A statistically based fatigue crack growth model for offshore structures

A statistically based fatigue crack growth model for offshore structures JULIAN WOLFRAM

Department of Ship and Marine Technology, University of Strathclyde, Glasgow, Scotland, UK

1. INTRODUCTION The conditions under which an offshore structure operates are subject to considerable uncertainty. The response of the structure to cyclic loading, usually estimated by fatigue calculations, is also subject to considerable uncertainty. This produces two effects. One, a need for general conservatism at the design stage and two, the requirement to inspect representative parts of the structure at intervals during its life to check for fatigue cracks. Unfortunately the fatigue calculations give no indication of what should be found on inspection at any intermediate stage during the structure's life. This leaves a number of awkward questions to be answered. Given the structure has been installed carefully and no extraneous damage or loadings have occurred at what stage are detectable cracks likely to occur? If a small crack is found over halfway through the structure's life is this a harbinger of early collapse of the member concerned? At the end of the nominal fatigue life a structure is still needed for secondary oil extraction. Can it continue in service if on detailed inspection no cracks are found? If so, for how long? To answer these questions a crack growth model is required wlfich can be related to the fatigue life of the structure. Fracture mechanics have been used extensively in the development of such models. Unfortunately most of these are of a deterministic nature and require welldefined inputs. In the case of offshore structures the important input parameters are not well defined in practice and the use of deterministic crack-growth models may well provide misleading results. There are a few probabilistic and statistically based crack growth models but they are not suitable for use in tackling the questions posed above. This paper describes a new statistical crack growth model based on the well-known Paris equation t specifically applicable to tubular offshore structures. In fact it has been developed as part of a wider research programme examining rational approaches to inspection and maintenance strategies for jacket structures. 2 The paper begins by addressing briefly the various sources of uncertainties of fatigue crack growth as they arise through the design, building and operating life of the structure. The following section first looks critically at the various calculation methods which have already been used to predict crack growth in the face of uncertainty before proposing a method based on a Taylor Series approach. Accepted September 1985. Discussion closes June 1986. 0141-1187]86]020089-10 $2.00 9 1986 Computational Mechanics Publications

To estimate the uncertainties, or statistical distributions of the various parameters in the new crack growth model appropriate data are required. The next section describes the analysis of data obtained both from the published literature and from other sources. The distribution of crack size at any time during the fatigue life for a particular ' T ' joint is estimated using the new crack growth model. This is compared with the equivalent fatigue calculation using Miners rule. Finally the scope, accuracy and limitations of the new crack growth model are discussed and a few conclusions drawn. 2. SOURCES OF UNCERTAINTY

2.1 This section catalogues the various sources of un. certainty which affect fatigue and fracture mechanics calculations for offshore structures. The uncertainties are of several different types: (a) The intrinsic random nature of structural steel at the microscopic level and ocean waves at a macroscopic level are examples of one type. (b) The systematic errors which exist in the mathematical models used to describe the fatigue process and perform the associated calculations. (c) Estimation errors occur when parameters are predicted from sparse samples of data rather than the whole population. (d) Finally there is an error involved in approximating the future by extrapolating the past. By looking at the whole fatigue process from the first design concept until final collapse, it is readily seen how the uncertainty increases with time.

2.2 One of the first design decisions is the choice of structural steel. This is usually specified in terms of minimum strength properties which must be achieved including yield stress, ductility and toughness. 3 The toughness can either be expressed in terms of Charpy Impact test values or crack tip opening displacement (CTOD). Empirical relationships have been develope d to relate these measures of toughness; 4 however, whatever measure is used specimens of nominally identical material show considerable scatter, not only in their toughness s but also in their yield strength 6 and fatigue crack growth rate parameter C, from Paris equation. 7 Purely on the basic material considerations therefore there is uncertainty concerning not only the rates at which cracks will grow, but also the critical crack size at which brittle fracture may take place and the ultimate load

Applied Ocean Research; 1986, Vol. 8, No. 2

89

A statistically based fatigue crack growth model for offshore structures: J. lr required to cause plastic collapse in the case of ductile failure. Fortunately in this area considerable data has been collected as a result of the United Kingdom Offshore Steels Research Project and similar efforts abroad.

2.3 For most fatigue calculation purposes the stress range histogram or spectra are obtained by assuming simple deterministic relationships between wave height or energy spectra and wave induced stress. There is considerable data available on wave height histories for various locations around the world. However, the a priori estimation of the long-term wave spectra for any new offshore location is extremely difficult and this is discussed again later. 2.4 When tile structure is being designed the 'hot spot' stress at joints must be calculated from the nominal stresses, obtained using design loads, by means of geometric stress concentration factors. These, whether they be obtained by experiment or theoretical calculations, show considerable scatter giving local stresses which may differ by up to 100%. 8-n 2.5 When the structure is under construction and the tubular members are joined together the metallurgical properties change as a result of the welding process. The precise extent of the change is difficult to quantifyJ 2 The welding process also gives rise to residual stresses in the welded region which will increase the rate of crack growth at least initially. Residual stresses can be reduced by heat treatment, grinding or peening. However, whether or not stress-relieving is undertaken the final state of 'built-in' stress, upon which the crack growth rate will depend, is not known exactly. The actual profile of the weld will also affect the local stress field; and these profiles even after grinding, i f this is undertaken, will show considerable variation from one weld to the next. This is another significant source of uncertainty in fatigue life estimation. 2.6 After fabrication all welds should be inspected, both visually and in critical regions by NDT. Welded joints are rarely perfect and inspection is not 100% reliable both due to the inspector and the techniquesJ 3 There is variety in both type and size of weld defects 14-16 which exist in offshore structures after they have been accepted as certifiable. This is a major source of uncertainty. 2.7 After fabrication the structure must be transferred to a barge, towed out, launched and piled in position. These activities involve loads, and particularly displacements which are difficult to quantify. Quite possibly elastic yielding occurs in highly stressed regions causing 'shakeout' (or 'shakedown') of residual stresses. Perhaps, equally possible, extraneous loads due to launching and piling may cause cracking. The net result is a structure, which, when finally installed, may have significantly different residual stress and defect distributions from those at the time of final post-fabrication inspection. Kallaby and Price 17 suggest a 10% allowance on fatigue lives for these effects - a guess in the face of uncertainty? 2.8 A fracture mechanics calculation for a joint in a structure assumes an initial defect size, local stress field and fracture toughness for the point under examination. However, all these parameters are subject to spatial variations throughout the structure which may be correlated in a complex manner. If a significant defect exists in a

90

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region of low local fracture toughness then cracking may occur even though the local stress is not a 'hot-spot' maximum. There is no information, to the author's knowledge, on the spatial distribution of defects, of a given size, and fracture toughness in relation to local hot-spot stress concentration.

2.9 The fatigue and crack growth calculations usually performed assume linear cumulative damage relationships which are independent of the relative order of stress of cycles of different magnitudes. However, it is well known TM that because of the large plastic zone generated at the crack tip as a result of a large stress cycle the damage caused by subsequent lower stress cycles is less than would occur under constant amplitude loading. Little work has been done in this area in the context of offshore structures although this problem has been considered in the aerospace industryJ 9 The associated problem of cycle-counting for a broad band spectral loading has been addressed by Wirsching 2~ who produced a correction factor, using the rainflow method, for the equivalent narrow band process. 2.10 The offshore environment will affect the rate of crack growth. Some work has been done on stress corrosion cracking and effects of cathodic protection and this is reviewed briefly by Schotz. 21 During the life of the structure marine fouling will accumulate and can increase drag coefficients by over 100%. 22 There is still considerable uncertainty concerning not only the effects of the marine environment but also the way in which these effects should be modelled. 2.11 Given this catalogue of uncertainties it is not surprising that fatigue and fracture mechanics calculations give very imprecise estimates of crack size and fatigue life. It therefore seems eminently sensible to use the results of periodic weld inspections that are made throughout the life of structure and which are required by certifying authorities, to revise and improve the estimation of fatigue life. To do this sensibly a crack growth model which takes some account of uncertainties must be used.

3. STATISTICAL APPROACHES TO FATIGUE LIFE AND CRACK GROWTH PREDICATION

3.1 Existhzg approaches Despite tile uncertainties enumerated above comparatively little work has been done on statistically based procedures to predict crack growth in offshore structures. Wirsching and his colleagues (see ref. 20 for further references) have done considerable work on probabilistic/ statistical approaches to fatigue life prediction for offshore structure using Miner's cumulative damage approach. Others 2a'24 have also worked in thisarea. Engesvik 2s has reviewed many of the uncertainties affecting the fatigue capacity of welded joints and, drawing on considerable data and the results of Monte Carlo simulation, has estimated distributions for many of the parameters. In the nuclear power pressure-vessel field Besuner and Tetelman 2s have used what they call probabilistic fracture mechanics to assess the fatigue lives of components. Their procedure based on the well.known Paris equation da --= dN

C(AK) m = C(YS N/rra)m

(1)

A statistically based fatigue c r a c k g r o w t h m o d e l f o r offshore stntctures: J.

where C = experimentally determined material crack growth constant; A K = range of stress intensity factor; m = experimentally determined exponent; Y = crack shape parameter; S = cyclic stress range;a = crack size;N = number of stress cycles. This is integrated to give Nx =

CS m yrn rrrn/2(m/2 _ 1)

(2)

3.2 A n e w approach

To avoid these problems equation (4) can be inverted and a reciprocal function of crack size considered: 1 a(xm/2_l )

1 a(om/2_O

NxCSmyrnrtm/2(m/2--1)

log N x = --log C - - m log S -- m log Y (3)

The parameters C, S and ao are then considered to be distributed log normally and the resultant log normal distribution o f N x is obtained. This simple 'Iog-linearising' approach allows the effect of the initial defect or crack size to be taken into account when calculating the fatigue life. ttowever, because of the necessary assumption for linearisation, ax >>ao, the number of cycles required to reach intermediate crack sizes cannot be estimated. Kozin and Bogdanoff, 27 working principally in the aeronautical field, have developed a probabilistic model for fatigue crack growth based on Markov chains. This model does allow the variation in time required to reach a given crack size to be predicted. However, it requires a particular form of relationship between variance of time to reach a given crack length, crack length itself and the stress range. In addition for practical purposes, such as correlation with inspection results, it is more useful to have the variation in crack size at a given time; rather than the variation of time to reach a given crack size. One method of obtaining the distribution of crack size as a function of time is to rearrange equation (2) to give 1

(6)

where r = l/a(xm#-l); d = l/a(oml2-1); T = the age of the structure in years (time); M = c y m z r m l 2 o n [ 2 - - 1); L = ~ n i S m where n i is the number of cycles at stress range S i per annum. This equation is much easier to handle because it is linear in the derived variables r, d, M and L. r represents a 'reciprocal' function of the crack size; when a distribution for r is found the corresponding distribution for the crack size a x can readily be obtained numerically. The term d is a reciprocal function of initial crack of 'defect' size. Values of d can readily be obtained from raw data on initial defect sizes. The terms L and M represent the 'loading' and 'material' responses respectively, and once again values can be obtained from oceanographic and experiment data. This equation can be used for a Monte Carlo simulation after distributions have been fitted to the derived variables, d , 3 I and L. However, apart from the problem of computer time mentioned above, largely arbitrary distribution assumptions will have to be made for these variables. The preferred alternative is to handle this equation using a Taylor Series approach. The Taylor Series has already been used in marine structural reliability 29 and also in maritime engineering economics 3~ to develop simple statistical models. From equation (6) closed f o r m expressions can be obtained for the mean, variance and higher moments of r as demonstrated below. For any variable y which is a function of independent v a r i a b l e s x i , x 2 , x 3 , . . . , x n tile Taylor Series yields

ax = [a(oX-m/2)_ N x CS m r m rrm/e(m/2 -- I)] 1/(m/2-1) (4)

Distributions can be assumed for the various variables on the r.h.s, and a Monte Carlo simulation used to find the distribution o f a x for any number of cycles N x . To simplify this exercise m is often taken as a fixed quantity, often 3 or 4, and all the variation in crack growth rate is accounted for by the variable C whichwill have a correspondingly greater coefficient of variation (standard deviation/mean). 24'28'29 However, a Monte Carlo simulation is time consuming on a computer, particularly when a simulation is needed at a range of intervals during the fatigue life to obtain a compre. hensive picture of crack growth with time. More importantly as demonstrated in the Appendix, equation (4) lacks statistical robustness when any of the terms in the denominator are considered as variables with significant coefficients of variation. The terms C, S m and ao, as will be seen later, all have significant coefficients of variation. In this case the mean value o f a x depends heavily on the higher moments of the distributions chosen for these parameters, particularly as the crack size becomes significant. Now the choice of distribution is a little arbitrary when the amount of data is small, and whilst the mean, standard and perhaps the third central moments will be the same or similar, the higher moments will usually be significantly different. It is also notoriously difficult to estimate higher moments accurately from small samples of data.

(5)

This equation can be written mor e simply as r = d - - TML

where N x = number of cycles to obtain crack size ax. It is assumed that all the variation in crack growth rate can be expressed by C, that m can be made constant, and that ao ~ ax. Then taking logarithms yields: --log {r:n/2(m[2 -- 1)} + ( l - - m / 2 ) logao

Wolfram

~~[ nay

y=f(ii~,u2,ii3 . . . . ) + L / : - - - /

(x,-ii3

i= lkOXillat

(x,-., + higher order terms where Il denotes a mean value. In the case of equation (6), where the terms on the r.h.s. are all independent, this yields r = Ila --TIlMIlL + ( d - - i l a ) - - T { I I M ( L - - # L )

+

IlLOs- IlM) +

(L-us.) (M--IIM)}

The expected or mean value o f t is then: (7)

E(r) = Ilr = Ila -- TIlMIlL

and the variance is given by E [ ( r - - i l r ) 2] = Or2 = E[((d -- Ilct) - - T { U m ( L - - i l L )

+ IlL (M--IlM) + (L --ILL) (M--gin))) 2] "

2

2

2

2

where Ox denotes the standard deviation o f x . This can be re-expressed in terms of coefficients of: variation

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A statistically based fatigue crack growth model for offshore stntctttres: J. Wolfram

1 2 2 ~2 2 , 2 i f 2 r-,2 t--~2~11/2 C r ~---- - [lid eel Jr i pl~flaL k L "-I-C~l -q- i.. M ~ L / J

(8)

Pr

where Cx = ax[Px is coefficient of variation o f x . The third, fourth, etc., higher central moments can be obtained by expanding E[(r--I.tr) n] when n = 3, 4, etc. in a similar manner. This yields the coefficient of skewness (S r = E(r--pr)3[o~ - nominalised third central moment, and sometimes denoted by V~I): 1

3 3 3

3 3

3

3

S t, = C 3 ~ r [IddC~lS~l - - T t l m l d L ( C L S L + C ~ I S M

+ ChC~ SM& + 3ChC2 SM + 3C~I C~ S L + 6C~ C~,t)] where S= =

(9)

E[(x--m,)3]/o~.

And finally the coefficient of kurtosis (X r = E[(r--pr)4][o4r

the non-dimensionalised fourth central moment which is sometimes denoted by/32): -

4 { 4 C~dC-M ..4 K~" -- ~41 144 4paC~tKa --T4p~,PM + C~,KL

Before continuing a few points should be noted about equations (7)-(10) and moments of probability distributions. The vast majority of distributions currently used are uniquely defined in terms of the first four moments, 31 certainly all those generally used in engineering. The Weibell, Gamma and log-normal distributions are defined by the first three moments and the normal distribution by only the first two moments. Another point worth noting is that even if all the variables d , M and L are assumed to be normally distributed, and therefore not skewed, the distribution of r will still be slightly skewed. This is clear from an examination of the final term in equation (9). 3.3 Extending the new approach to hwhMe more variables The model described above assumes that all the uncer. tainties related to crack growth can be gathered together under three headings and expressed as three variables, d, hi and L. In many cases it may be difficult to determine the moments of the distributions of these variables directly. However, the approach can be extended to cope with this problem in most circumstances. Consider, for example, the loading parameter L. This is directly related to the hot spot stress range which in turn is related to the waveheight, i.e.

(..rlJr

q

+ C~C~tKLKM

i=l

+ 4(ChC~&KM +C~ChSMKD + 6[,,-,4 ,,-.2 ,,+6

P~

2 2 + : 5 :2 +

- 2 - 2 T 2 (C~C~t

,-~'-L

12(chc].sM+c]. c~sL ~

where K~ = E [ ( x

= Am E lliHF'P

2 2 Ca2 C~I eL)

)}]

+ Ct~ C 3h & SM

(lO)

-pD4l/o~.

]. Estimate the mean, coefficient of variation, coefficient" of skewness (and perhaps the coefficient of kurtosis ) for the derived variables d, M and L using basic data. This is discussed further in the next section where appropriate data are presented, 2~ Using equations (7), (8) and (9) and perhaps (10)) determine the corresponding moments for r as functions of time T. 3 . From an examination of these moments select suitable distributions to represent r at a series of times T during the fatigue life. 4. For each distribution o f r at time T calculate numerically (or analytically if feasible) the corresponding crack size distribution. The index m from the crack growth equation is the same as the index in the fatigue equation - NS m = Constant. A constant value of 3, 4 or 5 is usually chosen as appropriate. The advantage o f this approach compared with the Monte Carlo method is that all the calculations can be performed to give a whole series of crack size distributions covering the entire fatigue life on a desktop microcomputer in a few seconds. Also, only one distribution has to be assumed compared with the Monte Carlo method where distributions must be assumed, for every basic variable. Finally the contributions of each variable to the uncertainty and shape of the final distribution can easily be investigated by parametric variations.

Applied Ocean Research, 1986, Vol. 8, No. 2

:

i=1

To obtain the crack size distribution at any time during the fatigue life the following steps are employed:

92

hi( aHp)m

i=l q

+C~C~IKL + C~r.C~,)

P L IIM

+

q

L = Z ni S m = Z

where A = transfer function relating hot spot stress to waveheight; H = waveheight; p = index, usually between 1 and 2. Now the transfer function A is unlikely to be precisely determined. It usually contains errors due to modelling assumptions and uncertainties related to its determination. Consequently it should be treated as a variable. The waveheight, of course, must also be treated as a variable. The term L then is the sum of the product A} n and H mp of at least two variables. This presents no difficulties provided L (and the same applies to d and M) can be represented as the sum (or difference) and[or product of a number of linear variables ( x b x 2 , etc.). So for the case above q

L = A m ~ ni Hmp =XIX2 t=1

where m and p are both fixed constants and the statistical moments o f A m and Z n l H mp can be found. The corresponding moments for 2 can then be found using the relationships below. For Y = A + B : P Y = P A + PB 2 2 _ 2 2 2 2 I d y C y -- I.IA C ~ -F IaB C ~

3 3 u3 r c r3s y __ - ~AC~SA + p~c~sB 4

4

.pyCyKy

4

4

=IdAC~KA

+P~C~K

B + 6 p A P B2C , ~2C ~2

and f o r Z = CD: P Z = PCIdD

c I = c$ + c~ + c~cb~ ~3C ~3S z = P C3 I A3D ( C3c S C

+

C~SD +CbC~ScS~

+ 3cbcgsc + 3cg CbSD + 6cbcg)

A statistically basedfatigue crackgrowth modelforoffshore structures: J. lr az Cz4 K z =laclao[CcKc 4 , 4 +C~KD-FC~C~KcKD

+ 4(cbc~KcS~ + c~cbK~Sc) + 6(cbC$Kc + c$cbKo.+ c$cg) + 12(c$CgSc+CgCbSD+ CbC~ScS~)] By using these relationships A, d, M and L can all be expressed, if necessary as linear functions of a number of independent variables. This allows more of the uncertainties outlined in section 2 to be represented in the crack growth model. It also allows terms catering for modelling error and bias to be introduced. 4. ESTIMATION OF PARAMETERS AND THEIR UNCERTAINTIES

4.1

There are three independent variables in the crack growth equation (6). A function of the initial crack or defect size, d, which represents the starting point of the crack growing process. The variable L represents the loading upon the crack and the variable M represents how the member responds to this loading. It is assumed that all the sources of uncertainty can be expressed by these three variables, either directly, or by combining a number of other variables as described in the last section. Each of these derived variables is now discussed in turn.

4.2 Loading L The cyclic loading which causes fatigue in offshore structures is caused primarily by waves and is calculated, usually, by means of Morison's equation. For members which are small in diameter compared to the mean significant waveheight (H) drag force loading predominates and the resultant cyclic strain amplitude is proportional to the waveheight Squared. For larger members the inertia force dominates and strain is proportional to waveheight. This has been found from full scale experiments a2 and is accepted by the offshore industry. 33 Assuming the stress isdirectly related to the strain then

s m = A m H mp

l
where p = 2 whenthe drag force dominates and p = 1 when the inertia force dominates. The term A includes many factors about Which there is uncertainty: the accuracy of the wave theory modelling and force coefficients even without marine fouling are not precisely established. The estimation of hot spot stress from nominal stress, however, is already largely catered for in the uncertainty associated with C. Using data from tubular joint fatigue experiments Iwasaki and Wylde a4 have performed regression analyses of cycles to failure on both measured and calculated 'hot spot' stresses. They found the residual errors to be very similar. Now C is estimated, see below, using measured stresses. If C had been estimated using the 'hot spot' stresses calculated from the applied forces used in the experiment, the residual error would not have been much greater. Hence the uncertainty involved in going from member force to calculated 'hot spot' stresses will be largely accounted for in the uncertainty associated with C. The uncertainty in L can readily be estimated using the equations of the last section to combine the uncertainties in A and ~ni Hmp, provided the uncertainties in the latter are known. The waveheight records for five locations around the UK, shown in Fig. 1, have been analysed on a year by year

.......---x..__,

Figure 1. data

Location of weather ships recordh~g waveheight

basis for periods up to 30 years. For each year at each location the product ~niH[np(H ') has been found where n i is the number of waves in waveheight band i. As waveheight records only are available, the relationship between waveheight and period given b~r Fang a n d Hogben as has been used to estimate corresponding wave periods. The values obtained for each location vary significantly from year to year and a set of typical results is shown in Table 1. The coefficients of'r skewness and kurtosis have been calculated for all the locations and are presented in Table 2 together with non-dimensionalised mean values. These results show some interesting characteristics. As would be expected the variation in mean value for the different locations is considerable and illustrates the importance of obtainingreliable wave data for the specific location of a structure. The drag force dominated regime (p = 2) shows significantly greater random variation than the inertia force regime, and also higher skewness. For similar reasons the higher value o f m shows the same trend. Clearly the scatter and skewness increase with the power to which H is raised. Even when wave data is collected for a specific location for a year or two the resultant wave induced stress parameter will be subject to considerable u.ncertainty. If the wave data in each year is assumed independent of the previous data, and there is nothing in the data to suggest otherwise, then the corresponding statistical coefficients for the mean value for a sample period o f n years are given by

c~ C//,mean=~n

szr

and SH'mean='~

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A statistically based fatigue crack growth model f o r offshore stntctures: J. Wolfram which implies that if a wave data sample of, say, two years is collected the corresponding distribution o f the mean value obtained would have coefficients around l / x / 2 times the values in Table 2. However, if the wave spectrum is estimated by interpolating wave, or even more uncertain wind, data from nearby weather ships or buoys then uncertainty is likely to be considerably greater. This sampling error should be taken into account when estimating the moments o f the distribution o f H ' . In effect there are two sources o f uncertainty: one due to the intrinsic random nature o f the wave process and the other due to a limited set o f measurements trying to represent the whole wave population in a particular location for the foreseeable life o f the structure. This large uncertainty in the estimated distribution o f H ' is offset b y the fact that it will be sampled each and

Table 1. Variation in the non.dimensionalised loading Eni Hmp from year to year vniIl~ np Year

mp = 3

mp= 6

mp = 4

mp = 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

0.808 0.964 0.905 1.160 0.894 1.020 1.001 0.878 0.937 0.661 0.476 0.533 0.651 0.827 0.992 0.061 1.112 0.827 1.046 0.666 1.070 0.820 0.900 0.647 . 1.592 1.563 1.990

0.534 0.885 0.538 0.849 0.469 1.236 0.726 1.431 1.358 1.430 0.144 0.214 0.306 0.437 0.872 1.037 1.247 0.576 0.732 0.292 l J061 0.406 0.511 1.369 2.348 1.977 3.973

0.725 0.928 0.830 1.119 0.779 1.122 0.934 1.025 1.047 0.727 0.350 0.419 0.539 0.700 0.965 1.069 1.181 0.764 0.992 0.536 1.054 0.689 0.796 1.653 1.815 1.747 2.505

0.320 0.830 0.334 0.495 0.219 1.089 0.500 1.633 1.742 2.883 0.043 0.085 0.138 0.235 0.738 0.886 1.114 0.346 0.419 0.140 1.032 0.203 0.256 0.994 2.848 1.830 5.700

Table2.

every year o f the fatigue life and it is the variation o f the mean o f this sample that is o f interest. So if the coefficients o f variation and skewness o f H ' are denoted by CH' and S H, respectively then the corresponding coefficients for a sample period o f T y e a r s will be

CH~=C~'

and

F o r a whole fatigue life Wirsching 2~ suggests the modelling error in estimating S has a mean value, or bias, o f 0.7 That is the actual value o f S m is 0.7 m times smaller than usually estimated in design.

4.3 Material response parameter M The parameter M is simply the parameter C from Paris' equation multiplied by a constant. The statistical parameters for C have been estimated using the results o f UKOSRP fatigfie tests on T connections as reported by Clayton. a6 The Paris' equation can be written in terms o f crack length, crack depth or even as the square root o f crack face area, and obviously the corresponding value o f C is different in each case. In 17 o f the experiments reported 36 the crack depth as well as the length was frequently measured. The values o f C corresponding to crack length, crack depth and the square root o f crack length multiplied b y crack depth have been calculated from this data set, together with the associated statistical parameters, and these are presented in the upper part o f Table 3. It is clearly seen from Table 3 that crack length (l) provides a significantly lower c.o.v, than crack depth (a) or (x/al). This is not surprising as crack length is easier to measure than crack depth. Crack depth also varies with position along the crack length, s7 As during in-service subsea inspection crack length only is measured it was decided to concentrate on crack length as a parameter. This allowed more experiment data to be employed and a further analysis was performed using 24 sets o f data. The results from this analysis are shown in the lower part o f Table 3. The values obtained with the enlarged data set are somewhat different from those obtained with the smaller data set but this is only to be expected when dealing with relatively small samples having large c.o.v.s. The method used for estimating C is different to those commonly used when m is a variable, ss Integrating Paris' equation yields

Statistical parameters for annual wave loadfllg term r~niHmPfor five different offshore locations Inertia regime (p = 1)

Drag regime (p = 2)

Location

rn

Mean

c.o.v.

S(x/#,)

K(/32)

1

3 4 3 4 3 4 3 4 3 4

1.0 1.0 3 5 5 9 15 39 58 216

0.35 0.47 0.27 0.33 0.15 0.25 0.49 0.62 0.38 0.48

1.14 1.44 0.46 0.88 0.39 0.88 0.42 0.60 0.03 0.43

4.04 5.19 2.92 4.12 2.86 3.21 2.78 2.84 1.76 2.31

0.33 0.43

0.49 0.85

2.87 3.53

2 3 4 5 Mean values

94

9 x/T SH' SH~.-

3 4

Applied Ocean Research, 1986, Vol. 8, No. 2

Mean

c.o.v.

S(x//31)

K(ta~)

1.0 1.0 14 37 33 123 224 1207 3131 44 344

0.80 1.22 0.50 0.73 0.57 1.05 0.82 0.99 0.82 1.28

2.04 2.38 1.32 1.22 1.61 1.98 0.96 1.46 1.69 2.67

7.90 9.03 4.67 3.37 4.54 5.41 3.45 5.07 6.19 10.81

0.70 1.05

1.52 1.94

5.35 6.74

A statistically based fatigue crackgrowth model f o r offshore stmctures:.J. Wolfram Table 3.

Statistical parameters for the coefficient C from Paris' equation ushlg three different crack size parameters (units N, ram)

m= 3 Crack parameter

tll = 4

Mean

c.o.v.

S (x/'~)

K(#2)

Mean

c.o.v.

S (x/~l)

K(O~)

Seventeen data sets Length (1) Depth (a) ~/al "

7.38 • 10 -~s 1.87 X 10 -~4 1.34 X 10-~4

1.18 1.67 1.72

4.50 3.55 6.63

6.54 4.76 8.31

1.99 X 10 -~s 4.92 X 10-~ 1.20 • 10 -t7

1~17 1.67 1.74

3.33 6.58 8.06

5.88 8.72 10.03

Twenty-four data sets Length (1)

8.28 X 10-~s

0.91

3.56

6.51

2.23 • 10-~s

0.94

2.50

5.67

llzU

~ - -iI~-"/2

Ni+x -- N i = c y m n m l 2 ( l - - m / 2 ) S m

(1 1)

and 11~(n/2 -- l i - m / 2

C Y m --

7rm/2(l , m / 2 ) S m (Nt +l -- Ni)

(12)

It is assumed in the values obtained above that any departure o f y m from unity is reflected in the uncertainty o f C, i.e. y m assumed equal to 1. A value o f C can be obtained for each pair set Ii, Ni; /i+a, Ni+a, obtained in a fatigue experiment, for a given value of m. A weighting procedure 3s is then used to calculate the average value o f C for each fatigue experiment. The statistical parameters in Table 2 are then calculated in the normal manner using the average C value from each experiment. The S values used in the equation above are the experimental ' h o t spot' stresses reported for each experiment. This is assumed 'characteristic' for the whole experiment and no attempt is made to modify S as the crack size increases. Again any such variations are represented by a contribution to the uncertainty in C. To obtain corresponding values o f M, C is multiplied by (m/2--1)Tr rot2. Whilst these values have been obtained for ' T ' joints, they can be used tentatively for other types o f joints in the absence o f sufficient data; just as the new D.En. ' T ' fatigue curve 11 is applied to other joint configurations. 4.4. Esthnation o f hdtial defect size The distribution of defects depths in the welded connections of offshore structures has been considered by Burdekin .qnd Townend, 16 Rodrigues et alJ 4 and Rogerson and WongJ s All collected post-fabrication weld inspection data and concluded that the weld defect heights followed a Weibull distribution. Burdekin and Townend take into account the reliability of the inspection technique (ultrasonic) but do not mention the effects o f repair o f substandard welding. Rogerson and his colleagues on the other hand take into account the effects o f repair but make no mention o f inspection technique reliability. The latter quote a three-parameter Weibull distribution as having the best fit, which implies that there will always be defects o f some minimum size: in this case 0.1 mm depth. Whilst all the authors have favoured a Weibull distribution the mean, and the other statistical parameters, for the various sets o f data are markedly different. Little is said about the spatial distribution o f defects and nothing about their location relative to ' h o t spot' regions. Clearly a defect which is in a ' h o t spot' region is more critical than a defect somewhere else in the weld. It has been observed a7 that a series of small but distinct defects occurring close together

Crack Size

/+

J

o

>

Number of cycles. Fi~tre 2. h z i t ~ l c r a c k s ~ e # l f e r r e d f r o m e x t r a p o ~ t e d c r a c k growth cttrve

Table 4. Statistical parameters for hdtial defect sizes deduced from fatigue e.vperbnent data (units ram)

Term m=3 Io

Range

Mean

c.o.v.

2.7-38

12.1

0.34

3.0-51

16.8 0.097

d = l/~/10 Ill : 4 1o d = ll.v/lo

S(3~)

K(#~)

0.67

1.26

4.78

0.35

0.63

2.39

0.67 0.81

1.29 1.51

4.80 4.64

may rapidly coalesce under cyclic loading to form a more substantial crack. This is not catered for in the simple Paris equation, although it is obviously important, particularly if length is used as a crack parameter. For these reasons the data just discussed has not been used in this case. Instead the crack growth equation used in the previous section has been extrapolated backwards to zero cycles to estimate the 'apparent' initial crack size, as illustrated in Fig. 2. From equation (12): lo = [1~- m / ' + C Y m S m rrrnl2 ( m / 2 - - 1 ) N i ] I/(1 - m12)

Any pair o f values Ii, N i could be used in the above equation but obviously values obtained close to the start of crack growth are likely to provide the best estimators. A procedure which recognises this 3s has been used in the estimation of the values o f 1o. The corresponding mean value and associated statistical parameters are given in Table 4. These 'apparent' initial crack sizes obviously provide an equivalent continuous crack size for a series o f small distinct defects, and any other crack initiation sites. Equally important they give the equivalent initial crack sizes in the critical 'hot spot' stress region.

Applied Ocean Research, 1986, Vol. 8, No. 2

95

A statistically based fatigue crackgrowth modelforoffshore structures: J. Wolfram 5. AN EXAMPLE CRACK GROWTH CALCULATION Consider a ' T ' joint in 32 mm thick steel designed using the new UK D.En. ' T ' curve for air. xl This fatigue curve is defined by

0.6

log (N) = 12.16 -- 3 log(S) for S~> 52.4 N m m -2

Cr

0.4

and log (N) = 15.61 -- 5 log (S) for S ~ 52.4 N mm -2 The wave data has been divided into two groups: that which produces mean ' h o t spot' stress ranges above 52.4 N mm -2 and that which produces stress ranges below 52.4 mm -2. The variation o f the following term is therefore considered: r"

9

513

+.;}

where the first term in the bracket corresponds to the waves which produce a mean stress range below 52.4 N mm -2 and the second term corresponds to the stress range above 52.4 N mm -2. This expression involves a few approximations. It is assumed that H~ and H~ are independent, which they are not, and that the transfer function A does not vary with waveheight, which it does. The first assumption is necessary but is unlikely to have a significant effect upon the results, as for most fatigue life of interest H~ is the dominant term. The second assumption could be avoided, or at least its effect made insignificant within the content o f the method proposed, by considering different mean values o f A for each waveheight band. However, for the purposes o f illustration the loading representation above is thought to be adequate. This yields //L

=

4.1 x l0 l~

Using the data compiled by Wirsching [op. cit] it is assumed that CA3 = 0.6 ant' SAs= 1.1. It is also assuraed that the distribution o f the waveheight term has CHI_ ~ = 0.7 and SHI_, = 1.5 which corresponds to m = 3 and the drag dominated regime in Table 2. N o w most o f the fatigue damage is caused by stress ranges above 52.4 N mm -2 so the statistical parameters for C, based on crack length, at m = 3 are selected from the lower part o f Table 3. In fact for all the tubular joint fatigue data reported by Clayton [op. cit.] the stress ranges were above 52.4 N mm -2. The values f o r m are then: Pm = 0.5 x ~" • 1.5 x 8.28 x 10 -is = 2.3 x 10 -~4 and Cm = 0.91, S m = 3.56 Finally from Table 4 the statistical parameters for d, the initial crack length term, corresponding to m = 3 are taken, viz. /au = 0.34 Ca = 0.35, S a = 0.63 Equations (7), (8) and (9) have been used to calculate the corresponding values of I~r, Cr and S r, and these are pre. sented in Fig. 3. This shows that the distribution o f r changes from having a positive skew to having a negative skew during the fatigue fife. A three-parameter log-normal distribution has been assumed for r for all values of T. This has necessitated parametric manipulat.ions because o f the change in skewness, but is thought a reasonable assumption, given the

96

Applied Ocean Research, 1986, Vol. 8, No. 2

~r

0.2 Cr

~r

Sr

& Sr

0

20

40

60

80

I00

TIHE

-0.2

(years)

-O.4L Figure 3. Changes in mean and coefficients o f variation and skewness o f r with thne

Table 5. Crack length exceedance probabil#ies for a nontflzal fatigue life oi"36 )'ears obtained using D.En. 'T" curve Crack length (mm) Year

5

10

20

50

100

500

1000

0 4 10 20 30 100

0.829 0.833 0.838 0.847 0.854 0.894

0.468 0.476 0.488 0.508 0.526 0.567

0.154 0.160 0.171 0.189 0.209 0.309

0.018 0.020 0.023 0.029 0.037 0.150

0.003 0.003 0.004 0.008 0.009 0.099

0.000 0.000 0.000 0.000 0.000 0.072

0.000 0.000 0.000 0.000 0.000 0.046

sparsity of data available. It does, however, introduce slight discontinuities in the calculated probabilities in regions o f high skewness and where skewness changes. The corresponding probabilities o f crack size exceedance for several values o f crack length have been calculated at intervals during the life and these are tabulated in Table 5 and presented graphically in Figure 4. It is interesting to note that the probabilities o f exceedance for long crack lengths are relatively similar. This is indicative o f the rapid growth o f large cracks. The nominal fatigue life, obtained by using the ' T ' curve above, is 36 years. At this time there is about a 0.1% chance of complete fatigue failure. If account is taken o f modelling bias, the actual loading L is about one-third (~0.73) o f the estimated value, i.e. /It. ~ 0.7 a x 4.1 x 10 l~ = 1.4 x 10 9. (The associated modelling uncertainty is largely accounted for in the CL value used above.) When the modelling bias is introduced the probability o f failure falls to around 0.01%. The modelling bias in effect constitutes a factor o f safety o f about 3 on loading which is sometimes thought necessary for fatigue calculations given the loading uncertainty. It is well reeognised that when uncertainties are large, factors o f safety must be high. In cases where m is constant irrespective o f the stress range the calculation is slightly simpler than the one above.

A statistically based fatigue crack growth model for offshore structures: J. Wolfram PROBABILITIES OF CRACK LENGTH EXCEEDANCE

p

.

o

~I

~,I

o

30

40

1000

5% Probability level.

100

CRACK

LENGTH

(mm) 10

1

0

10

20

In.IE (YEARS)

Figure 4.

Probabilities of crack length exceedance o f times duringthefatigltelife

In either case the Calculation is only as precise as the available data on which it is based. 6. DISCUSSION The first question the reader might pose is how accurate is this crack growth model. A precise answer is difficult. The data used to determine M,L and d are the same as that used to estimate fatigue life curves. I1 Although it is more restricted as crack length measurements are not made during all fatigue tests. However, as more data become available these can readily be incorporated and the precision of the modelling enhanced. The effect of the relatively small data sample on the uncertainty has not been quantified. The assumptions behind the model are those implicit in the Paris equation. The effects of broad band spectral loading, plasticity, crack closure and interaction between adjacent stress cycles are ignored. Irreversible cumulative damage is assumed. However, correction terms to allow for these effects, and the others discussed in section 2, can readily be introduced into the framework of the model. This is provided they can be expressed as linear independent multiplies of, or additions to, the existing terms. When a probability distribution is chosen to represent either a basic set of data or a series of moments an assumption is involved. Unfortunately moments are not very efficient estimators for distributions. 39 However, the distribution of r is not highly skewed in the region of interest which is a mitigating factor. In addition given the order of the uncertainties in the understanding and modelling of fatigue crack growth in offshore structures this is not a problem of major significance. As more data becomes available higher moments can be estimated with more precision and suitable distributions more closely defined. In spite of this the precision of the distribution, particularly in the tails, will always be open to some questions. This is not a problem associated with

this crack growth model alone. It exists for all statistically based models using limited samples of data. In more complex crack growth models the 'hot spot' stress is recalculated as the crack size increases. This requires precise knowledge of the crack size, shape and orientation together with the magnitude and direction Of all loads at any point in time. In practical cases for offshore structures this information is just not available and in addition the calculations involved are quite complex. In the model presented here the effects of crack size and shape upon 'hot spot' stress range are ignored;just as they are in fatigue calculations. The strength of this model, apart from its relative simplicity, is that the effect of each term and its uncertainty can easily be examined, and the model can be extended as and when data on other effects become available. It also has the ability to include in.service inspection results and environmental monitoring data to modify and up-date the crack growth predictions, a~ It is currently being used to examine the effects of fatigue cracking upon limit state strength or capability and to assess the variation in overall structural reliability over the fatigue life. 7. CONCLUSIONS A simple statistical cumulative damage crack growth model has been developed which can be directly related to cumulative damage fatigue life calculations. The model is flexible and can include refinements to allow for the numerous uncertainties which are associated with fatigue cracking as and when data becomes available.

REFERENCES 1 2

Paris, P. and Erdogan, F. A critical analysis of crack propagation laws, Trans. ASME, December, 1963 Wolfram, J. A techno-economic app~'oach to underwater inspection and maintenance strategies, Jour. SUT, Summer, 1983

Applied Ocean Research, 1986, Vol. 8, No. 2

97

A statistically based fatigue crack growth model f o r offshore structures: s lr 3 Rules for the Design, Construction and Inspection o f Offshore Structures, DnV, 1977 4 5

6 7 8 9 10

11 12 13 14

15 16 17 18 19 20 21

22 23 24

25

26 27 28 29 30 31 32

98

Johnston, G. D. Statistical scatter in fracture toughness and fatigue crack growth, ASTM STP 798, June, 1983 Wong, W. K. and Rogerson, J. tl. A probability estimate of the relative value of factors which control the faUure by fracture of offshore structures, 2rid hit. Conf. on Offshore Welded Structures, London, Nov., 1982 Rationalisation of safety and serviceability factors in structural codes, CIRIA Report 63, 1977 Gurney, T. R. An analysis of some fatigue crack propagation data for steels subjected to pulsating tension loading, Report 59[1978[E, Welding Institute, March, 1978 Gibstein, M. B. Parametric Stress Analk'sis of T Johzts, European Offshore Steels Research Seminar, Abington, Nov., 1978 Wordsworth, A. C. Stress concentration factors at K and KT tubular joints, Fatigue hi Offshore Structural Steel, ICE, London, 1981 Potrin, A. B. and Kuang, J. G. Stress concentration in tubular joints, J. Soc. Pet. Engrs, August, 1977 Background to New Fatigue Design Guidance for Steel It'eMed dohzts hz Offshore Structures, D.En., ltMSO, London, 1984 Ghonem, M. B. and Fenn, R. The effect of welding on the fatigue behaviour of a 0.5 carbon equivalent steel, 2rid Int. Conf. on Offshore Welded Structures, London, 1982 Forlie, O. and Pettcrsen. Reliability of ultrasonic and radiographic testing, Fitness for Purpose Validation o f WeMed Constructions, Int. Conf. London, 17-19 November 1981 Rodrigucs, R., Wong, K. and Rogerson, J. Weld defect distribution in offshore platforms and thek relevance to reliability studies, quality control and in-service inspection, OTC 3693, llouston, 1980 Rogerson, J . H . and Wong, W. K. Weld defect distributions in offshore structures and their influence on structural reliability, 0TC4237, tlouston, 1982 Burdekin, F. M. and Townend, P. H. Reliability aspects of fracture on stress concentration regions in offshore structures, hltegrity o f Offshore Structures, Applied Science, 1981 Kallaby, J. and Price, B. Evaluation of fatigue considerations in the design of framed offshore structures, Jourml of Petroleum Tech., March 1978 Brock, D. Elementary Enghleerhtg Fracture Mechanics, Noordoff Int., Leiden, 1974 Elber, W. Equivalent constant amplitude concept for crack growth under spectrum loading. Fatigue crack growth under spectrum loads, ASTM STP595, pp. 236-250, 1976 Wirsching, P. tt. Fatigue reliability for offshore structures, J. Struct. Eng., ASCE 1984, 110 (10), October Schotz, W. Procedures for the prediction of fatigue life of tubular joints, Int. Conf. Special and Plenary Sessions Steel in Marine Structures, Commission of the European Communities, Paris, 5-8 October 1981 Wolfram, J. and Theophanatos, A. The effects of marine fouling on the fluid loading of cylinders: some experiment results, OTCPaper No. 4594, llouston, May 1985 Manners, W. and Baker, M. J. Reliability analysis in fatigue, 2nd Sym. on Integrity of Offshore Structures, Glasgow, 1981 Bokabud, T. and Karlsen, A. Probabilistic fracture mechanics evaluation of fatigue failure from weld defects in butt welded joints, Fitness for Purpose Validation of Welded Constructions, Welding Institute, London, 17-19 November 1981 Egersrik, K. M. Analysis of uncertainties in the fatigue capacity of welded joints, Report UR-82-17, Dept of Marine Technology, The Norwegian Institute of Technology, Trondheim, December 1981 Besuner, P. M. and Tetelman, A. S. Probabilistic fracture mechanics, Nuclear Enghzeering and Design 1977, 43, 99-114 Kozin, F. and Bogdanoff, J. L. On the probabilistic modelling of fatigue crack growth, Eng. Fract. Mech. 1983, 18 (3) Ran, C. A. et al. Failure analysis and failure prevention in electric power systems, Nuclear Engineering and Design 1977, 43, 1-97 Flint, A. R. and Baker, M. J. Rationalisation of safety and serviceability factors in structural codes: supplementary report' on offshore structures, CIRIA Report UR9, October 1977 Wolfram, J. Uncertainty in engineering economics and ship design, Trans. North East Coast Inst. o f Engs. and Shipbuilders, 96, 1980 Ord, J. K. Families o f Freqitenc), Distributions, Griffen, London, 1972 Kenley, R. M. Measurement of fatigue performance of Forties

Applied Ocean Research, 1986, VoL 8, No. 2

33

34 35

36

37 38 39 40

Bravo, 0TC4402, 1982 Loader, D. J., Shorrock, P. A. and Staton, V. A. Implications of the new fatigue design guidance notes for the designer and platforn~ operator, 2nd Int. Conf. on Offshore Welded Structures, Welding Institute, London, 16-18 November 1982 Fang, Z. S. and Hogben, N. Long term statistics of wave heights and periods, The Naval Architect, February, 1984 Iwasaki, T. and Wylde, J. G. Recent research on the fatigue performance of welded tubular joints, 2 n d Int. ConL on Offshore Welded Structures, Welding Institute, London, 16-18 November 1982 Clayton, A. M. Assessment of UKOSRP crack growth data to investigate the remaining life of offshore structures following inspection, UKAEA Northern Division Report ND-R-852{R], November, 1982 Dover, W. D. and Holbrook, S. J. Fatigue crack growth in tubular welded connections, BOSS '79, London Wolfram, J. The analysis of fatigue crack data from tubular connections to obtain C values and estimate initial defect size, Project MASS Report, University of Strathclyde, 1985 Johnson and Kotz, Conthmous Univariate Distributions-I, ttoughton Mifflin, New York, 1970 Wolfram, J. Revision of fatigue life calculations and associated uncertainties in the light of in-service data, 9th Int. Conf. on Behaviour of Offshore Structures, Delft, ttolland, 1-5 July 1985

APPENDIX The rearrangement o f the Paris e q u a t i o n to give an expression for crack size, e q u a t i o n (4) in the paper, has the f o r m

I

Y = (X[-- TKx2xP3) q where x l , x 2 , x3 are distributed variables, p and q are positive n u m b e r s (usually), r is a negative n u m b e r (usually) and T represents time, and K is a constant. The n t h partial derivates to x l , x2 and x j , as well as cross coupled partial derivatives all exist for all values o f n . In a d d i t i o n these derivates are all positive w h e n

(X[ -- TKx2xV3 ) q > 0 Consider n o w the Taylor Series expansion o f y

k bxi/~ i

+I~ ( ~ )a2ym u (xi--"i)(x,--VJ) + finite higher order terms w h e n the e x p e c t a t i o n or m e a n o f y is e x a m i n e d at least all higher order terms o f the t y p e

~2(i+m)Y

ox~ l ox] m

where l, m = 0, I, 2, 3 ; . . .

remain, i . e . E ( y ) =f(~ul, P 2 , ~ a ) + f u n c t i o n o f higher terms. All the higher order terms have the form

f(x,,x~,xj) (x[ -- T K x 2 x f ) n where n is the order o f the term. N o w w h e n crack size increases as T increases, so does the magnitude o f the higher order terms. If the coefficients o f variation o f x t and x2 x x ~ are large (the general case) t h e n the higher order t e r m s m a k e a significant c o n t r i b u t i o n to the e x p e c t e d or m e a n value o f y as T approaches the nominal fatigue life. U n f o r t u n a t e l y estimates o f higher m o m e n t s f r o m small samples o f data are n o t o r i o u s l y unreliable.