Fatigue analysis of offshore structures under non-narrow banded stress processes

To be printed in ¡he Proceedings of the Fir.0 European Offshore Mechanics Symposium, Trondheint, Norway, 20-22 August ¡990.

FATIGUE ANALYSIS OF OFFSHORE STRUCTURES UNDER NONNARROW BANDED STRESS PROCESSES

HKaradeniz, Ph.D. Deift University of Technology

Dept. of Civil Engineering DELFT / The NETHERLANDS

ABSTRACT

This paper presents a probability distribution of stress ranges and a spectral fatigue damage calculation of offshore structures subjected to nonnarrow banded Gaussian stress processes. The stress range probability distribution is empirically formulated on the base of a number of stress range probability histograms obtained frôm the rainflow cycle counting algorithm using the Monte-Carlo procedure. Although the probability distribútion

introduced is general for both linear and nonlinear damage

accumulations, in the paper the linear damage accumulation (Palmgren-Miner's rule) is used with a multi-linear S-N fatigue model. This model can also be used for a nonlinear fatigue model applying a piecewise linear approximation. Formulation of the mean damages in short-term and long-term sea states-arcpresented in general. For a linear S-N model the customarily known damage correction factor is constructed. It is worked out that an empirical stress range probability distribution can be more generally used in the damage calculation than an empirical damage correction factor.

KY WORDS Offshore structure, Fatigue damage, S-N model, Stress range, Probability density, Stress spectrum, Nonnarrow banded process, Damage correction factor.

INTRODUCTION

In practical fatigue analysis of offshore structures it is often assumed that the stress process is narrow banded. Uñder fürther restriction of the process as to be Gaussian the mean fatigue

damage can be calculated precisely from the stress range

distribution which becomes Rayleigh. In this approach, the stress range is ässumed reasonably to be double of the stress amplitude. This approach is appropriate if the stress spectrum has a single peak which may be observed under certain conditions e.g. a) when the wave frequency and the natural frequency of the structure lie

within close proximity of each other or b) assumption of a

quasi-static response of the structure neglecting the effect of the natural vibration which is applicable only in sever sea states and if the structure is not dynamically sensitive. In realistic:situation, the contribution of aseversea state to the cumulative damage is

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considerably small in the long term since the dumulative damage is primarily produced by small sea states (sea states with small significant wave hàights). Because, a sever sca state has much lower probability of occurrence than a small sea state. En casc of

smáll sea states, although there may be some insignificant

flùctuations in a stress spectral shape of a complex strúcture, fundamentally the stress spectrum displays two major peaks occurring respectively around the wave frequency and at the lowest natural frequency of the scructurC. In this case, the stress

variation becomes a nonnarrow banded process so that the

Rayleigh distribution cannot be used to represcnt the distribution of the maxima, and also the stress range cannot be taken any longer as double of the stress amplitude. In order to improve the damage

under noinarrow banded processes a correction factor was

intrOduced by Wirsching and Light, 1980 using some rainflow

streis rángcs under a Gaussian process. Lutes er al., 1984,

Winterstein and Cornell, 1985 have also investigated the same problem for both Gaussian and non-Gaussian processes. They concluded that the nonnormality of the stress procçss. which is inherent because of the nonlinearity of the drag force term, gives rise to the damage and may be more significant than the bandwidth effect along. The importance of the nonnormality may be problem dependent. i.e. it depends on the degree of the nonlinearity of the process, so that it seems to be difficult to judge a solid decision from sorne-simple examples, for a complex structure which is both equally in drag and inertia regimes. Also it may be of practical interest to know individual error rates introduced by different assumptions in an analysis.Hence, the bandwidth effect is treated here as an important subject to be studied. The present paper introduces an alternative approach to the damage calculation under a nonnarrow banded Gaussian stress process using a multi-linear SN model. The procedure is based on the-experimental (rainflow) determination of the stress range probability distribution which is obtained independently from the parameters of the fatigue model used. This procedure is very suitable for the damage calculation using a multi-linear S-N model, and also it can be successfully applied to the fracture mechanics formulation of the damage assessment. Before it is introduced the following section, which presents some basic information, is thought to be useful.

DAMAGE ACCUMULATION RULE AND MULTI- LINEAR S-N FATIGUE MODEL

Fatigue damage is expressed as being a fúnction of the number of.

stress cycles and ranges. The damage caused by one cycle is

stated as,

dD-

(1)

where N(S) is the totai number of cycles to failure at the stress range S. By using Paris.Erdogan's law of the crack growth (Paris

and Erdogan, 1963) the number of cycles N can be expressed as, af

Cl

J

ay(a) Stm

(2)

where Cl and m are regarded as material constants, a is crack size,

ao and ar, are the initial and at the failure crack sizes. Y(a) is a

function of geometry and S is the stress rangc. For simple geometrical models, N can be calcúlated from Eq.2. In the case of random stress processes as assumedhere, a statement of N which is simila to Eq.2 was given byWirsching et al.. 1987, in which mean stress range powers wereused. For offshore tubular joints where gebmetriCal. environmentaland loading complexities exist,

the fracture mechanics approach to the damage accumulation is

not yet widely applicable. Instead, a function of N based on experimçntal data (S-N curve) seems to be more reliable and.sound

for practical purposes. Beyond that, as shown by Wirschingand Chen, 1988, the fracture mechanics formulation of N can be cast in a form of characteristic S-N curve. Therefore, a multi-linear S-N model, see Fig.I, is used in this paper. A particular segment

In 5h (NL) In (Sh)+1 In (S (1) in Fig.1 - A multi-linear S-N fatiguemodel.

of this model defiñed betweçn two stress ranges, shown by jin Fig.1. can be written as,

N=CS

(3)

where C and k are parameters which may be obtained from some experimental data or calculated from a piecewise linearization of

a nonlinear model, Sbis the local (hot-spot) stress range.

Assuming a linear damage accumulation rule (Miner's rule) the damage corresponding to the segment j of a multi-linear S-N model can be calculated from thesuperposition of damages due Lo

one cycles. This damage is written for variable stress ranges as,

D=5)

(4)

where nj is the total number of cycles of stress ranges lying

between the specified limits of the segment j in Fig.1. The total damage. D, for all segments of the S-N model will be calculated from the superposition of D for j=l to NL where NLis the number of scgments.

PROBABILITY DENSITY FUNCTION OF STRESS

RANGES OF A NONNARROW BANDED GAUSSIAN

STRESS PROCESS

Sïrcss ranges are closely related to maxima and minima of the stress variation. For a narrow band process with ¿cro mean, the

stress range is assumed to be double of the stress maxima

(amplitudes) and its prôbability distribution follows a Rayleigh function. For a nonnarrow banded stress process it cannot be precisely determined because of local variations of maxima and minima in a random fashion so that the stress range becomes a

random process dependent ort the maxima and minima. The

probability density fnction of the maxima of a wide-band

Gaussian process was given by Cartwright and Longuet-Higgins, 1956, as written by,

f,(u)=P(

.ucxp(

₎

cI(-)

(5)

where u is a variableof the stress maxima denoted by s, e and a arc

respectively the spectral bandwidth parametcr and the irregularity factor, a is the standard deviation of the stress process and (.) denotes the standard normal distribution function. Eq.5 includes

both (+) and (-) maxima. The afactor and the e parameter arc

calculated from,

= 2.... m22

where mo, m and m are respectively the zero, second and fourth spectral moments of the stress process (Karadeniz. 1989). It is worth noting here that the zero moment, mo. is equal to the stress

variance since a zero mean process is assumed. As a special case,

when e=O which corresponds to a narrow-band process. Eq.5 reduces to a Rayleigh function. Now, if we suppose that the stress

range, S. can be defined as,

S=AIsl

(7)

where A is a parameter with (+) value, sis the stress maxima Sand 't. I denotes an absólute value which is. used to obtain +S values. It follows from Eqs.5 and 7 that the density function of S can be

obtained as.

f5(y) = f1(y)-F f2(y) (S)

where y denotes a variable of the stress rangeS, fi(y) and f2(y) are

respectively. (6) 2e 1 __Y 2A2a2e2 (9)

y cp(

erf()

in which erf(.) denotes the error function. This density functionof S agrees very well with rainflow density histograms forsmall and large e values if A is suitably chosen.For moderate e values a poor

agreement is observed. Theresult can be improved considerably

if

different values of A in f i(y) and fi(y)of Eq.9 are used. In that

case Eq.7 is not any more valid. As a matter of fact, Eq.7 is used only to figure out globally the type of the probability distribution

As it is seen, the first part of Eq.9, f i(y), is a Gausianty;e and the second part. f2(y), is a modified Rayleigh type. The result can be further improved if a modificd Weibull type distribution is

used instead of the Rayleigh type given by f2(y) in Eq.9.

Henceforth, however, we assume that a normalized stress range. defined as

a

is used in the paper. The probability density function of this normalized stress range is written by, as similar to Eq.8,

f5(x)=f1(x)+f2(x)

(Il)

where fi(x) and f2(x) are respectively the Gaussian and the modified Weibull types which are.

ex f (x)

= (

(12)

f2x)=a()

cxp[_()d1

crf[?()C2}

In Eq.12. A, B and C are the parameters of the distribution to be

determined later in this paper. For the time being, it is supposed that these parameters are known. The distribution function of the normalized stress range s can be obtained from the integration of

Eq.l 1. The result is written as,

F5(x)= F1(x) + F2(x) (13)

where Fi(s) and F2(x) are respectively,

2

F1(x)=t Crf(-V)

2 C/2 C C/2

Fa(X)=

crf[()

J-

uexp[_()

J crf[?()

I

For a narrow-band process, i.e. e=0 and =l, as it will be stated

later that C=2 and B=2'2 , Eq.13 reduces to a

Rayleigh distribution written by,

F(x)1cxp( ..)

(15)

The moment generating function of x is also of interest in the mean damage calculation. This function is obtained in general as to be,

2 k42 k

Mk=

V(a)

Ak+ BkaJ xC crf(?I) edx

(16)

o

where It is a real number, integer or floating point, r(.)is the Gamma function. The mean fatigue damage can now be calculated using the stress-range density-function given by Eqs.1 I and 12.

CALCULATION OF THE MEAN DAMAGE FOR A

MULTI-LINEAR S-N MODEL AND THE DAMAGE

CORRECTION FACTOR IN A SHORT-TERM SEA STATE

As previously mentioned, the total damage using a multi-linear S-N model can be calculated from the superposition ofindividual damages of all segments of the S-N model. The damage of one segment has been given by Eq.4. The mean value of this damage can be written as, for the segmenti (sec Fig.1),

E (Dl= n1, E[__L_]

(14)

(17)

N(S)

or

using

Eq.3 with (ShSCF.S), where SCF is a stress concentration factor, and replacing S by S=a.x from Eq.I0. this mean damage will be,

E[Dl

= i. (SCF o )¼ J 5k f5(x) dx (18)

O

where EI.] denotes -an average value. Since the integral part of Eq.18 is the probability moment of x, Eq.18 can be rewritten as,

where Mk. is the moment generating function of z for k=k given by Eq.16.1The total mean damage can now be calculated from the superposition of E[D1 for j1 to NL. where NL is the total number of segments of the.S-N model. This total damage can be written as,

E-[D]= n - (SCFO)ki _Mk. (20)

where n is the total number of cycles for all segments (n=nj) in a sea state, P is the probability that a stress range lies betweenthe limits of the segmentj

in Fig.l. Thus, P=P1(S S

S+t}, or equally Pj=Pr(xj x x+i) where S and S.j are the nominal

stress range limits calculated from S=(Sh)j

I

SCF and Sj.1=(Sh)j+t / SCF. The limits of x are calculated from Eq.1O as to be x=S I-o

and xj.i=Sj+i / o. This probability can be written as,

= F( x,)-- F( s.) (21)

where F5(.) is the probability distribution of * given by Eqs.13 and 14. In this model of the calculation of the damage, it is experienced from the rainflow results that the total number of cycles n in a sea state is equal to T lTmwhere T5 is an assumed duration of the sea

state and Tm is the mean period of the stress maxima in this sea state. In a narrow-band case Tm is replaced by To. mean zero crossing period of the stress, since these two will be equal. In this special case, the total damage becomes,

E [D_{mb= :}

(2'ff

SCF

a)ki

( i + (22)

where (Pnb)j is the same as P for the narrow-band case, calculated from Eq.21 where F5(.) will be taken from Eq.l5 instead of Eqs.l3 and 14. A damage correction factor. , for a multi-linear S-N

model can now be calculated from Eqs.20 and 22 as

X= ELD] / E(D]nb with n = T5I Tm in Eq.20. For a linear S-N

model this correction factor will be, keeping in mind that Tm ITo=

Mk ₍₂₃₎

(2JT)k r(-+)

where Mk will be calculated from Eq.l6. lt is worth noting that for a linear S-N model and a given sea state, X will be a function of only k while, for a mulii-linear S-N model, it will be a function of k and C where j=1 to NL.

TOTAL DAMAGE IN A IJONG-TERM

PERIOD AND

ESTIMATE OF LIFE TIME

The mean cumulative damagein a long term is ultimately used to

estimatefatigue lives ofoffshore structures.The mean damagein

aparticular short-term sea-state is calculated aspresented in the

previous section. In a long term, e.g. a servicelife-timeT, there are many seastates with probabilities of occurrences so that the

long-termcumulative damage-will be calculated by superimposing

contributions of damages inshort-termsea states. Thus, it follows from Eq.20 that (nT / Tm, Ts is the duration of the short-term sea

state, Tm is the mean period of thestress maxima in that sea state). the mean cumulative damage using a multi-linear S-N model can be written as, NHs E [D J ₌ E LdD]5 (24) j=l

where NHs is thetotal number of sea states in thelong term T and

E (dD]5is a mean damagecorresponding to one stresscycle in a

short termsea state, whichis written from Eq.20 as,

E[dDl =

j-

(SCF0)k Mk (25)

Eq.24 can also be stated in termsof T (service life-time) as,

D01=T.D5 (26)

where D is the cumulative damage corresponding to a unit time in the long term, which can be written as,

NHs

Du=E[dD)s

(27)

InEq.27.Pu denotestheprobability that thesea states occurs.This modelis used for adiscretelong-termdistribution ofsea states.If, however, acontinuous lông-tcrm distribution is used, then Eq.27 becomes,

rr

I

j

J - E(dDI. fH,TL(h,Odhdt (28)

00

where fH5.T1(h,t) is the joint density function of the significant

waveheight(H5) and the zero-crossing period (T1) of waves. Ifthe

severity f the sea stateis characterized byH5 only, Lhcn Du will

be calculatedfrom.

D5

= 5

j

E[dD[5 fH,(h) dh (29)

where f5(h) is the marginal density function of H5, representing

the probability distribution of the long-term sea state. lt

is assumed that failure occurs when the mean cumulative damage D(oI reaches a value, Dr, which is taken to be equal to I as

commonly accepted. Using this failüre criterion a mean life-time will be estimated from Eq.26 as,

T1í=

(30)

CALCULATION OF THE PARAMETERS OF THE STRESS RANGE DISTRIBUTION

The probability distribution of stress ranges has been given by Eqs.l3 and 14. Its density function is also presented by Eqs.11 and 12 as parts of the Gaussian and modified Weibull types. In this section, calculation of the parameters A of the Gaussian part, B and C of the Weibull part will be presented. For this purpose, a number of probability-density histograms of rainflow stress

ranges, which are obtained by applying the Monte-Carlo

prOcedure. are utilized. The rainflow stress ranges are calculated

from the spectral function of a normal stress

of a monopod

structure shown in Fig.2, assuming that the inertia force

coefficient Cm is frequency independent. Using various significant

20m.

SWL

Fig.2 - An example monopod structure

wave heights (H1) and natural frequencies a number of spectra are generated, varying between uni- and bimodal shapes. In this way. a range of between 0.09 and 0.97 is produced. For each spectral shape, a number of 200 Monte-Carlo time-simulations arc used. The time period of each simulation is taken to be T=600 seconds. For a spectral shape, the extreme maximum and minimum of all rainflow stress ranges are found. Then, the difference between these two extremes is divided into a number of 50 equally spaced intervals. The number of stress ranges, say nj, lying between each interval are coünted. The ratio between n1 and the total number of stress ranges,sayn, identified by the rainflow algorithm gives the mean value of the probability that a stress range lies within the interval considered. In this way, probability-density histograms for all spectral shapes are Obtained. Based on these rainflow density histograms the parameters, A, B and C, of the analytical density function given by Eqs.1l and 12 are calculated by using a curve fitting procedure. A straightforward procedure is to usethe minimization of the total mean square error stated as,

50

E

=

[f1 f5(x1)]2 (31)

where f and f5(xj) are respectively the values of the density histogram and analytical function corresponding tothe interval i. Then, A, B and C can be calculated from E/aA= O

ÒE

/B

0,and E /aC= 0. Tbis procedure may apparently result in a best global shape. But, since the tail of the density functionis

much more influential than the global shape in the damage

calculation arc alternative procedure is used in this paper. The

parameters B and C of the Weibull part are calculated using the third and fourth probability moments calculated from rainflow histograms and the analytical function. The parameter of the Gaussian part, A, is calculated from the minimization of the mean square error as stated above. The third and fourth analytical

moments used are calculated from Eq.16 for

k3 and 4

respectively. This procedure requires an iterative solution. lt is experienced that the calculation is convergent. For someevalues,

the rainfiow density histograms and the corresponding analytical functions are illustrated in Fig.3. For each spectral shape, i.e. for cache value, the parameters A, B and Care calculated. In this way. scatters of che parameters A. B and C in the-Edomain are obtained.

From curve fittings to the scatters, these parameters

arc empirically calculated as being functions ofe. Using a nonlinear

regression analysis the following mean function is obtained for A.

1.164+ 1.123e°25 2.113 e4 (32)

The variance of A is calculated to be

aA=0.0014l5. The pärameters B and C are calculated in two parts. If e 0.80, the following mean functions are obtained.

TI

100 rn.

r

.0. 53

J

.0. 02

Fig.3

-where Bo.Bi and B, and Co, Ci and C2, arc some functions of e as presented below in Table I.

.0.051

.0.000

N

t

-,

TABLE I - FUNCTIONS OF THE PARAMETERS B AND C

FOR e> 0.80

0.00 I 59 .3 9 478 6.37 7.57

Strea. range _/Standard d,viat,n 130=

-

1.920e+ ¡.024

B= l.966c+ 1.311 1.678e+ 1.174

C_° -.1.685c+ 0.227

(I-c'°)

.0. 290

Probability Ona.ty r,n,t,on

H. - 3.00 C1= - 3.774c+ 1.764

(i

-c'°)2 C -9.474e... 6.004 2

(1-c'°)4

.0.241 .4_C -3.00_-0.565 #0.193 A - 1.8858 U - 2. 6899

C - 1.9601

The variances of B and C are calculated to be respectively

.0. 45 ais =0.001069 and ac2=0.003916. In che calculation of B and C

for e 0.80. a constraint is used to satisfy the narrow-band

.0.097 condition, RANGE=2'AMPLITUDE with a Rayleigh distribution

for AMPLITUDE. This constraint results in C=2.0and B=2!2 fòr

.0.040 e=0.0 as they are seen in Eq.33. The scatters of A, B and C. and

.0. 000 their mean functiòns arc illustrated in Fig.4. Since in the damage_{calculation mean stress-range-powers (probability moments) arc}

0.00 I 66 3.33 4.99 6.65 8.31

Str,,. roflg,

/

Standard dCviat,On of major concern, they are also calculated using respectively

rainflow histograms and the analytical function wich the

Prababilily danaity lnnctan empirical A, B and C parameters. These are demonstrated in Fig.5,

where the values are normalized respectively to (1.0*Max.V.),

.0.467 _{(1.54Max.V.) and (2.56Max.V.), Max.V. denotes the maximum}

.0.388

II, -7.00

-I, -3.00

C -0.790

value ofeach plot.

.0.311 A -_{8 -} .3500

2.5133 EXAMPLE.

.0.23

C -1.7742

This section presents examples of damage correction factors and the damage calculation of a monopod structure. In order co

.0. 56 compare the damage correction factors (DCF) with sonic others

cited in the literature, a linear

S-N

model is used. First, a

.0.079 comparison of the DCF is made for a number of k values (slopes

.0.000

of the S-N line) using uni- and bimodal spectral shapes shown in Fig.6. The DCF are calculated for a range of e values produced by

0.00 61 3 22 4.82 6.43 0.04

Streos range /Standard devotion changing the width of the unimodal spectrum from 0005 to

9.005, and the distance between the two modes of the bimodal

Probability dcflhity tvnctian spectrum from (0)2 - 0)1) =0.5 to 5.0. It is generally observed that,

fór large k values, results of Eq.23 and Wirsching and Light. 1980.

.1.068 _{are close to each other and considerably lower than those of}

Ho -15.00

ah -300 Wintcrstein and Cornell, 1985 for both spectral shapes. For small

+0. 890 _-0.930

and moderate k values, results of Eq.23 remain between those of

.0. 712 A -0. 72109 -3.1453

C -2.5036

Wirsching and Wintersteiri. Some numerical values arc presented in Table 2 where Wirs. and Wint. denote respectively Wirsching

.0. 534 and Winterstein. Illustrations for k=4.0 arealso shown in Fig.6 for

both spectral shapes. More exämples of the DCF and the

.0. 356 demonstration of thc damage calculation arc presented on a

monopod structure. The structural model and geometrical data are

+0. 178 _{as shown in Fig.2. Further, it is assumed that the wall thickness of}

the monopod is 0.20 meter, the critical damping ratio is.=0.01

.0.000

0. 00

1flh1Th1lILITTT

52 3.03 4. 55 6. 06 7.58 the inertia force coefficient iSCm=2.0. the mass of the deck is 2000

Stra,. range /Standard dCn,Ot ion ton (2*106 kg). The Picrson-Moskowitz (P.M.) sea spectrum is

used with a shape parameter of cs,=0.0081. The monopod is Illustrations of rainflow probability density histograms assumed to be full of water up to still water level. The dominant andanalytical functions. natural frequency is calculated to be 0)k=2.O rad/sec.

.0. 306

.0255 '0.204

2'I + 0.051 e05 - 1.419e3+ 1.011 p.(c)=2.0 - 0.277 e° - 1.995 E3 + 1.604

If £ > 0.80 che following mean functions arc obtained. 3(e)- 2.98Ó - 0.627 £ + 844.9 B0.- 674.7 B1 + 298.8 B2 2.064-0.391e + 7.815 C0- 2.607 C1+ 0.129 C2 (33) (34) / ji

Pr090b,Iity danoity fntian

Ho - 1.00 -1 -3.00 C -0.201 A - .5157 O - 2. 0269 - 2. 0929

2. 50 #2. 00 +1. 50

1.00

#0. 50 +0. 00 0.00 0. 20 0. 40 0. 60 0.80 1.00

Spectral bond-odth parameter. E

a - Parameter A of the Gaussian part.

#5. 00 +4. 00 +3. 00 +2. 00 + 1. 00 +0.00 -0. 00 0. 20 0. 40 0. 60 0. 80 1. 00

Spectral bond-width oorometcr.E

b Parameter B of the Weibull part.

#6. 00 +5. 00 44. 00 t3. 00 f2. 00 +1.00

Poromcter of the Causion port (A)

B parameter of the Weibull port

C parameter of the Weibul I port

variance et C -0.0039160

+0.00

-0.00 0.20 0.40 0.60 0.80 1.00 Spectral bond-width poromcter.E

1.00 .0.75 .0.SO .0.25 .0. 00 0.00 0.20 0.40 0.60 0.80 1.00

Spectral bond--idth peroecter.c

.0. 75

+0. 50

t0.25

c - Parameter C of the Wcibull part. . F(w) 1.0

2

F(w).e 1.65 cxp(-0.457w ) F(w)=11

Fig.4 Scatters and mean curves of the parameters of the stress w3

range density function in the e domain.

Iioent. o! the OtrOls-.o,gs oriobIe(e)

(nor.eIeed to I. 1.5.

2.5 o.i...

_oOI.00)

Mosente of the Steno-tinge variable (o)

(norlizwd to I. 1.5. 2.5 OO;+Oval+Cn)

.0.00

0.00 0.20 0.40 0.60 0.00 .00 Speetrol bønd-idth peroectet.

Fig.5 - Illustrations of the rainflow and analytical probability moments of the stress range.

TABLE 2 - COMPARISON OFX FOR UNI- AND BIMODAL SPECTRAL SHAPES

The spectrum of a normal stress at the bottom of this structure is analyticálly obtained as.

2

7.57 1&'[IO.2w2+2CXp(_IO.20?)_1]

S 5(w)- F2(w)S (w) (35)

4ri

w 22

2 w 2

w 1k'-() )

+ 4 _(ii;)

where F(w) is a frequency reduction function of Cm defined as.

and Sp(w) is a short-term P.M. sea spectrum given as.

k

Unimodal spcctrum Bimodal spectrum

¿ Eq.23 Wirs. Wint. t Eqî3 Wire. Wint.

0.223 0.922 0.868 0.968 0.218 0.92 b.811 0.981 4.0 0.425 0.377 08l6 0.951 0.449 0.859 0.807 0.953 0.531 0.851 0.804 0.942 0.745 0784 0.795 0.987 0.223 0.841 Ò:7n Ó.935 0.218 0.843 0.774 0.953 6.0 0.425 0.801 0.733 0.915 -0.449 0.795 0.730 0.956 0.531 0794 0.729 0.904 0.745 0.792 0.728 0.964 òi23 0.754 0.687 0.902 0.218 0.7-56 0.688 Ö.923 8.0 0.425 0.776 0.663 0.880 0.499 0.737 0.662 0.926 0.531 0.746 0.662 0.869 0.745 0.838 0.662 0.936

for 0.0 w 1.085

for l.085 w1.981 (36) for 1.981 C W

+1.00 .0. 75 .0. 50 .0. 75 +0.50 +0. 25

Damage correcto. factor forh -4.00

-

Present .0th WrIchiflq - WntCrOtein .0.25 +0.00 0.00 0.20 0.40 0.60 0.60 f00 SW)

Spectral band.idtff peremater. C

(A)

Damage corract,on foctor for h - 4.00

(A) - Prasent .0th e; riching

- lintaritoin

.0.00 0.00 0.20 0.40 0.00 0.60 t. 00

Spectral bénd.idtl parasotar. C

Fig.6 - Illustrations of X for uni- and bimodal spectral shapes for k = 4.0.

S,(w)= J. cxp( .1.LL)

(7)

In orderto calculate the rairiflow density histograms Eq.35 is used with in the whole frequency region, which produces baslc4lly uni .rnd bimoddl shapes .js depending on the choice of the values of Fl and Wk. The DCF arc also calculated using this spectral functibn with F(co)=I.0 for a number of k values. lt is generally obtained that Eq.23 gives a better fit, to the rainflow' results. lt is further observed that Wirschings X is somewhat unconservative in the region of small e values while it is.found rather conservative in the region öl high e values. As o speciál case when k=l.0 it has ben shown by Lutes et al., 1984 that X=1.0 irrespective of e values. This condition is verified by the rainflow results of this paper, and also satisfied by Eq.23. For k=1 .0and 5.0 the DCF are illustrated in Fig.7; For some sea states, sornek values and a natural frequency of Wk=3.O rad/sec, numerical resultsof.X are also compared in Table 3 where R.F. denotes the rainflow and Wirs. denotes Wirsching: It is worth noting that Wirsching'sX has been obtained as an average factor of four different spectralshapes while the results of Eq.23 are typical for offshore structûres with stress spectra having fundamentally uni- or bimodal shapes. Because, the parameters A. B and C in Eq.23 are empirically calculated in this paper using uni- and bimodal spectral shapes.

Spectral bofld.dth poroo.ctCr. E

+ I.00

.0.75

+0.-50

+0. 25

Domage correct ion factor

+2. 00 Presont oork Wirsching + Roin(loe +0.00 - - -0.00 0.20 0. 40 0. 60 0.80 1. 00

Spectral bor.doi'dth poromoter. E

Fig. 7- Illustrations of damage correction factors of a monopod structure for k = 1.0 and 5.0.

TABLE 3 - COMPARISON OF X OF A MONOPOD STRUCTURE FOR F(0)=l.0.

The damage calculation of' the monòpod structure is carried out using the frequency redúction function of Cm given by Eq.36. which results in á spectral shape with relatively broad banded peak around the' fundamental wavc frequency. Such a spectral shape is illustrated in Fig.8 for H=4.59 rn. and the natural frequency of the monopod, ok2.O rad ¡sec. Using a linearS-N model with k=3.0

and C=7.31° 1031 (stress unit isN/rn2)and assuming a SCF of 1.5,

mean damages are calculated for both -short- and long-term sca

states. For the short-term sea states the X values arc also

calculated. For the long-term sca states, a Weibull distribution for H0 with the density function of,

C

h-A Ce-1

h-A Cw

fH.(h)(j)

cp[(____-)

_J (38) where A=0.60, B1.60 and C= 1.20 (Battjcs, 1972) . is used. The' results of the calculations are presented in Table 4 where D0 denotes the damage for a short-term sea state, P0 is ihe probability

H, e

k=3.0 k=4.0

-k=5.0 R.F. Eq:23 Wirs. R.F. Eq.23 Wits. R.F. Eq.23 Wirs. 106 0274 0953 0948 006 0920 0912 0851 0880 0871 0801 2.45 0.503 0.906 0.893 0858 0.874 0.858 0.806 0.842 0.8250766 4.59 0.687 0.832 0.837 0.837 0.807 0.812 0.796 0787 0.801 0.761 7.50 O'805 0.762 0.750 0.830 0.745 0.7-32 0.794 0.733 0.737,0.76 I 11.35 0.885 0.698 0,698 0.828 0.689 0.686 0.794 0.683 0.685 0.761 16.71 0.945 0.653 0.678 0.827 0.645 0.6%) 0.7.94 0.632 0.661 0.76 41.50 .1.00 +0. 50 +0.00 0.

-e Present cOrk Wir irhg Roini 0e -. k1.O 00 0. 20 0. 40 0. 60 0. 80 I.00

str@ss spectrum ter lis - 4. 59 e +4. 23 .3. 53 .2. 82 .2. 12 +1.41 .0. 71 +0 00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 I rcucncy (rod/eec)

Fig. 8 - A typical stress spectrum of a monopod structure for H3 = 4.59 meter (a frequency dependent Cm is used).

TABLE 4- RSULTS OF THE DAMAGE CALCULATION OF THE MONOPOD STRUCTURE

that a sea state occurs, R.F., Wirs., N.B. and Pres. denote

respectively rainflow, Wirsching, narrowband and present work. As it is seen from Table 4, for this particular example, both present work and that of Wirsching underestimate the damages for all sea states. For small sea states the present work gives better results while for severe sea states the situation is reversed.

SUMMARY AND CONCLUSIONS

A formulation of spectral fatigue damages for offshore structures

is presented for a general Gaussian stress process using a

multi-linear S-N fatigue model. The formulation is based on using a probability density function of stress ranges. This density fúnction consists of two parts as being Gaussian and Weibull types. The parameters of the density function axe calculated on the base of a number of rainflow density histograms obtained from uni- and bimodal spectral shapes of a normal stress of amonopod structure. As it is more familiar, the damage correction factor is alsó presented. The applicability of the proposed probability denÉityfùnction to some other spectral shapes is investigated. Itis generally concluded that a) the proposed density function canbe

more generally used in the fatigue

analysis than a damage correction factor, b) this function is independent of a S-N line unlike a damage correction factor, e) for a linear S-N model, the damage correction factor is solely dependent on the slope of the S-N line while, for a multi-linear S-N model, it is dependent on both the slopes k and the constants C for j=I to NL, whereNL is the total number of lines of the S-N model, d) it reveals from the examples demonstrated that the empirical probabiJity distribution of the stress range can be practically used for the fatigue analysis

-

e-of e-offshore structures although it is basically derived from uni-añd bimodal spectral shapes, e) for large k values (slopes of S-N lines), e.g. k> 8.0, the empirical probability distribution produces unstable and non-monotonic damage correction factors in the region of high e vàlues. This situation can be explained in the light of rainflow density histograms so that, for high e values, the larger

histograms are concentrated around the zero point on the

stress-range axis while relatively smaller histograms arc spread in the tail which mainly contributes to the damage if the slope of the S-N line is large.

REFERENCES

Battjcs, J.A. (1972), 'Long-term wave heighi distributións a

seven stations around the British Isles," Deutschen

Hydrngrphischen Zeitschrift. Band 25, No 4, pp. 179-189 Cartwright, D.E. and Longuct-Higgins, M.S. (1956), "The statistical distribution of the maxima of a random function," ?rnc. Rnl. Soc. A, pp.212-232.

Karadeniz, H. (1989), "Spectral analysis and fatigue

reliability assessment for offshore structures," Proceedings of the 5th. International Conference on Structural Safety and Reliability, ICOSSAR'89, San Francisco, U.S.A.

Lutes, L.D., Corazao, M., Hu, S.J. and Zimmerman, J., (1984), "Stochastic fatigue damage accumulation "Jnurnil of structural F.rZineerinZ, ASCE, Vol.110, No.11.

Paris, P.C. and Erdogan, F. (1963), "A critical analysis of crack propagation laws," Journal nf Basic Enineerin Trans. ASME, Vol.85, pp.528-534.

Winterstein, S.R. and Cornell, C.A. (1985), "Fatigue and fracture under stochastic loading," Proceedings of the 4th. Internatiónal Conference on Structural Safety and Reliability, ICOSSAR'85, Tokyo, Japan.

7, Wirsching, P.H. and Light, M.C. (1980), "Fatigue under wide band random stresses," Tournai nf the Structural Div ASCE, Vol. 106, No. ST7.

Wirsching, P.H., Ortiz, K. and Chen, Y.-N. (1987), "Fracture mechanics fatigue model in a reliability format," Proceedings of the 6th. International Symposium on Offshore Mechanics and Arctic Engineering, OMAE'87, ASME, New-York. Wirsching, P.M. and Chen, Y.-N. (1988), "Considerations of probábility-based fatigue design for marine structures," Journal nf Marine Struci1Irs pp.23-45, Vol.1.

H, t

D, ('IO°) foiTsi ccc.

P,

R.F. Eq.23 Wire. N.B. R.F. PICS. Wirs. 1.06 0.289 0.961 0.945 0.902 0.722 0.694 0.682 0.651 4.559E-1 2.15 0.553 0.932 0.880 0.851 1.971 1.837 1.734 1.677 4.170E.I 3.59 0.702 0.914 0.831 0.836 4.449 4.066 3.697 3.719 1.I-34E.t 7.50 0.790 0.892 0.767 0.831 8.902 7.941 6.828 7.398 1.040E-2 11.35 0.855 0.882 0.731 0.829 18.31 16.15 13.38 15.18 2.610E-4 16.71 0.910 0.866 0.674 0.828 49.14 42.56 33.12 40.69 8.985E.7 D,5, (lang 1cm) 1.755 1.632 1.530 1.501