water load effects ~he ~oad co_,m.o bination coefficients diff?r significantly from oa_e However, for a strong truncation (TR = 0.~) :hey become d o s e to unity. This ~s because when there is a~ heavy truncation effect the stiltowater load effects a~'e ciose -~o their characteristic values so ores. tha~ I? becomes likely for the maximum waveqnduced ~oaC effect ~o occur when the still-wamr is a~ a high level A finai comment on the load combination fhci'~ors obtained is that they refer on!y to two ioads which m addition are of the sustained ~ype, i.e. during opera,son the structure is permanently subiected to both of ~hern~ despite the fact that high wave loads seldom occur. ~£fi~/e or six toads were considered as m the ease o f civil engineering smxctures, and if some were of ~raasient nature and short duration as are earthquake load~, smaller ic~ad combination Nctors could be expected. It is worth noting ehat ~:he eff%ct of' truncate.on tr @~;~c of the load models has not been treated previously in ar.~., of the toad combination studies regarding civil engmcer.ing load models] 93~-~4
5 C O N C L U D I N G RE}/~ARKS
This work has addressed the assessmen~ o f ti~e ~ong4.erm value of the combined stilt-water and wave-induced load effects ix snip structures. Solutions have been obtained
Combination of Primary Load Effects in Ship Structures by modelling both load effects as a sequence o f independent and identically distributed random variables and as two stochastic processes. In this latter case upcrossing rates of the combined process were determined and the probability distribution o f maxima was based on a bound formulation. L o a d combination factors have been introduced to express the combined load effect as a function of the characteristic values o f the individual processes. This approach, which is aimed at adoption in design codes, has been compared with the predictions of the combined load effects so as to determine the corresponding values of those factors. REFERENCES
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