Stochastic modeling of maximum still-water load effects in ship structures

Journal of Ship Research, Vol. 34, No. 3, Sept. 1990, pp. 199-205

Sto©Giasti© iilod^lmg ©ii' Sitescimym StööD-Water Load Effects

m Ship BMmMm^

C. Guedes Soares^

A stochastic model of an alternating renewal pulse process is proposed to describe the time depen-dence of still-water load effects in ship structures, extending previous studies that dealt only with random variables. The model proposed is applied to different ship types and the probability of occurrence of annual maxima is also determined in those cases. The results of a brief statistical analysis of load duration are also included as a basis to describe the time dependence of the process. Applications of this model to the development of code requirements are also included.

1. I n t r o d u c t i o n

DESIGNERS are becoming i n c r e a s i n g l y aware of the uncer-t a i n n a uncer-t u r e of m a n y of uncer-the design parameuncer-ters w h i c h r e q u i r e t h e use o f probabilistic methods to handle t h e m adequately. T h i s t r e n d has also been noticed i n the f i e l d of m a r i n e struc-tures, where r e l i a b i l i t y - b a s e d methods are becoming more accepted [1,2].^ These techniques are w e l l established now and have been used i n the past to assess the levels of safety e x i s t i n g i n d i f f e r e n t designs [3,4].

I m p r o v e m e n t of the e x i s t i n g approaches can be achieved o n l y w i t h a detailed m o d e l i n g of the s t r e n g t h of components, as reported i n [5] f o r example, or o f the load effects as con-sidered here. T h i s w o r k deals w i t h the m o d e l i n g of one of t h e two m a i n load components i n ship structures, a i m i n g at p r o v i d i n g the necessary b a c k g r o u n d to calibration, of design codes [6,7].

I t is common to d i v i d e the m a i n l o a d i n g on the ship struc-t u r e i n struc-t o struc-two componenstruc-ts w h i c h have d i f f e r e n struc-t n a struc-t u r e and v a r i a b i l i t y . The p r i m a r y ship s t r u c t u r e i s made up o f the continuous l o n g i t u d i n a l members t h a t deflect l i k e a beam. The s t i l l - w a t e r load effects t h a t act on i t are the v e r t i c a l shear forces and bending moments t h a t result f r o m the static e q u i l i b r i u m between t h e s e l f - w e i g h t of the ship, the cargo and the corresponding buoyancy forces. The hydrostatic pressures also induce h o r i z o n t a l forces on t h e h u l l sides, b u t those are resisted by secondary components of the s t r u c t u r e and thus w i l l n o t be considered here.

The wave-induced load effects are the a d d i t i o n a l shear forces a n d bending moments created on a ship at sea due to t h e effects o f t h e waves. T h e y r e s u l t f r o m the changes i n the d i s t r i b u t i o n of the buoyancy due to wave action and f r o m t h e i n e r t i a forces due to the wave-induced ship motions.

The t o t a l load effects on a ship s t r u c t u r e result f r o m sum-m i n g b o t h cosum-mponents w i t h due account to the c o r r e l a t i o n t h a t m a y exist. I f t h e t w o processes are n o t f u l l y coiTelated, t h e m a x i m u m t o t a l load effect w i l l be lower t h a n the s u m of the m a x i m a of the i n d i v i d u a l processes because t h e i r m a x i m a w i l l n o t occur at the same t i m e .

For design purposes, i n f o r m a t i o n is needed o n the ex-pected l i f e t i m e m a x i m u m v a l u e o f each load effect as w e l l

^Naval Architecture and IVtarine Engineering, Technical University of Lisbon, Institute Superior Tecnico, 1096 Lisboa, Portugal.

^Numbers in brackets designate References at end of paper. Manuscript received at S N A M E headquarters February 10, 1989; re-vised manuscript received July 24, 1989.

as on t h e i r t i m e v a r i a t i o n so t h a t load c o m b i n a t i o n studies can be conducted. T h i s w o r k addresses those topics t h a t con-cern the s t i l l - w a t e r load effects.

The s t i l l - w a t e r loads effects are d e t e r m i n e d b y the a m o u n t of cargo and i t s d i s t r i b u t i o n on board, since t h e w e i g h t of t h e ship is constant and t h e buoyancy forces w i l l be gener-ated as necessary to e q u i l i b r a t e the other w e i g h t s . T h e m a i n changes of cargo occur i n port, a l t h o u g h d u r i n g t h e voyages gi-adual f u e l consumption w i l l induce s m a l l changes i n the load effects. I f , i n a first a p p r o x i m a t i o n , these s m a l l and gi-adual changes i n load effects are neglected, t h e n one lon-g i t u d i n a l d i s t r i b u t i o n of shear forces a n d b e n d i n lon-g moments w i l l correspond to each t r i p . Each of the transverse ship sec-tions w i l l experience a d i f f e r e n t value of shear forces and b e n d i n g moments o n successive voyages. Previous studies have modeled these values as outcomes o f a r a n d o m v a r i -able, p r o v i d i n g i n f o r m a t i o n i n terms o f h i s t o g r a m s , m e a n values and s t a n d a r d deviations [ 8 - 1 0 ] .

Recently a n extensive study has been r e p o r t e d [11] w h i c h , i n a d d i t i o n to collecting data f r o m s i g n i f i c a n t l y m o r e ships and voyages, also considered the l o n g i t u d i n a l d i s t r i b u t i o n of such load effects. I n t h a t study, the m a x i m u m v a l u e o f the load effects experienced i n most ship types was w e l l below t h e design v a l u e prescribed by t h e codes. T h i s requires the p r e d i c t i o n of t h e l i f e t i m e m a x i m u m load effect so as to quan-t i f y h o w conservaquan-tive quan-the design is.

Whenever t h e predicted m a x i m u m l i f e t i m e load effects ex-ceed t h e m a x i m u m values established b y t h e c l a s s i f i c a t i o n societies, i t is necessary to consider the existence o f load dist r i b u dist i o n equipmendists on board, w h i c h w i l l p r o v i d e dist h a dist i n -f o r m a t i o n to t h e shipmaster, who i n m a n y cases w i l l redis-t r i b u redis-t e redis-t h e cargo redis-to avoid redis-t h a redis-t large v a l u e . However, i n redis-the data analyzed i n [11], values i n exceedance of the design value have been detected i n containerships a n d i n o r e / b u l k / o i l (OBO) carriers. To model the l i f e t i m e m a x i m u m i n these cases, a special f e a t u r e m u s t be included, as considered i n [12]. However, f o r a l l other cases, the analysis presented h e r e a f t e r w i l l be applicable.

2. S t o c h a s t i c m o d e l of s t i l l - w a t e r l o a d e f f e c t s

A r a n d o m phenomenom can be idealized i n d i f f e r e n t ways because t h e development o f a p r o b a b i l i s t i c model m u s t take i n t o account t h e objectives o f the study. F o r several purposes i t is adequate to adopt a r a n d o m v a r i a b l e f o r m u l a t i o n to rep-resent the i n t e n s i t y of the s t i l l - w a t e r load effects at a

spe-cific transverse section of a ship on d i f f e r e n t voyages. Each outcome, w h i c h corresponds to a departure s i t u a t i o n , is assumed to be d r a w n f r o m the same p r o b a b i l i t y density f u n c t i o n . I f the occurrences are assumed to be s t a t i s t i c a l l y i n -dependent, the probability d i s t r i b u t i o n of the m a x i m u m value i n n outcomes, Fmax.nC^). is g i v e n b y

Fr^^^M = [F^x)T (1) where Fxix) is the p r o b a b i l i t y d i s t r i b u t i o n of the i n d i v i d u a l

outcomes.

I f i n a d d i t i o h to load effect i n t e n s i t y one wishes to include i n t h e model the voyage d u r a t i o n , i t becomes necessary to use a stochastic process model. To establish t h i s model, con-sider the succession of voyages d u r i n g w h i c h the s t i l l - w a t e r load effects have a constant value, w h i c h changes a t every voyage, as shown i n F i g . 1.

Since t h i s process is characterized by pulses of constant i n t e n s i t y t h a t change a t specific points i n t i m e , i t can be adequately modeled by a renewal pulse process [13,14]. Each load occurrence becomes completely described b y its i n t e n -s i t y and d u r a t i o n , w h i c h are t w o r a n d o m variable-s f o l l o w i n g d i f f e r e n t p r o b a b i l i t y d i s t r i b u t i o n s . These processes are a special case of stochastic p o i n t processes [15] w h i c h trans-f o r m a continuous process i n t i m e i n t o one t h a t depends on r a n d o m occurrences at discrete points i n t i m e at w h i c h the process experiences a r a n d o m i n t e n s i t y .

The t i m e dependence can be modeled as a counting p o i n t process N{t), w h i c h is defined by Parzen [13] as a n integer valued process t h a t counts t h e number of points o c c u r r i n g i n a t i m e i n t e r v a l . The points, w h i c h are generated by a stochastic mechanism, define t h e end of one event and the i n i -t i a -t i o n of -the n e x -t one. The c o u n -t i n g process can be com-pletely defined either b y a direct representation of the number of points occurring i n a t i m e i n t e r v a l or b y m o d e l i n g the i n t e r a n - i v a l t i m e , t h a t is, the t i m e between t w o successive occuiTences. W h e n the i n t e r a r r i v a l t i m e s are independent, i d e n t i c a l l y d i s t r i b u t e d r a n d o m variables, they become a n or-d i n a r y r e n e w a l c o u n t i n g process. W h e n a r e n e w a l process has constant i n t e n s i t y between t w o successive points, i t is called a r e n e w a l pulse process Y{t) a n d can be represented by [13]:

mo

= 2 (2) where N{t) is a renewal c o u n t i n g process and Xi is a

se-quence of i d e n t i c a l l y d i s t r i b u t e d r a n d o m variables w h i c h represent the change i n the i n t e n s i t y of the process f r o m one period to the next, as i l l u s t r a t e d i n F i g . 1. Thus, t h e change f r o m one load period to the n e x t is obtained by a d d i n g a r a n -dom intensity t h a t w i l l make the process j u m p up or down.

One of t h e early applications of c o u n t i n g processes to t h e

x^(t)

Fig. 1 Illustration of a renewal pulse process

description of load processes is t h e F e r r y Borges-Castanheta, model [16]. They represented the load h i s t o r y X{t) as a se-»' quence of rectangular pulses of fixed d u r a t i o n . The i n t e n s i t y of the pulses Xi was described b y independent identically, d i s t r i b u t e d r a n d o m variables w i t h a g i v e n d i s t r i b u t i o n func-t i o n Fxix).

A generalization of t h i s process was considered b y L a r -rabee and C o r n e l l [17]. They adopted t h e r e n e w a l pulse pro-cess i n w h i c h pulse durations are allowed to vary, being modeled as i d e n t i c a l l y d i s t r i b u t e d and m u t u a l l y indepen-dent random variables. For several purposes the F e n y Borges-Castanheta model give the same predictions w h e n m a k i n g t h e i r fixed pulse duration equal to the mean d u r a t i o n of pulses o f t h e r e n e w a l process [18].

I n a p p l y i n g the renewal pulse process to t h e s t i l l - w a t e r loads, one m u s t recognize t h a t each period w i t h the load act-i n g starts w act-i t h a departure and fact-inact-ishes w act-i t h the u n l o a d act-i n g of the cargo i n the next port. Voyage d u r a t i o n s are r a n d o m because departure and a r r i v a l ports o f t e n change, b u t even w h e n t h a t is not the case, d i f f e r e n t w e a t h e r conditions w i l l m o t i v a t e d i f f e r e n t t r a n s i t times.

The i n t e n s i t y of the load effects is described by a proba-b i l i t y density f u n c t i o n fxix) w h i c h i t s e l f can proba-be a m i x e d dis-t r i b u dis-t i o n , represendis-ting d i f f e r e n dis-t load condidis-tions, as proposed i n [11]:

fxix)= fcic)-fx(x\c)dc (3) Jc

where fdc) indicates the p r o b a b i l i t y density f u n c t i o n of the load c o n d i t i o n C and fxix\c) is the density f u n c t i o n o f t h e load i n t e n s i t y conditional on the load c o n d i t i o n , w h i c h i n general is Gaussian [10]. T h i s representation is especially u s e f u l f o r t a n k e r s and other b u l k carriers of l i q u i d or solid cargo w h i c h have t w o clearly d i s t i n c t o p e r a t i n g modes, i n ballast a n d i n f u l l load.

A complete description of the t i m e v a r i a t i o n of t h e s t i l l -w a t e r loads m u s t include the t i m e t h a t ships spend b o t h i n p o r t a n d at sea. The period i n p o r t w i l l affect o n l y the t i m e v a r i a t i o n of the load effects since the i n t e n s i t y of t h e load effects i n p o r t is of no interest f o r t h e present purposes. H o w -ever, i t is i m p o r t a n t to account f o r t h e d u r a t i o n of t h e t i m e i n p o r t since i t influences t h e t o t a l n u m b e r of voyages t h a t a ship completes d u r i n g her l i f e t i m e and, t h u s , the n u m b e r of repetitions of load occurrence. T h e effect of t h e n u m b e r of repetitions on the p r o b a b i l i t y d i s t r i b u t i o n of m a x i m u m is i n -dicated by equation (1). I n t h i s context the load process is adequately modeled by sailing periods, d u r i n g w h i c h the load effects have a n i n t e n s i t y d i s t r i b u t e d as fxix), and by periods i n p o r t d u r i n g w h i c h the load is considered n o t to be acting or to have a zero i n t e n s i t y .

The o r d i n a r y renewal processes [13], w h i c h are a gener-a l i z gener-a t i o n of the renewgener-al pulse processes, gener-a l l o w for the pos-s i b i l i t y of periodpos-s of no load a n d of load acting. However, t h e y a l l o w f o r the possibility of consecutive occurrences of periods of the same type, w h i c h w o u l d not be v a l i d i n the case of s t i l l - w a t e r loads. A voyage is t e r m i n a t e d o n l y w h e n a port ( i n a generalized sense) is reached and, on the other h a n d , a period i n port finishes o n l y w h e n the ship i n i t i a t e s a voyage. T h i s implies t h a t t h e load-on and t h e no-load pe-riods always occur i n an a l t e r n a t i n g sequence. Moreover, i n most of the cases, the d u r a t i o n of periods i n p o r t f o l l o w s a d i f f e r e n t probabilistic d i s t r i b u t i o n t h a n do voyage durations. T h u s t h e assumption of i d e n t i c a l l y d i s t r i b u t e d pulse dura-t i o n s o f dura-the renewal pulse processes is also nodura-t v a l i d . There-fore, the s t i l l - w a t e r loads w i l l not be w e l l modeled by a n or-d i n a r y r e n e w a l process.

A n appropriate model to describe the t i m e dependence of t h e s t i l l - w a t e r loads must account f o r t h e a l t e r n a t i n g n a t u r e

X, (t)

t

Fig. 2 Illustration of an alternating pulse process

of the t w o types of pulses and f o r a d i f f e r e n t i n t e r a r r i v a l t i m e f o r each pulse type (see F i g . 2). Such a process, w h i c h is called by Cox [14] a n a l t e r n a t i n g r e n e w a l process, does not seem to have been used y e t i n load m o d e l i n g or i n load c o m b i n a t i o n studies.

Suppose t h a t the two types o f pulse d u r a t i o n i n a n alter-n a t i alter-n g r e alter-n e w a l process are dealter-noted by Di aalter-nd Dg. T h e alter-n , the process Z) = Z>i + is a n o r d i n a r y r e n e w a l process. Refer-r i n g to F i g . 2, the pRefer-rocess Di could be t h e one i n w h i c h the load is a c t i n g and the process D2 t h e one w i t h no load acting. I n a n e q u i l i b r i u m s i t u a t i o n , t h a t is, f o r a p p r o p r i a t e l y large values of t i m e t, the mean number of occmTences of any pulse type d u r i n g t h i s period is g i v e n by [14]:

E[N( Ö] = - + f - + 0(1) (4) where |x a n d a are the mean v a l u e and t h e standard

devia-t i o n o f D. T h i s expression neglecdevia-ts devia-t e r m s w i devia-t h an order o f m a g n i t u d e o f one, t h a t is 0(1). C l e a r l y , these variables are related to the i n d i v i d u a l processes b y |x = jxi + and CT^ = f \ + o-^2- Expression (4) indicates t h a t t h e mean n u m b e r of a r r i v a l s increases w i t h t i m e . T h u s , the asymptotic expres-sion f o r large values of t becomes:

Eimm - - (5)

IJ-l t shouIJ-ld be noticed t h a t i f t h e points occur according to a Poisson d i s t r i b u t i o n , the d u r a t i o n s D are e x p o n e n t i a l l y dis-t r i b u dis-t e d . T h i s i m p l i e s dis-t h a dis-t (J. a n d a are equal and dis-the asymp-totic expression (5) is also the exact one.

I n a n a l t e r n a t i n g r e n e w a l process the p r o b a b i l i t y t h a t the first pulse is on at a n a r b i t r a r y p o i n t i n t i m e t is g i v e n by [14]:

where (Xi and 1x2 are the mean d u r a t i o n s o f processes 1 and 2. The r a t e of occurrence of load pulses i n a t i m e u n i t , t h a t is, the a r r i v a l rate o f t h i s process, is g i v e n by:

1 1

\ = - = (7) M- (J-l + (X2

and the m e a n d u r a t i o n of the load pulses is (x,. T h i s is a special f e a t u r e of the a l t e r n a t i n g pulse process w h i c h has a n a r r i v a l r a t e t h a t is d i f f e r e n t f r o m the inverse of the pulse d u r a t i o n .

W h e n u s i n g t h i s model to represent t h e s t i l l - w a t e r load effects, the process type 1 is i d e n t i f i e d w i t h the load-on pe-riods, t h a t is, w i t h t h e voyage t i m e , and process 2 w i t h the t i m e i n p o r t .

C o u n t i n g processes become completely described b y N(t), the n u m b e r o f points i n a t i m e i n t e r v a l t, or by the i n t e r -a r r i v -a l t i m e X. According to equ-ations (5) -and (7), these q u a n t i t i e s depend on |x, the m e a n d u r a t i o n of load pulses, for large values of t. For shorter t i m e periods, equation (4) indicates the need to k n o w the variance o f the pulse dura-tions CT. Therefore, knowledge of the m e a n value a n d even-t u a l l y o f even-t h e variance of voyage duraeven-tions and of periods i n port w o u l d provide enough data to describe the s t i l l - w a t e r loads as a n a l t e r n a t i n g r e n e w a l process. The n e x t section presents t h e results o f such a n analysis.

3. A n a l y s i s of v o y a g e d u r a t i o n d a t a

To provide a n i n s i g h t i n t o t h e t i m e dependency o f s t i l l -w a t e r loads, a b r i e f s t a t i s t i c a l analysis o f load d u r a t i o n data was conducted. The data sets h a d enough voyages to provide a n i n d i c a t i o n about t h e type o f d i s t r i b u t i o n f u n c t i o n appli-cable to voyage durations.

Descriptive statistics o f voyage d u r a t i o n are presented i n Table 1 f o r selected ships. The ships i n Table 1 have the same i d e n t i f i c a t i o n t h a t was used i n [11], where C T stands f o r containership, B C f o r b u l k carrier, 0 0 f o r o i l / o r e caiTier, S T K f o r s m a l l t a n k e r s ( = 7 0 0 0 0 d w t ) a n d L T K f o r large t a n k e r s ( = 3 0 0 0 0 0 d w t ) .

The data set was available f o r only some o f t h e ships stud-ied i n [11], a l l o w i n g some tendencies to be defined concern-i n g representatconcern-ive duratconcern-ions for the d concern-i f f e r e n t shconcern-ip types. Some of the expected trends can be observed i n t h e results: Con-tainerships have m u c h smaller average durations t h a n larger b u l k carriers and tankers. A m o n g t a n k e r s there are signif-i c a n t dsignif-ifferences between t signif-i m e d u r a t signif-i o n of the s m a l l e r and l a r g e r ones. These differences have a n i m p o r t a n t i n f l u e n c e on the n u m b e r of l i f e t i m e load repetitions and t h u s on the l i f e t i m e m a x i m a , as results f r o m equation (1).

The d u r a t i o n of the periods i n p o r t is adequately modeled b y a n e x p o n e n t i a l d i s t r i b u t i o n , as can be concluded f r o m i n -spection of the s t a t i s t i c a l moments i n Table 1 and o f t h e em-p i r i c a l histograms of F i g . 3. F o r m a l tests o f fit, conducted i n other cases, d i d not reject this hypothesis, as reported i n [10]. The description of voyage d u r a t i o n is more complicated. Ships have well-defined t r a d i n g routes so t h a t t h e voyage d u r a t i o n s v a r y i n a s i m i l a r w a y to b o t h sides of the mean value. I n t h i s case, voyage d u r a t i o n s look l i k e the first t w o histograms i n F i g . 4 a n d can be described b y a n o r m a l

dis-Table 1 Mean value ((x) a n d standard deviation (a) of data on duration of v o y a g e a n d periods in port for c o n t a i n e r s h i p s ( C T ) , bulk c a r r i e r s ( B C ) ,

o r e / o i l carriers ( 0 0 ) , small ( S T K ) a n d large ( L T K ) t a n k e r s

Voyage Duration, days T i m e i n Port, days

Ship N O B S N O B S 0^2 C T 1 31 2.6 2.9 C T 7 22 1.7 2.7 '22 ï.'o 0,8 C T 8 19 6.1 5.6 B C 12 60 16.7 4.1 61 10.6 7.0 B C 13 45 14.3 8.9 46 13.1 11.1 G D I 114 27.9 10.3 111 3.1 2.4 S T K 6 201 9.6 9.8 211 5.7 8.7 S T K 22 145 15.2 12.1 168 5.5 7.3 S T K 23 155 13.7 11.5 162 3.8 4.5 L T K 28 103 24.2 9.5 104 4.3 4.3 L T K 39 45 22.7 14.3 63 4.6 7.5 L T K 40 21 21.9 13.6 30 5.8 10.6 A v g C T 72 3.7 3.5 22 1.0 0.8 A v g B C 105 15.7 5.3 107 11.7 8.8 A v g S T K 501 12.5 11.0 541 5.1 7.0 A v g L T K 169 23.5 11.3 197 4.9 6.3

CONTAINERSHIP C T 7 - T1M£ I H PORT BULK C A R R I E R GC 12 - T IHE IN PORT ORE C A R R I E R 0 0 2 - VOYAGE DURATION S U L K C A R R I E R B C15 - VOYAGE DüRft' ON

0 . 0 0 . 5 1.0 1 . 5 2 . 0 2 . 5 J . O 5 . 5 t . O T I M E I D A Y S l ORE C A R R I E R 0 0 2 - TIME IN PORT

10 2 0 50 1 0 5 0 5 0 T I K E (DAYS) TANKER TK 2 8 - T I M E IH PORT 2 0 1 0 6 0 BO T I M E ( D * r S I C O N T M N E R S H I P C T 1 - VOYAGE DURATION 2 0 2 5 50 T l h E ( O A T S ) TANXER TK G - VOYAGE OURATION

Fig. 3 Distributions of time In port

t r i b u t i o n . T h i s type o f s i t u a t i o n is more l i k e l y to be f o u n d i n large ships c a r r y i n g cargo i n b u l k , either solid or l i q u i d , and is the r e s u l t of t h e i r operational p r o f d e . F u r t h e r m o r e , because of t h e i r size, these ships cannot enter a l l ports and thus they have a s m a l l e r choice of possible routes, w h i c h leads to a smaller v a r i a b i l i t y o f the voyage d u r a t i o n s .

O n t h e other side o f t h e spectrum one f i n d s t h e s m a l l ships w h i c h can enter most o f t h e commercial ports a n d w h i c h are aimed at t r a n s p o r t i n g s m a l l e r amounts of cargo. I n general, they have shorter voyages a n d a m u c h more v a r i e d schedule t h a n t h e l a r g e r ones. These conditions i m p l y t h a t t h e en-countering of a p o r t w i l l be a r a n d o m event, i n w h i c h case the exponential d i s t r i b u t i o n w o u l d describe t h e voyage du-rations. I n fact, f o r these ships t h e e x p o n e n t i a l d i s t r i b u t i o n appears to be a n adequate m o d e l , as can be seen i n t h e t w o m i d d l e plots o f F i g . 4, corresponding t o a containership a n d a s m a l l t a n k e r . T h i s is also supported by the data i n Table 1, where one can observe t h a t n-i ~ CTI f o r these ship types. M i x e d situations can also occur. T h i s happens w i t h ships t h a t have t w o or t h r e e p r e f e r r e d routes w i t h d i f f e r e n t m e a n lengths and t h a t can also u n d e r t a k e e v e n t u a l shorter t r i p s . T h i s became especially applicable f o r m e d i u m a n d large t a n k e r s a f t e r t h e 1973 o i l crisis. Due to the l a c k of f r e i g h t s , some tankers were forced to accept contracts f o r p a r t i a l loads and f o r shorter distances. I n these cases t h e h i s t o g r a m m a y look l i k e t h e t w o l o w e r ones shown i n F i g . 4, w h i c h t e n d to indicate the presence o f a n e x p o n e n t i a l l y d i s t r i b u t i o n f o r t h e smaller d u r a t i o n s a n d possibly a n o r m a l d i s t r i b u t i o n cen-tered a t 21 a n d 33 days, respectively. However, w h e n a l l t h e data are considered together, they e x h i b i t t h e m a i n trends of a n exponential d i s t r i b u t i o n .

4. P r e d i c t i o n of l i f e t i m e m a x i m u m l o a d e f f e c t s For design purposes one o f t e n needs i n f o r m a t i o n about the m a x i m u m v a l u e of a load process d u r i n g t h e l i f e t i m e of a structure. W h e n d e a l i n g w i t h stochastic processes t h i s i n -f o r m a t i o n can be r e l a t e d to t h e upcrossing o-f g i v e n levels o-f load i n t e n s i t y .

The exact d i s t r i b u t i o n f u n c t i o n is d i f f i c u l t to derive, b u t relatively simple expressions can be based on a n upper bound to t h a t p r o b a b i l i t y . T h i s b o u n d is established by n o t i n g t h a t the p r o b a b i l i t y Q(a,D t h a t t h e process X{t) exceeds l e v e l a i n t h e period o f t i m e f r o m 0 to T is g i v e n by: 8 ( 0 12 I t T I K E I D A Y S ] C 0 N T A I N E R 5 M I P C T 12 - VOYAGE DURATION 2 0 5 0 I O S O T I M E ( D A Y S I TANKER TK 2 2 - VOYAGE DURATION

15 20 2 5 50 T I M E l O A Y S )

10 2 0 5 0 1 0 5 0 GO TIME I DAYS 1

Fig. 4 Distributions of voyage duration, illustrating tiie normally distributed, tfie exponentially distributed, and the mixed c a s e

Q(a,r)

=

The probability of one or more upcrossings i s always smaller t h a n t h e m e a n n u m b e r of upcrossings:

PVN{.oL,T) > 1] = 2 ™ a , T ) = j ]

< 2 > • P\Nioi,T} = j ] = £ [ M a , r ) ] (9) S u b s t i t u t i n g t h i s i n e q u a l i t y i n the previous e q u a t i o n results i n t h e f o l l o w i n g b o u n d [19]:

Q{a,T) < Q(a,0) + [ 1 - Q(a,0)] • E[NW,T)^ (10) W h e n considering l e v e l upcrossings, one is o f t e n i n t e r e s t e d i n levels t h a t are h i g h enough f o r t h e p r o b a b i l i t y of more t h a n one exceedance to be n e g l i g i b l e i n comparison w i t h t h e p r o b a b i l i t y of one exceedance. F o r a s t a t i o n a r y process sa-t i s f y i n g sa-t h e c o n d i sa-t i o n sa-t h a sa-t

2 j - P [ M a , T ) =j]<PVN{a,T) = 1] (11)

a good a p p r o x i m a t i o n to the mean n u m b e r o f l e v e l upcross-ings is g i v e n by:

where is the m e a n upcrossing rate o f level a, and T is reasonably large.

The mean upcrossing rate is defined as the l i m i t as M tends to zero of t h e p r o b a b i l i t y t h a t t h e process is below a at t i m e t and over a at i + A i :

v : = \ • [ 1 - Q{a,t)] • Q(a,i) (13) where \ is the a m v a l r a t e of the p o i n t process. The a r r i v a l

rate expresses the p r o b a b i l i t y t h a t there is a n occurrence of the p o i n t process i n the period between t and t + At.

Substituting this expression i n equation (10) yields the basic f o r m of the bound:

Q(a,r)

t h a t always s t a r t f r o m zero, i t takes the f o l l o w i n g f o r m [ 1 7 ] :

Q ( a , T ) < Q(a,0) + vtT (15) The upcrossing rate f o r t h i s type o f processes becomes:

v : = XQ(a,0) (16) The probability of the pulse process exceeding level a at t i m e

zero is determined b y the d i s t r i b u t i o n f u n c t i o n o f t h e process i n t e n s i t y :

Q(a,0) = 1 - Fx{a) (17) Substituting expressions (16) and (17) i n equation (15) yields

the f m a l f o r m of the bound expression f o r pulse processes t h a t start f r o m zero:

Q(a,r) < [ 1 - Fxiam + \ T ) (18)

T h i s p r o b a b i l i t y of exceeding l e v e l a is related to the prob-a b i l i t y of occurrence of the m prob-a x i m u m o f the process w h i c h is greater or equal to a, because o f the r e s t r i c t i o n of n e g l i -gible p r o b a b i l i t y of more t h a n one upcrossing, equation (11). T h i s simple expression requires o n l y the knowledge o f the a r r i v a l rate of the pulse process [equation (7)] i n a d d i t i o n to the d i s t r i b u t i o n of t h e pulse i n t e n s i t y .

T h i s bound on the p r o b a b i l i t y o f level exceedances can be applied to the a l t e r n a t i n g pulse process considered i n Sec-t i o n 2 by h a v i n g Sec-the a r r i v a l raSec-te defined [equaSec-tion (7)] and the d i s t r i b u t i o n of pulse i n t e n s i t y defined either b y the m a r -g i n a l or by the c o n d i t i o n a l d i s t r i b u t i o n i n equation (3). The a n d v a l rate of equation (7) can be estimated f o r d i f f e r e n t ship types f r o m the results of Table 1 i n Section 3. The dis-t r i b u dis-t i o n of dis-the pulse i n dis-t e n s i dis-t i e s can be modeled w i dis-t h dis-the results of the s t a t i s t i c a l analysis reported i n [10] or prefer-ably f r o m the more extensive analysis of [11].

E x a m p l e calculations were p e r f o r m e d for d i f f e r e n t ship types so as t o show the effect of d i f f e r e n t voyage d u r a t i o n s and o f d i f f e r e n t p r o b a b i l i s t i c descriptions of load i n t e n s i t i e s on the l i f e t i m e m a x i m u m values o f the load effects. O n l y

b e n d i n g moments were considered, and t h e i r p r o b a b i l i s t i c n a t u r e is described b y the s t a t i s t i c a l moments obtained i n [11] and summarized i n Table 2. I n the cases of b u l k car-riers, and o r e / o i l can-iers, the bending m o m e n t statistics are the average values corresponding to the ship types. I n t h e case of tankers, w h i c h are considered i n small and lai-ge sizes, the m e a n v a l u e and standard d e v i a t i o n o f t h e b e n d i n g mo-m e n t have been determo-mined f r o mo-m the regression equations reported i n [11]:

BM = 114.7 - 105.6 W - 0 . 1 5 4 L aBM= 1 7 . 4 - 7 . 0 W-H 0.035 L

w h i c h were calculated f o r ships w i t h a l e n g t h L o f 220 and 330 m (722 and 1083 f t ) , respectively, and i n a f u l l load con-d i t i o n (W = 1.0).

The b e n d i n g m o m e n t intensities have been n o r m a l i z e d by d i v i d i n g t h e m by the design value prescribed i n t h e rules of the classification societies, so as to make t h e design value equal t o 100 f o r the h o g g i n g condition, i n w h i c h tension is induced i n the deck o f the ship, and - 1 0 0 f o r the sagging b e n d i n g moments, w h i c h induce tension i n the b o t t o m .

The ship types indicated i n Table 2 have t w o d i s t i n c t op-e r a t i o n a l modop-es, i n f u l l load and i n ballast. T h u s , rop-esults arop-e g i v e n f o r both load conditions. The bound defined i n expres-sion (18) has been used to predict the p r o b a b i l i t y o f exceed-i n g the desexceed-ign value o f h o g g exceed-i n g bendexceed-ing m o m e n t d u r exceed-i n g one year, w h i c h is denoted as Q ( 1 0 0 , l ) i n Table 2. The period o f one year was recently shown to be the appropriate one to be able to account f o r t h e t i m e v a r i a t i o n of the r e l i a b i l i t y dur-i n g the shdur-ip's l dur-i f e t dur-i m e [20].

The p r o b a b i l i t y of exceeding the design v a l u e e i t h e r i n t h e h o g g i n g or sagging conditions, w h i c h is denoted as Q ( ± 1 0 0 , 1) i n Table 2, is the s u m of the two r e f e r r e d p r o b a b i l i t i e s . F u r t h e r m o r e , t h e p r o b a b i l i t y of exceeding t h e design value e i t h e r i n h o g g i n g or sagging and e i t h e r i n loaded or b a l l a s t condition, Q<,(± 100,1) is the s u m of the exceedance proba-b i l i t i e s Q ( ± 100,1) f o r the loaded and f o r t h e proba-b a l l a s t condi-tions.

The i n t e r a r r i v a l r a t e \ was determined b y a p p l y i n g equa-t i o n (7) equa-to equa-the m e a n d u r a equa-t i o n s indicaequa-ted i n Table 1. The as-s u m p t i o n waas-s made t h a t the i n t e r a r r i v a l t i m e f o r voyageas-s i n ballast was equal to the one of voyages i n the f u l l load condition. T h i s corresponds to the n o r m a l mode of operation, b u t other operational p r o f i l e s could easily be accommodated i n the c a l c u l a t i o n .

A n a l y s i s of t h e results of Table 2 indicates t h a t exceed-ance p r o b a b i l i t i e s can be s i g n i f i c a n t l y d i f f e r e n t depending on w h e t h e r one considers b e n d i n g moments i n t h e h o g g i n g or sagging conditions. For example, t h e y e a r l y p r o b a b i l i t y of exceedance f o r b u l k carriers i n ballast is 0.0393 f o r hog-g i n hog-g moments and 0.0031 f o r sahog-ghog-ginhog-g moments. T h e prob-a b i l i t y o f exceeding prob-any o f the types of moments is 0.0424.

Table 2 A n n u a l probabilities of e x c e e d a n c e of d e s i g n v a l u e of bending moments (100) in individual s h i p s of different type

Ship Load Bending Moment _{Q Q Q Qa}

Type Condition Y e a r (100,1) (-100,1) ( ± 1 0 0 , 1 ) ( ± 1 0 0 , 1 ) B u l k ballast 13.5 33.7 6.7 0.0393 0.0031 0.0424 0.0458 can'ier loaded - 7 . 9 27.6 6.7 0.0003 0.0031 0.0034 O r e / o i l ballast - 2 0 . 4 26.4 5.9 0.0000 0.0090 0.0090 can-ier loaded - 6 2 . 5 17.7 5.9 0.0000 0.1173 0.1173 0.1263 S m a l l ballast 29.7 21.7 10.4 0.0068 0.0000 0.0068 tanker loaded - 1 5 . 7 18.7 10.4 0.0000 0.0000 0.0000 0.0068 L a r g e ballast 12.8 25.6 6.5 0.0023 0.0000 0.0023 tanker loaded - 3 2 . 6 22.6 6.5 0.0000 0.0105 0.0105 0.0128

Noticeable differences i n the p r o b a b i l i t i e s of exceedance ex-ist f o r most ship types between the ballast and the f u l l load condition. For example, i n the case o f b u l k carriers these are 0.0424 and 0.0034 f o r the ballast and f u l l load condition, re-spectively.

The t o t a l p r o b a b i l i t y of exceeding the design values i n one year Q o ( ± 1 0 0 , l ) i n any load condition is the s u m o f t h e prob-a b i l i t i e s Q ( ± 1 0 0 , l ) f o r eprob-ach loprob-ad condition. Tprob-able 2 shows significant differences i n t h i s probability for the various ship types. W h i l e for b u l k carriers i t is 0.0458, for o r e / o i l carriers i t is 0.1263, for s m a l l t a n k e r s 0.0068 and f o r large t a n k e r s 0.0128.

The differences t h a t are apparent between the results f o r s m a l l t a n k e r s and f o r large t a n k e r s are a consequence of the d i f f e r e n t a r r i v a l rates, and o f t h e values used to describe the i n t e n s i t y o f the bending moments i n b o t h cases (Table 2). T h i s comparison shows t h a t even f o r the same ship type the p r o b a b i l i t i e s of exceedance can d i f f e r by a factor of 2, de-p e n d i n g on shide-p size.

5. A p p l i c a t i o n i n d e s i g n c o d e s

The development of probabilistic models f o r load effects takes i n t o consideration i t s a p p l i c a b i l i t y i n r e l i a b i l i t y anal-ysis and i n the development of p r o b a b i l i s t i c a l l y based struc-t u r a l design codes [6,7,10]. Therefore i struc-t is u s e f u l struc-to analyze t h e i m p l i c a t i o n s t h a t d i f f e r e n t f o r m u l a t i o n s of l i f e t i m e max-i m u m s t max-i l l - w a t e r load effects can have on code requmax-irements.

Codes can have a wide scope, b e i n g applicable for example to a l l ship types, or t h e y can be more l i m i t e d i n scope by c o n t e m p l a t i n g o n l y one type of ship. The decision about the scope of the code involves a balance between safety and economy, as discussed i n [22]. General codes m u s t be appli-cable to a l l ship types. Thus, w h i l e appropriate f o r t h e rtiost severe cases, t h e y are conservative a n d t h u s uneconomical f o r the other cases.

Codes o f a more l i m i t e d scope a l l o w more economical and r a t i o n a l designs i n the sense t h a t a l l ship types can be de-signed to the same n o t i o n a l p r o b a b i l i t y level. They are, how-ever, more cumbersome to apply because instead of one de-s i g n r e q u i r e m e n t t h e y m u de-s t include ade-s m a n y requirementde-s as the d i f f e r e n t types of ships considered. I n reference [10], a study was made i n w h i c h the effect of ship type was mod-eled by p a r t i a l safety coefficients t h a t should be i n c l u d e d i n the e x i s t i n g s t i l l - w a t e r b e n d i n g m o m e n t r u l e r e q u i r e m e n t . The r e s u l t i n g requirements reflected the same safety level for the d i f f e r e n t ship types.

W h e n f o r m u l a t i n g a code r e q u i r e m e n t applicable to a spe-cific ship type, i t is necessary to adopt a probabilistic de-s c r i p t i o n of the load effect i n t e n de-s i t y applicable to the whole class of ships, w h i c h is d i f f e r e n t f r o m the one t h a t describes each i n d i v i d u a l ship i n t h a t class. I n fact, t h e standard de-v i a t i o n of the load effects m u s t be l a r g e r so as to account f o r the v a r i a b i l i t y of mean values between d i f f e r e n t ships o f t h e same type, i n a d d i t i o n to the v a r i a b i l i t y of the load effects on successive voyages of t h e same ship. However, the mean v a l u e is the same i n b o t h cases. I t can be observed t h a t t h e standard deviations of b e n d i n g moments applicable t ó ship types, w h i c h are reported i n Table 3, are d i f f e r e n t f r o m the ones i n Table 2, w h i c h are applicable to i n d i v i d u a l ships.

A p a r a l l e l s i t u a t i o n exists between the standard d e v i a t i o n applicable to one ship type a n d to a l l ship types. I n codes of l i m i t e d scope one w o u l d use the s t a t i s t i c a l description of the s t i l l - w a t e r b e n d i n g moments i n d i c a t e d i n Table 3 f o r each ship type. I n a general scope code the bending moments would have the s t a t i s t i c a l moments i n d i c a t e d i n T a b l e 3 f o r a l l ship types.

Results o f t h e a p p l i c a t i o n of the a l t e r n a t i n g r e n e w a l pulse process to d i f f e r e n t ship types are presented i n Table 4 u n

-Table 3 Average of mean value a n d standard deviation of bending moments (hogging being positive) at m o s t loaded t r a n s v e r s e section for

different ship c l a s s e s

Ship Load Mean Standard Classes Condition V a l u e Deviation B u l k ballast 13.5 42.1 earner loaded - 7 . 9 34.1 O r e / o i l ballast - 2 0 . 4 40.4 carrier loaded - 6 2 . 5 19.3 Tankers ballast 33.0 30.4 loaded - 2 6 . 3 38.3 A l l ballast 29.1 41.7 Types loaded - 3 . 7 49.9 all 4.3 50.3

der the h e a d i n g o f QT- Included also are t h e results o f u s i n g load effect statistics applicable to a l l ship types, i n w h i c h case the y e a r l y p r o b a b i l i t y of exceedance is denoted by QA-T h i s example is intended to i l l u s t r a t e the i m p l i c a t i o n t h a t the choice of the scope of t h e data set has on the design v a l -ues of the load effects t h a t w i l l be incorporated i n the re-q u i r e m e n t s of s t r u c t u r a l codes.

I n a l l cases QA was l a r g e r t h a n QT as a consequence o f t h e l a r g e r variance o f the load effects, w h i c h now include the v a r i a b i l i t y o f t h e average loads between i n d i v i d u a l ships. These results show clearly t h a t to have a m a x i m u m s t i l l w a t e r load r u l e r e q u i r e m e n t common to a l l ship types i n -volves the a d d i t i o n a l conservatism of u s i n g QA instead of Q T , w h i c h is the appropriate one w h e n there is a r e q u i r e m e n t for each ship type. Table 4 also includes the y e a r l y exceed-ance p r o b a b i l i t i e s of a l l load conditions applicable to ship types, QTa, or to a l l ships, QAU- The l a t t e r is about t w i c e the f i r s t one i n most of the cases.

Comparing the f i n a l results i n Tables 2 and 4 shows clearly t h a t exceedance p r o b a b i l i t i e s f o r ship types are s i g n i f i c a n t l y h i g h e r t h a n f o r i n d i v i d u a l ships. Therefore, code r e q u i r e -ments w h i c h are applicable to ship types m u s t necessarily be more conservative t h a n requirements based on direct cal-culations on a specific ship. These results can be used to q u a n t i f y t h a t difference. For example, w h i l e the a n n u a l p r o b a b i l i t y of exceedance of the design v a l u e is 0.0458 f o r an i n d i v i d u a l b u l k carrier, i t is 0.215 f o r t h e whole class of b u l k carriers a n d 0.702 w h e n u s i n g a load model apphcable to a l l ship types.

6. C o n c l u s i o n s

A stochastic model has been developed to represent the t i m e dependence of s t i l l - w a t e r load effects. A n a l t e r n a t i v e r e n e w a l pulse process has been used to represent the succes-sion of voyages and periods i n p o r t t h a t ships are subjected to.

T h i s stochastic model has been s h o w n to be u s e f u l f o r pre-d i c t i n g the p r o b a b i l i t y o f exceepre-dance o f l i f e t i m e m a x i m u m s t ü l - w a t e r load effects. T h i s v a r i a b l e , w h i c h is i m p o r t a n t to establish the design values f o r s t r u c t u r a l codes, has been shown t o depend on the t i m e d u r a t i o n of t h e loads.

Predictions have been made o f the a n n u a l p r o b a b i l i t y o f exceeding the design value s t i l l - w a t e r b e n d i n g moments f o r various ships, s h o w i n g t h e effect o f the differences i n the probabilistic description o f load i n t e n s i t y a n d of load dura-t i o n . Predicdura-tions were also made f o r ship dura-types a n d a dis-cussion was provided on t h e a p p l i c a t i o n of t h i s model to the development of code requirements, e x p l o r i n g the possibility of h a v i n g d i f f e r e n t code scopes.

Table 4 Annual probabilities of e x c e e d a n c e of the nondimensional d e s i g n value (100) of still-water bending moments for various s h i p types u s probability distribution of load intensity corresponding to the s h i p type c o n s i d e r e d ( T ) or to all s h i p types {A), in hogging (100) a n d s a g g i n g ( - 1 conditions or in both ( ± 1 0 0 ) , in ballast or loaded or both situations (a). T h e s e values a r e to be c o m p a r e d with the probabilities in Table 2 w h i c h

applicable to individual s h i p s D a t a Set \ / Y e a r QT (100) QA (100) QT ( - 1 0 0 ) QA ( - 1 0 0 ) QT ( ± 1 0 0 ) QA ( ± 1 0 0 ) QT. ( ± 1 0 0 ) QA. ( ± 1 0 0 ) B C ballast 6.7 0.155 0.343 0.027 0.008 0.182 0.351 B C loaded 6.7 0.006 0.145 0.027 0.206 0.033 0.351 0.215 0.702 0 0 ballast 5.9 0.010 0.308 0.168 0.007 0.178 0.315 0.702 0 0 loaded 5.9 0.000 0.130 0.177 0.185 0.117 0.315 0.355 0.630 S T K ballast 10.4 0.159 0.508 0.000 0.011 0.159 0.519 0.630 S T K loaded 10.4 0.006 0.214 0.312 0.306 0.318 0.520 0.477 1.000 L T K ballast 6.5 0.104 0.335 0.000 0.008 0.104 0.343 1.000 L T K loaded 6.5 0.004 0.141 0.206 0.201 0.209 0.342 0.313 0.685 7. A c k n o w l e d g m e n t s

M o s t of t h i s w o r k was performed d u r i n g the author's stay at the D i v i s i o n of M a r i n e Structures of the N o r w e g i a n I n -s t i t u t e of Technology. The a u t h o r i-s g r a t e f u l to Profe-s-sor T o r g e i r M o a n f o r the i n s p i r i n g discussions and encourage-m e n t provided d u r i n g t h e course of t h i s w o r k . T h i s w o r k is i n the scope of the project Development of Codes f o r the De-sign of M a r i n e Structures, w h i c h the a u t h o r is conducting at I N I C , the N a t i o n a l I n s t i t u t e f o r Scientific Research, at its Centre for Mechanics a n d M a t e r i a l s of the Technical U n i -versity of L i s b o n ( C E M U L ) .

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