Probabilistic fatigue damage calculations of offshore structures under narrow and broad band stress processes

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Dr. H. KARADENIZ

TU DeIft

Deift University of Technology

Faculty of CivilEngmoo.ing

Group of Structural Mechañics

i - INTRÖIXTCTION 3

2 - PAL11GRF-MINER'S RULE AND S-N CURVE 5

3 - PROBABILISTIC FATIGUE DAMAGE FOR CYCLE OF A NARROW-BAND

STRESS PROCESS 8

'4 - PROBABILISTIC FATIGUE DAMAGE FOR ONE CYCLE OF A BROAD-BAND

STRESS PROCESS 15

5 - DrtINA.TION OF DAMAGE CORRECTION FACTOR FOR BRQAD-BAND

STRESS PROCESS Vi

6 - CUMULATIVE FATIGUE DAMAGE tIJR ING A SEA STATE AND IN A LONG

TEÑ4PERIOD 30

7 - DONSTRATION OF FATIGUE DAMAGE CALCULATIONS BY AN DANPLE

STRUCTURE '40

8 - DISCUSSION AND CONCLUSIQNS. 56

REFERENCES... 59

APPENDIXA ...62

I - Tirne-HistoPy simulation of the stress process 62

2 - Rain-Flow counting method 644

3 - Monte-Carlo method 65

APPENDIX B 68

A numerical integration procedure to calculate total fatigue damages 68

i - Normal Distribution 69

2 - Lognormal Distribution ₇₀

3 - Wibull Distribution 71

APPENDIXC ...73

Calculation of. variance of A for limited number of Monte-Carlo

experiments ...73

Normal Distribution 75

the formulation, the Palmgreri-Miners damage-accumulation rulé is used in combination with an experimentally determined S-M curve allowing for

non-¡lnear'itieS in logarithmic scales in which case a piecewise linearization technique is employed.

Mean fatigúe damages under both narrow- arid broad-band stress processes aré f iFst formulated for one cycle. The randomness of the damage due to

broad-band stress process is represented by a random variable which is

estimated statistically by using a cycle cunting method (Rain-Flow) fPöm the time-domain simulation of the stress process, given in a spectral

form, and the Monté-Cárlo procedure.

The cumulative damage during a sea state is calculated in terms of the

damage for one cycle and the total number of stress cycles, which is assumed to be randöm for brOad-band process, in the sea state. Then, the cumulative damage for the long term sea state is calculated by the

superposition of damages with probabilities of occurrences of individual séa states. On this basis1 a probabilistic fatigue life-time estimation is also explained given that the probability of failure or equivalently the

rèliability index B is known

Finally, an example is worked out to demonstrate the calculation procedures presented. FOr this example, statistical characteristics of

random variables of the fatigue damage arid the total number of stress

cycles in the case of broad-band process are expressed in functional f orms of the mean perio4s of zero-crossings and maxima of the stress pFocess. For practical purposes, a correction factor representing randomness of the

damage due to broad-band process is also calculated statistically for a

I - IWrIXXTI

Fatigue is an important phenomenon that causes failure in structural

members wlien loading conditions are time dependent and continuous as

happening in offshore enviPônments. Structures to be built

in such

environments are imposed on Wmd, waVe, curreht and earthquake exposures. Among them, waves play a mäjor role in fatigue failure due to their

continuity in time in ràfldom sequences, as being tiny, moderate and

sometimes catastrophical Recent developments of offshore energy

explorations and rodúctions require movements to more hostile and deeper ocean environment vhere structUres to be built are relatively flexible and dynamic sénsitive. Fôr such structhres, the fatigue failure becomes mOre important and needs to be studied in a sophisticated way. However, fatigue

is a complex phenomenon that a little is known about its cumulätive excursion in time under random loadings. It depends on loading types,

structural material, joint configuration, initiál conditions and imperfections of the geometry, etc... In the theoretical point of view,

methods of the fracture mechanics are not yet available to solve this

complicated problem in a general sense.. But,

they may

give some explanations and result in formulations for very simple cases that mostly far from the reality concerning offshoré structural engineering. In the

experimental point of view, laboratory tests have been carried out to obtain fatigue data for design calculations, see e. g. refs. Cl] and [2].

However, most of. data are produced from constant amplitude tests, and few

from variable amplitude tests (random loading). In all loading conditions,.

some specific joint types have been tested, and therefore the data available are subjected to some limitations of the joint types as well as the loading conditions. In the environmental practice., no limitation does exist concerning loadings and other environmental factors that can not be

represented by a laboratory set-up. The joint configurations in practice are also much more compléx than those used in laboratories. Therefore, a

prObabilistic analysis seems more eligible than a deterministic analysis

for estimating the fatigue life of. tubular joints of offshore steel structures. Such an analysis can be carried out by making synthesis of

available fatigue data from experiments and the theóPetical tools that

available so far. This report is devoted to the analytical calculation o the probabilistic fatigue damage and life-time estimation of offshore

steel structures assuming that the S-N curve of the fatigue model, which

into account. It is also assumed that ztPess rangez are statistically

known,which can be obtained by pèrformitig a spectral añalysis that not

explained in this report. The probabilistic damage and the associated life time are formulated for both narPoW-band and broad-band stress processes.

In the formulation, nön-iinéaP S-N curves defined in logarithmic scales

are taken into account by using a piecewise linearization tehriique.

However, the main pürpöse of this report. is to fothu1ate a general

probabilistic fatigue damage to be used for estimating the fatigue life

and for assessing the reliability index. The inteñtion is to provide

präctical and easily calcúlatable statements by using computers rather

than to produce some difficult and cumbersome. analytical expressions.

Computational difficulties and the solution algorithms are also mentïoned. For broad-band stress processes Monte-Carlo method is employed to estimate the fatigue damage, and consequently a correction factor is determined.

This factor is treated in the analysis as a random variable. The stress

conentration factor (SCF) to convert the nominal stresses to hot-spot

stresses may also be treated as a random variable when the reliability

D;

n(S)

i=T

N(S1)

where,

n(S.) : Number of stress cycles at stress range S1,

N(S1) : Number of stress cycles to failure at stress range S1, q : Number of applied stress blocks.

This fatigue model states that the damage caused by a cycle of a variable amplitude stress history is equal to that of a cycle of the constant amplitude stress history. Then the cumulative damage is obtained by the

linear summation of contributions of äll stress cycles to the damage. It

is assumed that fai.luEe. òccurs when the cum.i1ätive damage,

b,

reaches a

value which is equal to unity as commonly accepted. InvestigatiÒns to

check the validity of this value have revealéd that it varies considerably under realistic circumstances, e.g. for welded structures a range of

0.5

-2.0 has been considered to be appropriate . Therefore, it is assumed in this study that the damaße to failure i a random variable. The damage for one cycle at the stress range S1 is written by,

dD1 (2.2)

This damage can be calculated if the function N(S:). is known. For very

simple cases, ari analytical function for N(S) can be found by using the

Paris-Erdogafl crack-growth lawt63 which s stated as,

--(.3]

Fatigue life prediction was first carried out by Palmgren , arid later by

MineP'using

a linear fatigue damage-accumulation rule subjected to some

restrictions. This rule is stated as,

(2. i)

where a is the crack size, C and k are regarded as material constants and

X is the stress intetisity factor. oweveP, for offshore tubular joints where geometricàl, environmental and loading complexities exist, it is

very difficult to assign a formulation to N(S) by using fPactúre mechanics at the present time. Instead, a function based on experimental data seems

more reliable and convincing. Or, to be more accurate, a combination of

the Facture mechanics and the experimental data may prOduce a reasonable

function that can be used in design consideration. Such a fuñction is riot

yet available, at least to the author#s knowledge, for complex cases like offshore tubular joints. Therefore., a S-M curve obtained röm experimeital data is used in this study with some uncertainties that taken into account by means.. of random parameters The S-H curve is expressed as,

N c (2.4)

where C and k arè parameters obtained from the experimental datá. Sh

the local stress range (hot-spot stress range). In (2.4), all variables

(C,k,Sh) are random in nature. Because, Sh is a function of joint typè and random loading1 C nd k are obtained from curve fitting to experimental data that always inclùde uncertainties. A detailed study for the determination of C and k is presented in a previous reportE7], arid

therefore it is not explained here. However, it is assumed thät thèse two

parameters are correlated. random variables with knòWn statistical characteristics. A typical S-N line with the. probability density-cupve for ¿ given hot-spot. stress range is as shO in Fig.2.i, in the

logarithmic scale.

InC

inÑ

nmm=

(1-P;k)

_nC at the stress range corresponding to,

inC,k °i2nC

The logarithm of the S-N curve given by (2.'i) represents a line written

by,

in N = in C - k in Sh (2.5)

Since C and k are assumed to be random, N becomes also random. From (2.5), the mean value and the variance of inN can simply be calcUlated as. function of in S1.. Thus, for a

ivenS

OrflN= (In Sh) d - 2 (inSh) 0inC,

°iflC k ...(2.7)

where p denotes the mean value, a denòtes the standärd deviation and p

denotes the correlätion coefficient. As it is. seen from (2.. e)) and (2.7)

that the mean value of in H is a linear fUflction of in Sh while the variance is a quadratic function. The minimum value of the variance of

in N is calculated to be,

(2.8)

(29)

For the fatigue life estimation, the méan valUe of the S-N line given by

(2.5) will be used. In the Peliability calculation or in the sensitivity analysis, the. randomness of C and k will be taken into account to

represent inherent uncertainties. The reliability and sensitivity analyses are not discussed in this report since it is devoted to the calculation of the probabilistic fatigue-damage and lifé-time estimation.

iriN irkC k Sh

3 - PBABILISTIC FATIGUE DMIALiE FOR, C»&E CYCLE OF A N4R-AND STRESS

CESS.

In offshorestructùral enginèering, it is often assumed that the stress

process due to random ocean waves is narrow-band. The stress process is

related to the wave process by sme linear transfer operations provided that the now-linear drag force term is somehow lInearized. However, since it is assumed that the wave process ìs Gaussian with zero mean, the stress process becomes alsO Gaússian because of the linearity between these two processes8. This feature. of the stress process enables a probabilistic stress range calculation which is used to estimate the fatigue damage analytically by applying the S-N curve. A typical narrow-band stress process is as shown below in Fïg.3.i.

I'

s

Fig. 3. 1 - A narro-band stress processi

The stress range S is simply calculated as being double of the stress

amplitude, which is also equal, to the stress maximum under the narrOw-band

assumption. Thus.,

A

S= 2s

(3. 1)

A

where, s denotes the stress amplitude. This nominal stress range will now

be transformed to the local stress range (hot-spot stress range) by means of stress concefltration factor (SCF). Thus,

finite element method in general,. For practical purposes, some empirical

expressions are also availàble9. In any case, this factor contains some uncertainties that must be con.sidered when the reliability analysis is

carried out. For the fatigue. life prediction, this factor can be

considered as being detèrministic. The stress amplitude, s is naturally a

random variable being thé envelope of the stress maxima in time domain,

see Fig.3.1. It can be shown that, f ôr é narrow-band Gaussian process, the

probability distribution of the maxima. is a Rayleigh distribution, see

Fig.3.2, tated 2 s

1s

fA(S) - exp(-

....) s o s s Rayle i g h s

where is the variance of the stress process calculated from,

(3.3)

.00 0.75 1.50 2.25 3.10 - 3.75 1L51 .21 5.00 s

FAg. 3.2 - Probability distribution of the stress amplitude for narrow-band process.

(w) dw . (3.4)

In which S5(w) is the stress spectrum. By using eqs. (2.2), (2»4) and (3.2) the fatigue damage for one stress cycle can be written as,

A

which is a function of the random variable s. The mean value o thi damage. can be Writteñ as,

E(dD]

=

f

(2 SCY s

fA(S)

dz s

in which the function

A()

is given by (3.3).

s

If a single S-N line shown in Fig.2.i is used, the integration given by (3.6) will result in,

EEdD] = (2 I4 SCF

O5)k r(14)

k rj+i k. (2 SÇF) j S fa(s) ds C. j S si. (3. 6) (3;7)

whePe ) is the Gamma function. If, however, more than one S-N line

between O n S is used, see Fig.3.3, the integration given by (3.6) can not so simply be evaluated as it is done for one S-N line. In this

case, contributions of multiple S-N lines will be calculated separately and the final result will be obtained by linear superposition. Thus, from

(3.6), it is Written that,

(3.8)

where C. and k. are the fatigue parameters of the S-N line denoted by j, and s.1 are the integration limits calculated from (3.2) as,

2

o.

s SS

(S

hj+i

- 2 SCF

in which and (Sh).1 are shown in Fig.3.3. In (3.8), NL denotes the

nurnber of S-N lines to be considered.

N Fig. 3.3 - Multiple S-N lines representing a non-linear

S-H curve.

Dy using a variable transformation written as,

(3.8) can be stated in a more familiar form as,

NL

\

(243})kJ

r k. k.

E tdD] S

y.) - r (

i-, y. )j .. (3. 1 1)

I

C. 2

2J+

j=i

in which (.,. ) denotes the incomplete Gaa function defined by,

(3.9b)

r(x,b) exp(-u) du (3.12)

In (3.11), Yj and

y1

are calculated from (3.10) and (3.9) as,

yi = r i 12

vI

SCF o J s r (S ). ia

hj+1

L2

y

SCF

s

(3.13b)

where S denotes the hot-spot stress range as defined before., and a is

the standard deviation of the stress process calculated from (3.4).

E(dD] =

(2

'.4

SCF a

s k

r(141 .y)

Fig. 3.4 A S-R line with an endurance limit

As a special case, for a S-N line with endurance limit shown in Fig. 3.4,

the mean fatigue damage. of a stress cycle given by (3.11) will be _reduced to the simple form written by,

(3. 13a)

(3. 14)

where C, k and y are shown in Fig. 3.4.. 1f the. endürance limit varies linearly, instead of being a constant, then eq. (3.11) will be used to

calculate the mean fatigue. damage of a single stress cycle. Un order to

calculate the damage f röm (3.14) or (3.11), the incomplete Gamma function must be calculated first. A númericäl procedure to perform this caiculatioh is presented here for the completeness of the text. By using a variable transformatiOn written as,

u = z .. b (3.15)

the incomplete Gamina function given by (3.12) will be stated in the

following foPm. r(x,b) = exp(-b) f(z+b)' exp(-z) dz

\-/ r(x,b) = exp(-b) ) W. (z1+

b)Cl

1=1 (3.16)

As it caribe recognized the integrand of (3.16) is of the Gauss-Lagùerre

type, see ref. (1.1]. Thus, it cari be written that,

whére z. (i=1,...,rl) are the abscisss (Zeros of Laguerre Polynomials), W

are the weight factoPs and n is the ntaber of integration points. From a numerical investigation it has been found that, for the fatigue parameter

k 10 , a three point integration produces very good results. The

abscissas and weight factors are given in Table 31.

Table 3.1 - Abscissas and weight factòrs of Gauss-Lagtierre integration for n=3.

(3. 17)

It. is worth noting that, in the case of endurance limit vaPying linearly,

some herical

difficulties are endountered due to the fact that the

z.

i i

- 0. 41577 &56 0.71109301

2. 29428036 0. 27851773

slope, k, of the endurance line becomés large and the y value (see

Fig.3.4) is relatively small. In this case, the difference o Gamma

functions written in (3.11) will be (r(x)-r(x,b)) with a large x arid small b values so that it will be numerically unstâblé. This problem is övercome by using a series solution of the différöncé as itten by, see re-f. [il],

r(x) r(x,b) =

_(.)ñ!

(..)" n=O

(3.16)

A few terms are sufficient to obtain satisfactory results for large x and

small b values.

As a s'ary of this section, the meañ fatigué damage of a single stress

cycle under a narroW-band process is calcúlated f ro the statement given

by (3.11) for a fatigue model of multiple S-N lines. The related

incomplete Gamma function is calculated from (3.i7). If one S-N line with

an endurance limit is to bé used, then the statement given by (3.ii) will be applied to calculate the damage. In the followi section, attention will be paid tö the case under a broad-band stress process.

0.00 0. '10

- P)BABILISTIC FATI(JE DAMAßE FOR ONE CYCLE OF A BAD-BA) STRESS

Although â narrow-band stress process is assumed in design of offshore

structures, it proves in reality that the stress process is mostly non

narrow-band, especially when a fatigue analysis is concerned A stress process, in the spectral form, of an offshore structure is like that as

shOwn in Fig.4h1 where the first peak corresponds to the significant wave freqUency (peak of the sea spectrum) and the. second peak correspoñds to the lowest natural frequency of the structure (peak of thé structural transfer function). There may be some other f luctuatións in the stress

spectrum. But, they are of minor impor-tance, see ref s [i2-ii].

S55(w) I 80 I'. 20 1.60 2.00 2. '10 2 80 3' 20 - FREQUENCY wp "k

Fig.4.1 - .A schematic stress spectrum.

The existence of twO important pêäks in the streSs spectr reveals that

the process is broad banded. The degree of the brôàdness depénds on the.

severity of the sea state and the natUral fre4uency of the structure as well.. For moderate sea statés and natural frequencies of structures

3 rd./sec. where most fixed type structurés fall into, the amplitudes of

the stress spectral peaks are about in the samé oPder. in such a casé, the bandwidth effect of the stress pPocess m.tst be takeñ into a'ccöunt when the fatigue damage is calculatéd1 especially when the reliability òr the.

sensitivity analysis is conducted..

The tithe

history of a broad-band stress process is like that as shown in Fig. 4.2..

Sit)

1+) Maxima

-

-f I

I

k

\

\

/

\_.

--V

I)

Maxima

Fig. 4.2 - A broad-band stress process.

This process displays some local maxima and minima in both positive and

negative sides of the stress. In this caSe, the stress range is not any

longer double of the stress maximum. It is, in fact, a random variable

with an unknown probability distPlbution. Iñ order to facilitate the

calculation, lt can bé assumed that the stress range is related to the

stress malmum with a random variablé as written by,

I

where A Is the random variable and Isi is the absolute value of the stress maximum. Since A is assumed to be a positive random variable, an absolute

value of the stPess maximum is used here to guarantee that the stress

range remains alWays positive. The random variable A is to be calculated by equalizing the fatigue damages obtained from. a cycle counting

method15

and analytically-calculated. On the other hand,

it

is also possible to define a random variable for the broad-band fatigue damage as

a ratio between the damages obtained from the cycle coutitthg method and

analytically calculated. This procedure is identical to that intrOduced by WirschingEtO], _{except that the analytically calculated refePence damage is} not necessarily based on the narrow-band assumption in this study. Thus,

1

Ak

dD

(2 SCF Isi)

assuming that the experimentally determined S-N curvé under a broad-band process is aváilable The probability distribution of the stress maxima is not any more Rayleigh, and it is found rather complicated. The density

function is obtäined to

be17,

fa(s) =

exp(-2mn:2)

exp(

_-(4.5)

asstiflg that the stress process is Gaussian with zero mean. This distribution includes not only

(+)

maxima b.ut also (-) maxima, see

Fig.4.2.

In eq. (4.5), the parameter, e, is defined as,

the mean damage ôf a stress cycle under a broad-band process cari be

written as,

ED

= a EEdD]

(4.2)

where a isà random varIable whichwill be determined later and ECdDa] the mean calcúlated reference-damage. Here, attention will be paid to the

calculation of _E[dDa] Assuming that À=2 fròrn (4.1) it can be written that, as similax to (3.2),

Sh = 2 SCF II

(4.3)

Where, once more, S is the hot-spot stréss raflgè, S is the stress

concentration factor and Isi is the absolutè. value, of the stress maximum.

FPòm (2.2), (2.4) and (4.3) the damage. dDa can b writtèn as,

(4.4)

(LL6a)

i

Th2'

E

-(x) =

JCexp_

du

-(4.b)

in Wbich Tm is. the period of maxima and T0 is the period of zero

crossings. In (4..5) and (4.óa), rn0, m2 and m4 denote respectively the

zero, second and fourth spéctral. moments of the stress process, which are calculated from, for i0, 2, and 4,

m. S35(w) d( (4.7)

o

In this statemént, S55 Ks.') is the stress spectrum. When 1=0, it is seen that rn0 is equal to the variance given by (3.4). In (4.5), (. ) denotes

the standard ñormal. distribution function defined by,

For some valués of e and m=t, the probability density function written

by (4.5), is shown in Fig.4.3.

-3.00 -2.25 -1.50 -0.75 .00 0.75 1.50 2.25 3.00 3.75

s

Fig. 4.3 - Probability density function of mäxirna ôf a

broad-band process.

As Illustrated in this figure, it can easily be verified that when -, 0,

(4.5) redUces to the Rayleigh density function which corresponds to the narrow-band pPocess. However, if the spectral width - 1, -then (4.5)

reduces. to the Gaussian density function. By using the probability density function of the stress maxima, defined by (4.5), the mean damage _EtdDa]

can be calculated from,

EEdD]

=

LdDa

fe(S)

ds

which is valid for the fatigue model of a singlé SN line. Here, the intégration is taken from - to +, ecaúse the process Inc ludes also negative maxima. Having intrOduced (4.4) intO (4.9), the resultof the

integration can be written as,

A closed form solution of I is obtained as similar to the cäse of närrow-band Process. This is,

k+2 k/2

E._(2m0)

r,,k+i -' 'v5 (4.9) (4.12)

A general closed form solution of 12 definéd by (4.11b) is not possible

due tO its complexity with the noPmal distribution function (, ).

Therefore, a numerical integration procedure must be used to evaluate

this

integration. By using a varïable transformation, _{(4.11b) dät be reduced to} where I

-k

EEdD]

(2 CF) ' ds (4.10') (4. lia) (4.11b) and 12 are defined

e = ('11+12) by, '' k / s' V2iffi' _J O

-Isi

exp(\-2me

Q = S

_s

exp(- . ds

the form of the Gauss-Hermite ìntegration. Alternatively, after sorne

manipulatíóñs are perforthed, it can älso be reduced to the form of the

Gausa-Laguerre integration. The sécond one, for this particular problem,

seems more efficient in the sense that the'rón-lïnearity of the integrand

is considerably dècreased by the variablé tratisfoPrnation, leading to the

use of less

ntber

of integratiòn points. Therefore, this procedure is

adopted in this study. The integrand of (4.lib) is denoted by F(s) which

is,

F(s) = 151k

exp(-This function is in the form of that as shOwn in Fig.4.4. Having

introduced (4.13) into (4.11b), the statement of I becomes,

12 =

Ç7F1oe

ds

m0

oe

From Fig,4.4 this statement can alsp be written as,

j[F(s)+F(-s)J ds

o

o 12

=i/?' (2m0)k

J

yk' erf(/(1

') exp(-y) dy

e

(4. 14a)

(4. 14b)

(4.16)

in which erf(. J denotes the error function defined by,

erf(x)

4-

rexp(_u2) du (4.17ä)

Now, by introducing (4.13) into (i.14b) and using a variable

transfoFmation wPit ten by, 2

s =2

y (4.15)

the expression of ₁₂ will become to a well known form of the

k = 3.8, see eq 4.13)

ds

C C

Fig.4.14 Functioñ, F(s), defined by (4.13).

This function

Cn

also be Written in terms of the normal distribution function defined by (4.8) as,

erf(x) 2 4(v'2x) i _(4.17b)

Thus, ₁₂ can easily _{be calculated numerically from (4.18) by using the}

Gauss-Laguerre integration procedure. The final expression is Written by,

k/2

12

1E

Cath0.

>

WI

/2 y.1

(4.18)

where n is the number of integratioñ poïnts, W. and y. (1=1,.;. ,n) are

weight factors and abscissas of the Integration. As _{a speciâl case, when} approaches zero, this expression produces the resUlt of the narrow-band

process in which case given by (4.12) becomes zero.

o In o 0 o Ôo - 1.00 2.00 300 'LOO 5.00 s s ds o u, C o e In

So far, the mean fatigue damage EEdDa] defined by (4.10) is discussed

under the assuffiption that the experimental fatigue model is represented

by a

single sN line.

However, in the case of multiple S-N line representation of the fatigue model, the calculation procedure is somewhat similar to that explained in sectioñ 3. To do this, the definition of I

and

_I2

see (4.10), are slightly changed in the sense that the integration domain with infinite limits is now finite. The integration limits of a

particular S-N line, say line j in Fig.3.3, are (S& and (Sh)jj. For

this particular SN line, the parametèFs I and ₁₂ are calculated and the results äre presented below.

in which k is the slope of the S-N line j, X. and x.1 are calculated

from, k/2 k+1 k+2 (2m0)

- r(--x1)

(Sh)j

2V2i e SCF

r _'S 2 I ' hj+1 i+1= [

v21

e SCF

The parameter I

_can

be Written as,

I

ecP(-Y,) :ii

in Which A, ₁₂₁ and ₁₂₂ are defined respectively by,

z1+ y )k/'2erf(Ad/Z+

Y)

(14.19) (14.20a) (14.20b) (4.22b) (2m0) ₂₁

- 122)

(4.21)

122= exp(-y.1

where V1

_and

z., (1=1,. . ., n), are respectively weight factors and abscissas of the Gaüss-Laguerre integration,

and

y.

and

y are as defined by (3.13). Having, calculated the paPameters t1

_{and '2}

for a S-N

line, the total mean reference-damage of one stress cycle will simply be

calculated by the superposition

as,

E(dD]

(

_{(I+ 1a)j}

W.(z.+ Y+i)'2eT(AVz+

... . (4.22c)

(4.23)

The mean fatigue damage under broad-band stress process will be estimated

from (4.2). Here, it is Worth noting that only one random variable a in the case of multiple S-N line representàtion of the fatigue model is used

as in the. same way of the one S-N line fatigue model. - The méan fatigue damage, EEdD], defined by (4.2), is dependent on the random _{variable cx. Iñ}

the following sectiOn, a Pòcedure to detérmine this random variable is

time domain. Thñ, for an assumed period Of time., the total damage is

calculated by using the Pairngren_NineraS, Pule written by (2.

1)

assuming that each stress cycle -is considered as a constant amplitude stress block. Thus, the damage in the period assumed will bej,

where n denotes the number of cycles in the period. Having divided this damage by n, the mean damage for one cycle will be obtained as,

n

E[dD] =j:j>N(S)

OP by using S-N. relation given by (2.4),

n E(dD]

=

_.>

(SCF

1=1

in which ScF is the stress concentration factor and S1 is the nOminal

stress range for thé cycle i, and j denotes a S-N line. By introducin.ß

(5.6j- and the statement Of E[dD] given by (4.10) into (5.1), a,. will be obtained as written by,

a. =

j k

k1

2 '

(I+I).

If the number of cycles, ri, and the reciprocal stress ranges aré knOwn,

the calculation Of a. will be carried out frOth (5.7). In order to find out the number of cycles and the stress ranges, the time històPy of the _stress

process is requlPed. This can be produced in twO ways, either by. carrying out a time domain analysis or by making a time history simulation from the Stress spectral values. The time-domain analysis is pPobably the most (5.4)

(5.5)

(5.6)

direct and general one allowing for non-Linearities in wave forces, at the

expense of còpuatioh times. In this pPocedure, fOr a given sea state,

the. surfacé elevation will be simulated fPOm the sea spectrum, then the time. dOmain änalysis will be carried oUt. In the second way o producing

the. stress time. history, â linearity between the sea surface elevation and the stress is assumed. First, Stress spectral values are calculated by using a spectral method, see e.g. ref. (13] then the simulatiOn is applied. This can be written

N

s(t) a. cÓs(w.t+p.) (5.8)

1=1

where s(t) is the stress in time domain,

M

is thé

total nber of

simulätion terms and a. is the amp litudé of the i tri, simulation term

calculated from,

in which S(w.) is the valué of the stress spectrum at the frequency w. and is a small frequency band in the vicinity of w1, see Appendix A. in (5.6) is a rañdom phase angle uniformly distributed between 0-Cu. For given, phase angles

,

(i=1 ...,n), and for a time period assumed,

the number of cycles and the reciprocal stress ranges will be identified

from (5.8). Then, by using (5.7) the required random parameter, a, will be determined. For the identification of stress cycles and ranges, there

are methods among which the rain-f loW còúntirig rias been reported as

being the most cönvehient16'20. Therefore, this method is applied in this Study. Its detail is presented in Appendix A. Since the simulated

stress process is a ftthction of p., see (5.6), the random variable a. defined by (5.7') will also be a function of these random phase angles. Its estimate will be obtained by making artificial experiments regarding to p, using the Monte-Carlo procedure, see Appendix A. It ïs however assumed that a large number of Monte-Carlo simulations are carried out t9 predict the random variable a.. Its mean value can be written as,

in which R is the number of Monte-Carlo simulations and (a.) is the

jr

realization of a. from the r th. simulation. When R -

,

then

- j

'

R -* a.. This calculation will be repeated for possible variations of

a3 j

the stress spectral shape, which can be obtained by using different sea

states and natural frequencies of the structUre, in ordér to construct a

relationship between C,

R and the bandwidth parameter of the stress j

spectrum, e. The reason of usïng a dual variation of the sea state and the natural frequency of the structure to determine the. stress spectral shape

may be such that different spectral shapes with the. samé. e. can produce different fatigue damages. Having calculated point by. point population of and e, as illustrated in Fig. 5.1, the mean value of a. will be

j j

estimated from a regression analysis. This Will be. denoted by

.

for a

specific S-N line indicated by j, see section 7. Then the meân value, ,

will be calculated. from However, this. calculation can be Pepeated

for a number of k (slope of the S-M line) values to relate the méan, , to

the slope of the S-N line,. .k, so that, in practice, for a given k. value

the reciprocal can easily be calculated. Having determmed the mean.

correction factor, ¡, of the damage under a broad-band stress process, the mean damage for one stress cycle Will be calculated from (5.î) as

=

_ECdD]

. (5.11)

in. which EtdD.] is the mean Peférence damage for one stress cycle défined by (11.23). Here, it is worth noting that this damage o öne cycle is

obtained for one sea state. Thus, it. can be chsidéréd, as a function of

the significant wave height and the mean Zero-crossing periOd of. waves. The variance of will be estimated from, see Fig.5.1,

j

where n s the total number of points of the scatter of Since the

J

calculation is carried out for all values this variance will be

independent of the sea state. But, it will be dependent on the slope, k.,

of the S-N line denoted by j.

E

Fig. 5.1 - Scatter of _(lia

R and the representative mean curve

i

a. versus .

J

The variancê of awill be calculated in general by using the statements (5.1) and (5.2) as a summätion assumïng that a., (i1,...,NL), are independent random variables. This is written as,

2 a- (2 a

/

C. EdD]

j a j=1 (5.13)

Sj

_{i2> ['aR}

_- (5. 12)

which is dependent on the parametèrs of S-N lines. It is obvious, however, that in the casé of one S-N line fatigue model, a will be equal to a

J So far , thé fatigue damage due to one stress cycle has been discussed. In

thé following section, the cumulative damage during a sea state and in a given service life time, T, will be discussed in detail.

6 - !JIÀTÏVE FATIGJE IHAGE flJRING A SE& StATE AND IN A L(G 1I PIOD

In previous sections, a fatigue damage caused by one stress cycle has beeh formulated Ari, general for both narrow-band and broad-band stress processes. In this section, the total damage for a given sea state and in

a long term period will be calculated. on the basis of. one cycle damage fOrmulations. The total mean damage during one sea state which is indicated by i can simply be calculated as,

i5=

1

(6.1,)

where n1 is the. total nuber of stress cycles during the sea state and is the mean damage. of one stress cycle to be calculated from (5,. 11).

Generally speaking, n1 is a random variable. For a narrow-band process, n. can be calculated from,

n. =

I

in which T is the duration of the sea state and T is the zero-crossing.

period of stress process in the sea state. For broad-band stress _process,

this statement is not any longer valid. For large e values, local stress

cycles associated with local maxima play an impóPtarit rolé in the damage accumulation. In this case, n1 may be estimated by using Th, mean period

of maxima, instead of using To In (6.2). For moderate e values, lt may nt

be. predictable whether or not a c.lósed f Orm statement of n. exists.

Therefore, it is assumed that this factor (n.) is considered _here as a

random variable defined by,

(6.2)

(6.3)

where y. is a parameter representing the randomness. of n. for the sea

state. i. In genèral this parameter may be denoted by 'i being a fúnction

of the spectral shape of the stress process, thus a variable of e. The

èxplained in the previous section, assuming that the duration of the sea

state, THs

lfl

(6. 3) is replaced by the. Period TR in which the rain-f low cycle counting is performed. Hence,. from (6.3) it is written that,

where n is the number of cycles identified by the Pain-flow counting in the period of time, TR. This is dependent on E. From the scatter Of y in

the domain a mean cUPie can be found to represent this parameter as a

continuous function of e. This is denoted by V. Thus,, for a given sea

state. i the cumulative damage, can be expressed as, from (5.11), (6.1)

and (6.3),

= V1 _E[dDa]i (6.5)

or with a single damage correction factor defined by,

1

v a.

(6.6)

the damage. will be,

-T

s () EEdD]1

(6.7)

Here, again,

'L

is the duration of the sea state i, E(dD]. is the mean reference damage given by (4.23) and T0 is the zero crossing periOd of the stress process for the sea state denoted by i. The zero crossing period is calculated from,

¡mo

T0

= 2ff

_(6.8)

y

in which m0 and rn2 are the zerò and second spectral móments of the stress process. For a deterministic sea state, i.e. for a given Hs (significant

V = n (.9.) (6.4)

wave height) ant Tz (zero-crossing périod of waves'), of the duration THSJ

the i.ulatiVe fatigúe damage will be ôalculated from (6.7).

So far, the câlculations of

T.

and E[dD J are explainéd for shOrt term

i

ai

-descriptioñ of the sea state represented by the spectrum of surface

eievationt2l]. _{However, in a long térm period of time when the cumulative}

fatigue damage becomes very important, there may bè many sea states With

some probabilities O occurrencés. In this' case, the total damage Will be calculated by using the linear accumulation criterion which states that the damages due to différent sea stätes will be superimposed. Thus, the

total damage in a life time T will be,

NHS

D0

> 5

where H is the number of sea states occurred in 'the period of service,

or life time, T. The damage i's 'as given by (6.7). Having introduced

this statement into (6.9) will be,

Dt0t )

i

() E[dD]

j

j_ I

(6. 10)

in which TB denotes the sum of the durations of identical sea states

indicated by i during, the service time T. This can be expressed in terms Of T and the fraction of time that ' the sea state i occurs, i.e. the probability that the sea state i occurs in T. Thus,

Having uséd this stätement in (6.10), the total damage will be stated as,

D T (è--). E[dD.]. P.

tot

_/

_Toi

ai

i1

It is clear that the sum of P. where i=1,.. .,N will be equal to 1. This probability of occurrence of a particular sèa. state is calculated from a

wave scatter diagram which represents the long term description of. the sea state together with the short term descPiptiorl. The scatter diagram

provides ntbers of occurrences of sea states. for some Observed Hs and Tz

values. These two parameters chaPacterize the severity of the sea

statest22. A typical scatter diagram is as showriin.Fig.6.1.

L

Tz (sec.

Fig..6. I - A wave scatter diagram and marginal distribution of Hs.

The probability P. ca bè stated An terms of the joint probability of Hs

and Tz as,

Where fTz(h,t)

is the density function Of the _{joint probability}

distribution of Hs and Tz äs demonstrated in Fig.6.2. Having introduced

(6.13) into (6.12) and replaced the summation by the integration, the

total. damage for the long term period becomes as,

E 10 ci,

I

8 7 6 5 4 3 2 1

Total number of occurrences.

41

4

7

2.1 10 5 37 9 2 32 63 8 3 1 84 63 10 1

13168

66 8 15 58 162 70 14 3 10 22 45 15 2 1036 2 10 12 14 16 = f,Tz(hi,ti) 4h At (6. 13)

D

tot 2ir

= T

JJ

ECdP] fiis,Tzh1t) dh dt (6.14)

which is the damage due to the continuous modelling of the long term sea

states. However', for the discrete form of the seä states eq. (6.12,) will be

applied to calculate. Dt0.

Hs

Fig. 6.2 - Illustration of a joint pPòbability density function of Hz arid Tz.

It is worth noting that this damage is formulated here under the

assumption of dèterministic wave direction. In order to account for the.

randomness of the wave direction, (6.14) must be integrated once more. If it is assumed that the wave direction is uniformly distributed between

O-?rr, then the total damage is stated as,

2ir

j1]'

fHs,tz,t) dh dt d0 ....

(6.15)

o

óo

where is the principal wave direction-angle measured from the global X axis of. the structure. (.14) and (6.15) can alsó be interpreted as mean fatigue damages for the long term. structural response during the life time . If, however, the severity of the sea state is characterized by the

D

to t

significant wave height (Hz) only,, the lotig ter description of the sea

state will be represented by the probability distribution of' Hz and the

short term wavè statistiös. In this case, the joint probability of Hz and Tz will be replaced by the marginal pPobability of Hz to calculate the

damage.. The. dèñs.ity function òf the marginal disti'ibution can be obtained

from the scatter diagram, as shown in ig.5.1, and defined by,

f.(h)

_r

f,Tz(ht) dt

0

In this case, the damage given by e.g. (.14) will be,.

E[dD 3 f (h) dh

T0 a Hz (6.17)

From the wave records analysis, it has been reported that fH(h) fits a Weibull distribution in general. In the case Of the continuous modelling of the long term sea state, the integrals in (ô.l4) (5.15) arid (6.17) cati

be reduced to the Gaussian fos foP cePtàin cotiditions so that they cri easily be carried out numePically. During this study, it häs been

investigated that. a few Gaúss integration points suffice tO prOduce good results, The calculation proceduré will be expläined in Appendi B iti

detail.

The total fatigue damage, given by (5.14), (6.15) Or (617) in the.

continuous modelling of the sêa state and by (5.12) in the discrete.

modelling, can be used to estimate the ultimate life time of a. structural member ur joint as well as to assess a fatigue reliability measure.. It is assumed that a failure occurs when the tOtal damage, reaches a 'value

equal to 1 as cothmotily accepted. Howevèr1 it has been reported5 that this value varies in the range between 0.5-2.0. Therefore, it may be considered as a random variable denoted by Pf which is defitied to be the

damage to fàilure. t.f. the ultimate life time is to bé deterrnined, the mean

value f Df will, be used. In general, when 'Dt0t Df theh failure occurs.

Df Z =

in(

D -tot

From this definition the safe and failure regions are t be,

Z > O - - Safe region

Z O -

>

Failure region

In the reliability calculation, this faLlure function will be used. It

can

reasonably be assumed that Df is lognormally distributed. Values of

statistical meàsürès of this random variable may be bést estimated by

engineering judgément, e.g. the mean may be

Df= 1.0 and the standard

déviation o= O.IÖ

as depending on the degree of assumed uncertainties. From the inspection of presented so

far,

it can .be seen

that

there is a linear relation between the time period, T,

and

the total. damage, Dt0t. By defining a random variable, Y, which contains all uncertainties in the

failure function, it cän be written

that,

ZY-inT

(6.19)

where Y is a multivariate random function in the design variables space so that,

Z(X1,X2 X)

=Y(x1,x2,..

,x) - in

T . ...(6.20)

in which

X.,

(i=i2,...,n),

are the

design variables. From the definition of the reliability index, ₌

_P/a

it can be written that,

ii-in

T

Y

(6.21)

Where

is

the mean

and

a,1, is the standard deviation of the random fúnction, Y. From (6.1:) it is also seen

that

the reliability index, , is a linear function of in T as shown in Fig.6.3. For a given failure probability which, in terms of , is stated in general as,

"y

4, Df Y = D1 (6.22)

the life time can be estimated from (6.21) as,

T1.

exP(P-k

o)

(6.23)

This is the life time during that the probability of failure

where (. ) denotes the standard normai distribution functiOn and

is a

given reliability index. Reversely, if the lifé time, '1', is defined

beforehand, then the reciprocal failure probability is calculated from

(6.21) and (ó.22).

(6. 2'4)

where D1 is the total damage for a unit life-time, e.g. for one second

time, defined in general by,

o _"Y In T

Fig. 8.3 - Reliability index versus In T.

In these calculations, the statistical méasures of Y, and cr,,, are

assumed to be known. In practice, these. measures can be calculated from

the first oPder approximation of the random function Y in the design variables space. This function can be written from the definition of the

T

in which Dt0t is given by (6.l'fl, (6.15) or (6.17). If the unit of time is

defined as being One year, then the result of (6.23) will be in years. As an example of the calculation of the probabilistic fatigùe life time, the

[25]

example presented in a previous report , SAPOS, is worked out here. The

result reported are,

Damage for T=1 sec. 0. 60591E-11

Damage for T=1 year = 0. 19121E-3

The statistical values of the failure function Z and the reliability index for T=25 years are,

ij

= 5.363 = 1.076

B 4.984

From (6.19) it is seen that a= Thus, 1.076. From (6.21), is calculated as i..; in T + B which gives, 8.5817. The life time based

on deterministic calculation is found to be, Tlife 5230years. This life

time corresponds to a B index of B0.018i8 which, in terms of the failure

probability, equals PFZ Ö.50. However, if the life time is calculated,

given that the failure probability is 1.44124E-3 which corresponds to

B=3.O, from (6.23) it is found that TlifeZ 211 years.

SinCe the function Y defined by (6.24) is a non-linear function of the

design variables, its statistical measures are approximately calculated by using a first order Taylor series in

practice2627. Thus,

1=1

(6.25)

Y z

y(*)

in which is the vector of design variables evaluated at the

linearization point, x. denotes a design variable and is the

partial derivative of. Y calculated at . From (6.26) it can be written

that, assuming independent design variables,

z

y(*)

+ * ay p

- x.) - *

x. i ) (6.27à) (6.27b)

where p arid a are respectively the mean and thestandaPd deviation of

x1 X1

the design variable, x.. It is worth noting here that, for a correct estimation of , the linearizátion point must be on the failure surface.

However, for a rough estimation; the mean value approximation can also be

used. For further details, see e.g. ref. [281. In the following section _an

7 -

ISFRATIC

OF FATIJE flAJIACiE CALJLATIOHS BY K

CM1PLE STI1URE

The calculation procedures for the estimatioñ of the fatigue damage

presented in previous sections are demonstrated in this. sedtion. For this

purpose, a single pile is chosen as an example since it is simple and an

analytical formulation of the stress spectrum is readily availableC29],

which is more or less representative o the stress spectral shape of more complex structural conf iguratioIs, at least around the frequencies of

spectral peaks The example. structuì'e. is as shown in Fig.7.1.

/9'

Fig.7.i - An e,cample pile structure.

By changing the mass of the deck, natural frequencies of the pile can be

generated to obtain different stress spectral shapes. For this example, the spectrum of the stress at the base can be formulated as, for the deép water condition and without the frequency reduction of the inertia force.

term,

ri : Water elevätion

h : WaLl thickness of the pile d : Water depth

SS

s

(w) = qQ s (w) SS a g 4a g . exp(-

J)

w

Hsw

2 r (L)- 212 2 (A) h I1-(----) I + q,

()

L

wk.j

.

where a is the parameter of the sea spectrum which is taken to be 0.0081

and Rs

is the significant wavè height. If ït is assumed that d=100..Om., and'h=0.20., ari.by intröducing (7.2) intO (7.1) the Stress

spectrum will be obtained as,

2 _-

3jj3

exp(- :

4)

Hsw

r 212 . 2 .2

I1-(---)

I + 4F

()

L

wk.

. rin

(7.2)

(7. 3)

Here, by chaflging _Wk and Hs different shapes o the spectrum will be

generated The damping coefficient is taken to be a constant equal to

0.0i. A it has been mentioned in sectiOn 5, the time history simulation of the stress process will be made by using the spectral function given by

(7.3) For this particular example, the frequency range is chosén to bé

0.45/4

w 2.7cA). The lower bound is determined dependently on Hz in In which,

3

p : Density of water (1024 kgAn

2

g : Acceleration of gravity (9.81 rn/sn Cm : Inertia force coefficient (2.0)

d Water depth

h Wall thickness of the pile

¡ Natural frequency of the pile

Damping ratio to the critical

S(w)

Spectrum of the water elevatiön, Q

Here, the .Pierson-Moskowitz sea spectrum is used for S(w) which is given by,

2

1.1 2

order to prevent a possible numerical instability and the upper bound is

just chosen by considéring that the functibn S() dies away very rapidly

in the frequency région beyond the natural frequency. For the values of

Hs=9.Om. and 3.0 rad./sec., the stréss spectrum is illustrated iti

Fig.7.2 where the first peak corresponds to the fundarnentäl wave fPequency and the second peak is dUe to. the natural frequency.

O o o e e o-o z-e o o O e O e %oo - 0.50 1.00 1.50 - 2.00 2.50

3.003.50

.b0.50

w0 w1 FREQUENCY 2 %

Fig.?.2 - Stress spectrum of the example structre for Hs=9. 0m. and 3.0 rad./sec.

In order to generate time history simulations of the process, the frequency region of the spectrum is divided into four mih subregions, one àröund each peak, one between the peaks and one. in the tail. The bounds of these subregions are defined dependeñtly on Hs and Wk as,

w0 =0.4I5/V1

W1 2.5w0+ O.

=

=

For the illustration shown in Fig.7.2, thesè frequency bounds are, = 0.15 rad./sec. = 0.75 radi/sec. = 2.85 rad./sec. 3.15 rad./sec. A)4 = 8.ÌÖ. rad./sec.

Each subregion is randomly dividèd into 15 intervals, for an interval see

Fig.A. i in Appendix A, so that a total number of M=60 frequency intèrvals are used. För the definitiOn of N, see (5.8) or (A.1.i) in the Appendix A.

The reason of using random frequency intervals is to prevent possible periodicity in the simulation of the stress time-histoPy if, however, all frequency intePvals are equal, the time history procesill be periodic with a period corresponding to the minimum input fPequency of the spectrum, see e.g. ref. [20]. Having generated samples of random values for the phase angle (each sample contains 50 random values between 0-2ir, the

stress time history is calculated for each samplé as explained in the Appendix A. A typical stress tiè history òbtained from the. spectrum shown

in Fig.7.2 is illustrated in Fïg.7.3 o o o o

r

o o o 'J o

ii

O b. V

I

't

ji

I

Fig.7.3 - Illustration of a stress time history associated with the spectrum shown in Flg.7.2. o-J D

o

zo o u, 'O u, u.' IO 00 o o o r.

For demonstration purposes only, the duration of the record of each sample

is restricted to give a total number of 200 extrema (minima and maxima), which varies in time between 3.4-3.8 minutes for this specific problem. On this basis, the rain-flow algorithm is applied, and it is experienced from the calculation that approximately a number of 100 (in all cases) stress

cycles are counted. For each variation of Hs and

k' i.e. for each form of

the spectral shape, a total number of R=25 Monte-Carlo experiments are

carried out to estimäte the random parameters a., y and A, see section 5

and 6. For this particular problem, the slope of the S-N line is taken to be constant and equal to

kf

3.8. FOr some sea states and natural frequencies, the resùlts of the spéctral analysis are presented in Table 7.1.

Tablé 7.1 - Results of streSs spectral analysis

r/s Hs m. (10 ) (1.0 ) (10 ) sec. sec. 1.0 0.8202 6.6530 57.9162 2.2061 2. 1296 0.2611 3.0 5.0 1.2894, 1.8539 7.2938 7.5739 60.4935 60.8257 2.6418 3.1086 2. 1817 2.2172 0.5639 Ö.701Ö 3.0 7.0 2.3994 77607 609531 3.4937 2.2420 0. 7669 9.0 2.8580 7.8893 61.0189 3.7817 2.2593 0.8Ö19 11,0 3.2346 7.9798 61.0575 4.0003 2.2715 0.8231 13.0 3.6085 8.0493 61.0829 4.2069 2.2809 0.8403 1.0 1.5222 8.9623 54.7697.2.5894 2.541'? 0.1912 2.1230 10.0931 59.4461 2.8816 2.5890 0.4391 5.0 2.7233 10.4281 59.9664 3.2109 2.6202 0.5?8Ö 2.5 7.0 3.290Ô 10.6339 60.1476 3.4949 2.6419 0.6546 9 0 3 7626 10 7716 60 2356 3 7135 2 6570 0 6986 11.Ö .1490 10.8673 60.2858 3.88232.6677 0.7265 13.0 4.5302 10.9400 60.3174 .0432 2.6759 0.7497 1.0 3.1013 12.0421 47.6065 3. 1886 3. 1601 0.1334 3.0 429Ó2 14.9261 57.3889 3.3686 3.2043 0.3084 5.0 4.9944 15.4482 58.40Ö1 3.5726 3.2316 0.4264 2.0 7.0 5.6126 15.7136 58.7214 3.7551 3.2503 0.5008 9.0 6.1.1.68 15.8786 58.8684 3.899? 3.2632 0.5475 11.0 6.5240 15.9892 58.9484 4.0135 3.2723 0.5790 13,0 6.9207 16.071.1 58.9977 4.1232 3.2793 0.6062

Results of statistical values of the damage parameters aré presented 1h

Table 7.2 whePé (w)R and _(u)R denote respectively the mean and the

standard deviation of the parameters, a, y and X obtained from the Monte-Carlo analysis. These statistical estimates are calculated from the

statements given by (A.3.3) in Appendix A.

Tablé 7.2 - Statistical values of the damage parameters for k=3.8 and R25 Monte-Carlö simulations.

The parameter X is defined in general as

À =a

V

r/s Hs

m

a broad-band u narrow-band y X broad-band.

°&R

'aR

aR

'vR

°vR

°ÀR

1.0 0.2611 0.9072 0.2966 0.8757 0.2863 1.0311 0.0152 0.9355 0.3082 3.Ö 0.5639 0.8384 0.1978 0.6950 0.1640 1.2121 0.0266 1.0175 0.2453 5.0 0.7010 0.7802 0.1568 0.5661 0.1138 1.4108 0.04149 1.1019 0.2281 3.0 7.0 0.7669 0.7685 0.1458.0.5110 0.0970 1.5738 o.Ó543 1.210± 0.2356 9.0 0.8019 0.7122 0.1448 0.4484 0.0912 1.6876 0.0539 1.2011 0.2409 11.0 0.8231. 0.6703 0.123,1 0.4069 0.0748 1.7804 0.0591 1.1919 0.2109 13.0 0.8403 0.6402 0.1060 Q.3766 0.0623 1.87.5 0.064 1.18790.1918 1.0 0.1912 0.9161 0.3203 0.8991 0;. 3143 1.0172 0.0089 0.9320 0. 3265 3.0 0.4391 0.8633 0.2422 0.7763 0.2178 1.1183 0.0210 0.96610.2738 5.0 0.5780 0.8176 0.1886 0.6701 0.15445 1.2363 00232 1.0118 0.258 2.5 7.0 0.6546 0.7969 0.1807 0.6087 0. 1380 1.3331 0.0338 1.0645 0. 2501 9.0 0.6986 0.7892 0.1646 0.5742 0f1198 1.4093 0.0377 1.1134 0.2361 11.0 0.7265 0.7771 0.15540.5463 0. 1092 1.4621 0.0447 1.1375 0.2347

1300 7497 074880 1551 051030 1057 1 51830 04'9

1 136302318

1. 0 0 1334 0 9044 0 3339 0 8963 0 3310 1 0095 0 0069 0 9137 0 3403 3.0 0. 3084 0.9Ö76 0. 2732 0. 6633 0.2600 1.0509 0.0134 0.9542 0.2892 5.0 0.4254 0.8725 0.2444 0.7897 0.2212 1.1105 0.0182 0.9699 0.2755 2.0 7.0 0. 5008 0.8475 0.2273 0.7348 0. 1971 1.1641 0.0184 0.9871 0.2658 9.0 0.5475 0. 8580 02089 0.7201 0. 1754 1.2029 0.0230 1.0334 0.252 11.0 0.5790 0.8216 0.2069 0.67280.16944 L2361 0.0226 1.0168 0.2587 13.0 0.6062 0.8345 0.1849 0.66770.1479 1.2644 0.0243 1.0554 0.2327

The statistical mean and 'iariance of A, aré approximately calculated by using the first order Taylor set-les assuming that a and y are independent random variables.. The results are,

(7.5a)

2 2 2

z (p a ) + (p a

A

av

V a

which give quite good results. The approximate values of and

calculated from (7.5) are cmparèd to those given in Table 7.2, in Table 7.3.

Table 7.3 - Comparison of _{(i.JA)R and (aA)R} calculated statistically (Table 7.2) and approximately from (7.5).

(7.5b)

r/s m

E

From Table 7.2 Approximate (from 1.5) (aA)R (aA)R 1.0 0.2611 0.9355 0.3082 0.9354 0.3061 3.0 0 5639 1.0175 0.2453 1.0162 0.2409 5.0 0.7010 1.1019 0.2281 1.1007 0.2240 3.Ò 7.0 0.7669 1.2101 0.2356 1.2095 Ó.2332 9.Ö 0.8b19 1.2011 0.2409 1.2019 0.2474 11.0 0.8231 1.1919 0.21Ö9 1.i934 0.2227 13.0 0.8403

1.1879 0.19181.1892

0.2011 1.0 0.1912 O.932Q. 0.3265 0.9319 0.3259 3.0 0.4391 0.9661 ô.2738 0.9654 0.2715 5.0 0.576Ò. 1.0118 0.2358 1.0i0 Ò.2339 2.5' 7.0 0.6546 1.0645 Ö.2501 i.06V1 0.2424 *.o Ö.698 1.1i34 0.2381 1.1122 0.2339 11.0 0.7265 1.1375 0.2347 1.1362 0.2299 13.0 0.7i97 1.1363 0.2318 1.1369 0.2378 1.0 Ö.1334 0.9137 0.3403

0.9130 0.337

3.0 0.3084 0.9542 Ö.2892 0.9538 0.2874 5.0 0.4264 0.9699 0.275 0.9689 0.2119 2.0 7.0 0.5008 0.9871 0.2658 0.9866 0.2651 9.Ö 0.5475 1.0334 0.2552 1.Ö320 0.2521 11.0 0.5790 1.0168 0.2587 1.0156 0.2564 13.0 0.6062 1.0554 0.2327 i.0551 0.2347

band assumption (see Table 7.2), I O, and _.12 C(1+k,'2) are used

in (5.7), which are obtained from a narrbw-band stress process.. The mean

values, (w)R based on broad-band and narrOw-band assumptions arid

are graphically shown in Fig.7» whéré the continuous lines are the. best

curve fittings.

o

c

o

Fig.7.L - Mean values of a. and y Parameters.

From the scatter of averagé välues,

'R'

obtained from the Monte-Carlo

analysis it can

be expressed that the mean values are in the form of, see.

also Fig. 5.1,

O.I2 0.211

determined from the least square error.

036 0.116 0.60 072

EPSILON (E) 0.811

whère=

_E((J)R]

and

V.= E[(JJ)R]P

A

and B parameters are

denotes a S-N line. It is

+ B. (7. a)

obvious from Fig.7.4 that there are two sets of A and B, one is for the càlculätion based on broad-band assumption and the other one is for that based on naFrow-band assumption! From the closed examination of (7.6) it cari be séèn that,

where Tm and To are respectively the mean periods of maxima and zero crossings of the stress.process. For a storm period, say THsP the total

nUmber of stress cycles n can be calculated from (6.3) and (7.7b) as to

be, or,

= A () + B

V.

= A (.1!) ₊

W

j v Tm V

[A()

+BJ

.f

TH

n z A (d)

+ B

A = 0.542997 = 0.385543 A = 1.154690 a

B =-025i308

a

-A; 1.01965

B =-0.01834, t Broád-band assumption Narrow-band assumption (7.7ã) (7. 7b) (7.8) (7.9)

from which it can be concluded that the total number of stress cycles can be best estimated as a linear cómbination of the total number of maxima

atid zerò crössingz in a storm. For a narröw-band stress process when

Tm - T0, it is Obvious that A+ Bz-i.0. For the example demonstrated

here, the A and B parameters are calculated as liSted below.

Iñ oPder to estimate fatigue damages under a broad-band stress process, it

is seen from Flg.7.4 that the narrow-band assumption can also be used

successfully. The. corresponding correction parameter is À ;

V

where is

the damage due to broad-band stress process given by (6.7), is calculated on the base of the narrow-band stress assiimption in which case,

are obtained. These will be used in (4.23) and (5.3) to calculate E[dD]

and ¡. dr, having introduced these relations into (4.23) and (5.3) it is

òbtained that, 'EtdD] = Ç (2m0)3 _{r(1+k /2)} in which .

IS calculated from the narrow-band assumption. For the example studied here,, this parameter is obtained as,

= 1.15469 () - 0.251308

j . T0

Then, 5 i and from (6.7) the bPoad-band damage will be,

D=

k. k./2 (2SCF) ' _{(2m0) rc1+k/2)}

/

j a j=1

-) EEdD ]

T0 a

From an investigation of Table 7.2 it is seen that the multiplication, where j is based on the narrow-band assumption, is populated around

based on the bPoad-band assumption, see Fig.7.5.

_This

can be verified analytically by using the identity that,

(7.10)

(7. 11)

(7. 12.)

(7.13)

a

EEdD]bb =

v arib

E[dD]

(7. 14.)

O .(7.9a)

k /2

12 = (2m0)

'

whéPe. thé. indices bb denotes broad-band and nb denotes r arrow-band. u, 4100 Curve fitting o + o + 4. 0.12 0.211 0.36 0.118 0.60 0.72 0.611 096 EPSILONE

Fig.7.5 - Comparison of and obtained from

the Monte-Carlo analysis, where bb and nb denote respectively broad-band and narrow-band.

Having introduced (Pa)bbZ into (7.1A,) it can be obtained that,

Since LEdDa] is calcúlated from (.23) in general as depending on I and

from (7.15) it can be obtained that,

r(1+k/2)

[ek+2r(+)

2:j Y2erf(/

')e _di]... (7.16)

The

first

term in the L] on the right hand side of (7.16) is relatively small. Having neglecting this term and replacing the erPor function by approximately (7.16) will be,

E (dDa]

r(lk/a) z

_{y T0}

r(1+k/2)

Sincé p z T / 'It, (7.17) will be satisfied. This is mofe accurate for

small e valués as it is alzò seén from Fig. 7.5.

It is worth noting hepé that A and B parameters of

.

and

V

presented, in (7.7)

are

calculated for only one k3.8O value of the S-N line in this demonstration. It is however obvious thät these parameters are functions

of the k parameter (slope) of the S-N line. Different k values _{result in}

different A and

B pärameters. In order to determine functional f oFms of the A and B parameter with respect to k, these parameters will bé calculated for a number of k values just as explained so far for k=3.80.

Then a non-linear regression analysis will be performed to find smooth

curves for A and B versus k parameter. This specific investigatiOn is _not

contained in this report since the purpose here is to outliné the,

calculation procedures. Finally, a demOnstration of fatigue damages of the example structure is presented in Täble 7.4 for the natural frequencies

3,0, 2.5 and 2.0 rad./sec. In this table EEdD] is calculated by using thé narroW-band assumption, thus, from (7.10). 5: is calculated fròffl

s:

V

assuming that is baSed on the narrow-band calculation. For _this

particular exarnp1e, 5: is found to bé,

1.18199 - 0.25625 () - 0.02118 ()

The mean fatigue damage is simply calculated _as,

E(dD] = 5: EEdD ] -a

(7. 17)

(7.18)

(7.i9)

In the last two columns of Table 7.4,

(Dt) and D.are respectively total damages in the long term period, calculated from _{(6.17), for 5:i} (narrow-band assumption) and 5: is calculated fPom (7.18) (broad-band _assumption). In the calculation, _{a. three parameters Weibull distribution}

(see Appendix B) for the probability distribution of the long term seä states is uséd.

The parameters of the Weibull distribution are assumed to be, see

ref. [23],

A =0.60,

_w

B =1.67 and

_w

C =1.20

w

Table 7.4 - Fätigue damages of the example structure for k=3.80 and in C = 85.87, ströss concentration factor SCF=1.0

The values of total damages for a unit time of one second written in the

last two columns of the Täble 7.4 are obtained by using the trapezoidal

integration rule, and those written in (. ) are obtained by using Gaussian type integration with four integration points, see Appendix B. These

results are used to estimate life times assuming that failure occurs when Dtot reaches unity. The life times calculated for the natural frequencies cónsidered are written, in Table 7. 5.

r/s Hs m T0 sec

E[dD]

-9

10 E[dD]

-9

10

(Dt),T

-9

10

Drr

-9

10 3.0 1.0 3.0 5.0 7.0 9.0 11.0 13.0 2.2061 2.6418 3.1066 3.4937 3.7817 4.0003 4.2069 1.0359 1.2109 1.4020 1.5583 1.6738 1.7611 1.844422.0672 1.3221 3.1229 6.2257 10.1630 14.1692 17.9261 0.8961 0.8542 0.8076 0.7691 0.7404 0.7187 0.6979 0.0126 0.0145 0.0167 0.0185 0.0195 0.0209 0.0219 1.1847 2.6676 5.0280 7.8162 10.4912 12.8832 15.4002 - -0.9013 (0.9625) 0.7728 (0.8262) 2.5 1.0 3.0 5.0 7.0 9.0 11.0 i3.O 2:5894 2.8816 3.2109 3.4949 3.7135 3.8823 4.0432 1.0188 1.1130 1.2254 1.3229 1.3976 1.4553 1.5110 4.2809 8.0544 12.9273 18.5140 23.8921 28.7687 33.9977 0.9001 0.8778 0.8507 0.8270 0.8087 0.7945 0.7808 0.0124 0.0134 0.0146 0.0157 0.0166 0.0173 0.0179 3.8534 7.0698 10.9972 15.3108 19.3215 22. 8572 26.5447 2.1882 (2.3553) 1.9242 (2.0771) 2.0

LO

3.0 5.0 7.0 9.0 11.0 13.0 3.1886 3.3686 3. 5726 3.7551 3. 8997 4.0135 4.1232 1.0513 1.1055 1.1553 1.1951 1.2265 1.2573 1.009016.54860.9024 30.6573 40.9209 51.0766 60.1483 67. 9835 76.0523 0. 8925 0.8796 0.8676 0. 8580 0. 8504 0.8430 0.0123 0.0127 0.0133 0.0138 0.0143 0.0146 0.0150 14.9342 27. 3600 35. 9918 44.3164 5t.6066 57. 8152 64.1091 6.8280 (7.5797) 6.1058 (6.7813)

Table 7.5 - Demonstration of life times for the example structure.

Esti

tAon of t.he variance, of the correction factor

There are twò kind of numerical uncertainties in the correction factop of

the damage. One is introduced by the Monte-Carló experiments and the other one is due to the curve fitting to the mean values obtained from the Monte-Carlo. analysis. Standard deviations of the parameters concerning the Moñte-Carlo experiments are presented in tables 7.2 and 7.3 for the

example presented here. It is clear howevep that this kind of

uncertainties can

be reduced

as small as wanted by increasing the number of Monte-Carlo experiments. In the extreme case, when the number of

experiments tend to infinity, the coñèrning uncertainty vanishes practically, thus it can be neglected in the calculation of the _variance.

Here, the variance of A will be estimated for the _{uncertainty due to the}

curve _{fitting. Since A is defined by (7.4) its variance will be estimated} as, sÍmilar to (7.5b),

2

-

2

-

2

OEA = (a s) +

(y s0)

2 2

in which S and S

will be

calculated from (5.12), see also Appendix C..

These variances are. independent of e parameter unlike

since

_{it is a}

function of e by means of the average parameters and V. As a

2 2

demonstration, and s _{calculated from (5.12) for the example are,}

a 2

-4

s, = 1.3033 10 and S 1.3708 IO

where

_s

is due to the narrow-band assumption. Having introduced _these values and (7.7) into (7.20), and using the appropriate values of A and B parameters, the variance of the random Variable 'A will be obtained as,

(7.20) (rad./sec. )

T1f

in years

3=i

(narrow-band) 1 (brOad-band)

3.0 2.5 2.0 32.9 13.5 1.2 38.3 15.3 4.7