*Corresponding author. Tel.: #44-0207-679-7182; fax: #44-0207-383-0831.E-mail address: l.etube@meng.ucl.ac.uk (L.S. Etube).

Marine Structures 12 (1999) 565}583

Review of empirical and semi-empirical > factorsolutions for cracked welded tubular joints

L.S. Etube*, F.P. Brennan, W.D. DoverDepartment of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK

Received 1 April 1999; received in revised form 20 October 1999; accepted 29 November 1999

Abstract

The practical use of fracture mechanics has been established for use on large turbine andelectric generator rotor components used in the atomic power generation and the aircraftindustry. Application areas in the o!shore industry have also been identi"ed. Fracture mechan-ics is currently used at the design stage of o!shore facilities. It provides the basis for fatigue lifeprediction, steel selection and tolerance setting on allowable weld imperfections. Fracturemechanics is also used during the operational stage of a structure to make important decisionson inspection scheduling and repair strategies and as a tool for establishing limits on opera-tional conditions. Linear elastic fracture mechanics relies on the use of the stress intensity factorconcept. The stress intensity factor is a very important fracture mechanics parameter. Therefore,the accuracy of any fracture mechanics model for the prediction of fatigue crack growth ino!shore structures for example will depend very much on the accuracy of the stress intensityfactor solution used. Several empirical and semi-empirical solutions have been developed overthe years with varying degrees of accuracy. This paper presents a review of some of thesemethods and attempts to assess their accuracy in predicting> factors for welded tubular jointsby comparing predicted results with experimental data obtained from fatigue tests conductedon large scale welded tubular joints. The experimental results were conducted under simulatedservice conditions, using a jack-up o!shore standard load history (JOSH). A comparisonbetween the experimental and predicted results shows that there may be other factors, whichin#uence fatigue crack growth under variable amplitude conditions. Some of these factors havebeen identi"ed and discussed in this paper. ( 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

The S}N approach is used extensively to design welded o!shore tubular joints andother welded connections for o!shore applications. However, the S}N approach

0951-8339/99/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 9 5 1 - 8 3 3 9 ( 9 9 ) 0 0 0 3 3 - 7

cannot be used in assessing the structural integrity of cracked tubular joints in service.In this regard fracture mechanics is used and, at present, it is the most powerful anduseful technological tool available for describing and solving fatigue crack problems.It is also used in practical engineering applications to make important decisions oninspection scheduling and repair strategies. Fracture mechanics is used as a tool forestablishing limits on operational conditions.

Some of the existing fracture mechanics models, used in the prediction of fatiguecrack growth in o!shore welded tubular joints, are examined in this paper. These fallin the category of empirical and semi-empirical models and adapted #at platesolutions based on "nite element analysis. Results obtained from each of these modelsare compared with experimental results. The accuracy of these models in the predic-tion of fatigue crack growth in welded tubular joints under variable amplitude loadingconditions is assessed on the basis of the predicted > factors. Emphasis is placedon the e!ect of service loading and a consideration of sequence e!ects on the accuracyof existing models when used for fatigue crack growth prediction in o!shorestructures.

2. The concept of stress intensity factor

Irwin is one of the many researchers who made a great deal of contribution to thedevelopment of fracture mechanics concepts. He extended Gri$th's theory [1] forductile materials and postulated that energy due to plastic deformation should betaken into account in evaluating the energy associated with the creation of a newcrack surface. He also de"ned a quantity, G, the strain energy release rate or `crackdriving forcea, which is the total energy absorbed during cracking per unit increase incrack length per unit thickness.

Perhaps his most signi"cant contribution came in the mid-1950s [2], when heshowed that the local stresses near the crack tip can be expressed in the form

pij"

K

J2prfij(h)#2, (1)

where r and h are the cylindrical co-ordinates of a point with respect to the crack tipand K is the stress intensity factor.

In general the mode I stress intensity factor for a centre crack of length 2a in anin"nite plate subjected to a uniform stress "eld, p, (Fig. 1) is given by

K"pJpa. (2)

Eq. (2) gives the stress intensity factor (SIF) in the absence of all boundaries of a formapplicable to the mode of loading and specimen geometry. Cracks in welded tubularjoints are usually in a complex stress "eld. This complex stress "eld is generallydi!erent from the case of a uniform stress in an in"nite plate. SIF solutions for cracksin tubular welded joints must therefore include various correction functions toaccount for boundary e!ects due to loading and specimen and crack geometries. This

566 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Fig. 1. Crack in an in"nite plate subjected to a uniform stress "eld.

leads to a stress intensity factor solution which can be expressed as

K">pJpa, (3)

where > is the stress intensity factor correction function with the following generalrecommended [3] form

>">s>

w>

e>

g>

k>

m, (4)

where >s

is the correction for a free front surface, >w

the correction for "nite platewidth, >

ethe correction for crack geometry, >

gthe correction for non-uniform stress

"eld, >k

the correction for the presence of geometrical discontinuity and >m

thecorrection for changes in structural restraint.

Over the years, di!erent analysis methods have been used to determine the > factorsfor cracked tubular welded joints. This has led to the development of several SIFsolutions for semi-elliptical surface cracks. Some of these are empirical and semi-empirical solutions obtained from experimental results and those based on "niteelement analysis results. This paper presents a review of these methods and comparestheir accuracy with experimental results obtained from large-scale tests conductedunder realistic environmental conditions.

3. Experimental results

Calculation of experimental stress intensity factors can be carried out using experi-mental crack growth data. With the increasing accuracy in the measurement of

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 567

experimental fatigue crack growth rates using data from NDT techniques, such as thealternating current potential di!erence (ACPD) [4], it is possible to determineexperimental> factors with reasonable accuracy. This approach has been used in thepast to develop empirical > factor models.

Fatigue crack growth data obtained from large-scale fatigue tests [5,6] have beenused to determine experimental >-factors. The fatigue tests on >-joints outlined inRefs. [5,6] were conducted under representative variable amplitude loading condi-tions using a simulated jack-up o!shore standard load history (JOSH). These experi-mental values are used as a benchmark for comparing the accuracy of other modelsexamined. The procedure adopted in determining the experimental > factors ispresented here.

The determination of experimental > factor relies on the use of a suitable crackgrowth law such as Paris law

da

dN"C(*K)m, (5)

C"2.72]10~12, m"3.532 and *K">*SJpa,

where a is the crack size, *K is the stress intensity factor range and *S is the hot-spotstress range. By assuming that Paris law applies, experimental > factors may beobtained from

>"A1

*SJpaB A(da/dN)

C B1@m

. (6)

The experimental crack growth rates were determined from the fatigue tests conduc-ted for this study. These growth rates are similar to those reported in [7] for testsconducted on single notch bend specimens (SENB) specimens using the same steel.The experimental crack growth rates were used to calculate the corresponding> factor curves. These are shown in Figs. 2}5. The irregular nature of the >-factorcurves is due to the variable amplitude nature of the loading sequence and theresulting retardation e!ects.

The accuracy of the experimental > factors depends on the Paris law materialconstants C and m. The C and m values shown above were obtained from compacttension tests performed on parent plate in air by Creusot Loire Industrie (CLI) [8].A summary of the data supplied for both parent plate and the heat a!ected zone isgiven in Table 1. Where appropriate data are not available it is not recommended touse arbitrary values from PD 6493 [3] as misleading results can be obtained.

The results from CLI [8] were comparable to those from tests conducted on thesame steel at Cran"eld University [7]. It is important to note that the accuracy of theexperimental > factors presented depends greatly on the values of C and m used inanalysing the experimental results. However these values are sensitive to mean stresse!ects and often this leads to di!erences in empirical solutions as a result of the scatteron the C and m values. The data shown in Table 1 were used for comparingexperimental results with those predicted using available > factor models.

568 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Fig. 2. Experimental > factors for >1.

Fig. 3. Experimental > factors for >2.

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 569

Fig. 4. Experimental > factors for >3.

Fig. 5. Experimental > factors for >4.

570 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Table 1Paris law air data for SE 702

C m

Parent metal (PM) 2.715]10~9 3.5320Heat a!ected zone (HAZ) 3.872]10~9 3.1687

4. Empirical Y factor solutions

Empirical models have been developed for rapid and accurate analysis of crackgrowth data. Some of these models have gained wide acceptance and have beensuccessfully used in the analysis of fatigue crack growth in tubular welded joints.These models which include the two phase, the average stress and the modi"edaverage stress models are presented here and their performance is compared withexperimental results.

4.1. Equations of Dover et al.

Irwin's in"nite plate solution predicts that the stress intensity factor at the deepestpoint for a semi-elliptical crack is always greater than that at the surface. However, ithas been observed experimentally that, crack growth on the surface may be higherthan crack growth at the root as the crack aspect ratio changes. It was thereforenecessary to derive corrections for the "nite dimension e!ects on the surface cracks ina semi-in"nite body.

Holdbrook and Dover [9] carried out a series of fatigue tests on #at platespecimens of "nite dimension containing semi-elliptical cracks in a tensile stress "eld.This led to the development of equations which accounted for e!ects arising from"nite cross section area, "nite second moment of area, load eccentricity and anychanges in the position of the neutral axis. These equations were found to providegood correlation with crack growth data for surface cracks in plates.

Dover and Dharmavasan [10] extended the work to tubular joints using anexperimentally based method for the determination of stress intensity factors of cracksin tubular joints. The approach adopted was based on a stress intensity factorexpression of the form

*K">*SJpa. (7)

The correction factor > was obtained through the use of an experimental technique.The values of the correction factors obtained for "ve >, ¹ and K joints were used forthis study. Based on the results obtained, the following equation was recommendedfor deriving > factors for tubular joints.

>"(1.18!0.32S) ¹(0.13~0.02S) A¹

a B(0.24`0.06S)

, (8)

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 571

where ¹ is the chord wall thickness and S is a non-dimensional average stressparameter. S is the ratio of the average stress concentration factor, SCF

av, to the hot

spot stress concentration factor, SCFHS

, at any location of the joint intersection. It isgiven by

S"SCF

avSCF

HS

, (9)

SCFav"

1

p Pp

0

SCF(/) d/ (for axial loading and OPB) and

SCFav"

1

p Pp@2

~p@2

SCF(/) d/ (for IPB),

where OPB and IOB denote out-of-plane and in-plane bending, respectively. This setof equations assumed that the stress intensity factor only depends on one dimension ofthe crack and the average stress concentration factor was taken over the entire jointintersection instead of over the instantaneous crack length. This made the predictionsof stress intensity correction factors using these equations not to be as accurate asdesired. In a later investigation by Dover and Connolly good predictions wereobtained for crack shape development in plates subjected to bending loads. Othermore accurate models have since been developed by Dover et al. to predict the> factor. The more recent models are the average stress model [10], the modi"edaverage stress model [11], and the two phase model [12,13] which accounts for sizee!ects on the early growth and the propagation phases of crack extension.

4.2. The average stress model

The average stress (AVS) model [10] was proposed after testing large-scale 16 mmtubular joints. This model made use of several stress intensity modi"cation (>) factorsand assumed a thickness correction for joints other than 16 mm. The > factorpredicted by this model is given by

>"AA¹

a Bj, (10)

A"0.73!0.18S and j"0.24#0.06S.

This model has been used to predict experimental crack growth rates in tubularwelded joints with discrepancies only occurring during early growth where the crackdepth is less than 25% of the chord wall thickness. The > factor predicted by thismodel is compared with experimental results in Fig. 6.

4.3. The two-phase model (TPM)

The two-phase model [13] was based on published crack growth data andwas developed mainly to consider crack growth a!ected by joint size. It is given

572 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Fig. 6. Comparison of experimental > factors with AVS prediction.

in the form

>"MIB A

¹

a Bk, (11)

where B and k are functions of size and average stress parameter, S, and p is the earlycrack growth phase controlling parameter. M

Iis taken as 1 for the propagation phase

(a/¹'0.25) and (0.25¹/a)~p for the early crack growth phase (a/¹(0.25).

p"0.231A¹

0.016B~1.71

b0.31S0.18HS

, B"(0.669!0.1625S) A¹

0.016B0.11

and k"(0.353#0.057S) A¹

0.016B~0.099

,

where b is the ratio of brace to chord diameter. The early crack growth phasecontrolling parameter was produced by assuming that early crack growth behaviourcan be treated as an extrapolation of the propagation phase modi"ed by an exponen-tially decaying e!ect determined by the wall thickness, the diameter ratio and thehot-spot stress. It has been argued [11] that the thickness correction exponent whichdetermines the value of the early crack growth phase controlling parameter, p, is suchthat it imposes a very severe dependence of crack growth on thickness. Therefore,making this model more sensitive to thickness e!ects than has been observed

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 573

Fig. 7. Comparison of experimental > factor with TPM prediction.

experimentally. The predicted > factor curve obtained using this model is showntogether with the experimental results in Fig. 7.

4.4. The modixed average stress model

The modi"ed average stress model (MAVS) is the most recent empirical stressintensity factor solution for cracked tubular welded joints proposed by Austin [11]. Itwas proposed after testing large-scale 16 mm tubular joints. It is an extension of theaverage stress model and was developed by applying a 15% reduction factor to theoriginal AVS model. The reduction factor was based on the assumption that rain#owcycle counting provided a higher degree of correlation with constant amplitude datathan range counting on which the original AVS model was based. Austin [11]suggested the 15% reduction factor after noting that the equivalent stress determinedfrom rain#ow counting was higher than that obtained when simple range countingwas used for the representative double peaked spectrum originally used to develop theAVS model. This factor was found to be 1.15. A modi"cation to the AVS model wasthen proposed based on this di!erence which required that the > factor predictedby the AVS model be reduced by 15%. The > factor predicted by this model isgiven as

>"0.85A A¹

a Bj. (12)

574 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Fig. 8. Comparison of experimental > factor with modi"ed AVS prediction.

All the variables are as de"ned for the AVS model. The degree of accuracy obtainablefrom this model depends largely on the accuracy of the original experimental data onwhich it is based. Under variable amplitude conditions this may equally be a!ected bythe method of cycle counting used and the detail contained in the crack growth data.Fig. 8 shows how the results predicted by this > factor model compare with experi-mental data.

5. Adapted plate solutions

Stress intensity factor solutions for plates cannot be applied directly to tubularwelded joints. This is as a result of the di!erences in the existing boundary conditions.They are however important in that plate solutions can be used to provide estimatesof stress intensity factors for other geometries by applying the appropriate boundarycorrection functions. For instance #at plate solutions may be used to obtain stressintensity factors for semi-elliptical cracks in ¹-plates by introducing a correctionfunction to account for the in#uence of the weld detail and the attached plate.

Di!erent researchers have used di!erent approaches over the years to model thee!ect of the weld detail on the #at plate solutions and develop stress intensity factorsolutions for welded connections. These range from methods based on weight func-tions to those based on "nite element analysis carried out on welded joints. Theseapproaches fall within three broad categories of methods generally used to determinestress intensity factors. These include classical solutions for idealised geometries,

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 575

numerical methods and semi-empirical solutions based on a combination of experi-mental and analytical data. The more widely used of these solutions is that due toNewman and Raju [14].

5.1. Newman}Raju SIF solution for surface cracks

Newman and Raju (NR) [14] derived a stress intensity factor solution for a semi-elliptical crack in a #at plate. The proposed solution gave the stress intensity factor fora surface crack of depth, a, and surface length, 2c, in the form

KI">

NRJpa, (13)

>NR

"

Fmpm#F

bpb

U, (14)

where Fm

and Fb

are the correction functions for the tension and bending stresses,pm

and pb, respectively. U is an elliptical integral approximated by

U"S1#1.464 Aa

cB1.65

.(15)

The correction functions for tension, Fm, and for bending, F

b, are given as

Fm"CM1

#M2A

a

tB2#M

3Aa

t B4

D fw

and

Fb"C1#G

1Aa

t B#G2A

a

tB2

D CM1#M

2Aa

t B2#M

3Aa

tB4

D fw,

where

G1"!1.22!0.12 A

a

cB, G2"0.55!1.05 A

a

cB0.75

#0.47 Aa

cB1.5

,

M1"1.13!0.09A

a

cB, M2"!0.54#A

0.89

0.2#(a/c)B,

M3"0.5!A

1.0

0.65#(a/c)B#14A1.0!Aa

cBB24

and fw"SsecA

nc

wJa

cB.The function, f

w, is the plate width correction function for a plate with a "nite width,

w. Even though this #at plate solution cannot be applied directly to welded tubularjoints, it is very important in that it can be used to provide estimates of stress intensityfactors for other geometries by applying the appropriate boundary correction func-tions.

The Newman}Raju solution has been shown [15] to yield results which agreeclosely with experimental tubular joint> factors for cracks of a/¹'0.15, by applyinga moment release function to account for the stress redistribution which accompaniescrack propagation in tubular joints.

576 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Di!erent researchers have used di!erent approaches over the years to model thee!ect of the weld detail and crack geometry on the #at plate solutions and developstress intensity factor solutions for use in the analysis of cracks in tubular weldedconnections.

A semi-analytical model based on the Newman and Raju #at plate solution forpredicting > factors in welded tubular joints was proposed by Monahan [16]. Thismodel included the following:

1. A non-uniform stress correction (NSC ) to account for weld geometry,2. A linear moment release (LMR) to account for load shedding and3. A crack shape correction (CSC ) factor to account for the e!ect of crack geometry.

The proposed equation is given by

>NR`NSC`LMR`CSC

"

(Fm>

g(1!B/¹)#F

b>

g(B/¹)(1!a/t))W

U, (16)

where >g

is the non-uniform stress correction factor, W is the crack shape correctionfactor and B/¹ is the bending to total stress ratio.

The non-uniform stress correction factor,>g, accounts for the in#uence of the stress

concentration produced by the weld detail. This factor can be obtained usinga method proposed by Albrecht and Yamada [17]. Using Albrecht's method, thisfactor is given by

>g"

2

nn+i/1

pxi

p Asin~1Axi`1a B!sin~1A

xi

a BB. (17)

This was derived for a non-uniform stress distribution, p(x), that remains symmetricalabout the crack centre line as shown in Fig. 9. The crack dimension, a, used herecorresponds to half the length of a through crack, which is de"ned as the crack depthfor surface cracks as expressed in Eq. (13).

The crack shape correction factor, W, introduced by Monahan was included toaccount for the in#uence of crack aspect ratio. This factor was obtained by comparingexperimental > factors with those obtained by the Newman and Raju #at platesolution which included a non-uniform stress correction factor and the linear momentrelease model such that

W"

>Exp.

>NR`NSC`LMR

. (18)

Monahan [16] used curve "tting through the values given by Eq. (18) and showed thatW could be approximated by the following equations:

W"1 fora

2c)0.05,

W"

1

1#0.7(a/2c!0.05)0.4for 0.05(

a

2c(0.26. (19)

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 577

Fig. 9. Schematic illustration of Albrecht's method for determining >g

[17].

The above crack shape correction factor was derived from experimental data obtainedfrom tests conducted on a combination of X and multi-braced tubular joints. Thismeans that it may not be directly applicable to other joint geometries. The reason forthis is that the crack shape evolution curve depends greatly on both the joint geometryand the mode of loading. This is also demonstrated by the fact that the crack shapecorrection factor shown above is unity for the range of crack aspect ratios obtained forthe >-joints tested for this study. It is therefore possible that, a wide range of crackshape correction factors can be obtained, depending on the geometry of the specimenstested. This was shown to be the case in a recent study [18] on ¹-joints under axialloading. Myers [18] used a similar approach adopted by Monahan [16] and obtaineda crack shape correction factor applicable to ¹-joints under axial loading given by

W"0.9 fora

2c)0.05,

W"A1

1#0.7(a/2c!0.04)0.4B!1 fora

2c'0.05. (20)

578 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Fig. 10. Myers' data used in deriving CSC factors for NR solution [17].

The data from which this correction factor was derived are shown in Fig. 10. The "rstpart of the curve can be considered to represent an upper boundary for a/2c)0.05.However, the crack shape correction factor used for a/2c'0.05, is outside the scatterband for the data used. This would make it di$cult to use this type of semi-empiricallyderived solution for other joint geometries.

The brace and chord thickness of the>-joints used in this study are the same as thatused by Myers [18]. However, both the geometry and mode of the loading aredi!erent. As a result, the crack shape evolution curves obtained from the two studiesare di!erent. This di!erence is shown in Fig. 11 where the best "t curves obtained forthe two geometries are compared. The predicted > factors obtained by using thecorrection factors given in Eqs. (19) and (20) are compared in Fig. 12 with theexperimental results from >-joints. The sensitivity of > factors to crack aspect ratio isdue to the semi-empirical nature of the model. As a result, there is some degree ofuncertainty in the applicability of the model to the prediction of > factors for casesother than those from which the crack shape correction factors were derived. In orderto avoid this uncertainty, a crack shape correction function which also accounts forthe e!ect of joint dimensional parameters and the mode of loading employed needs tobe introduced.

There is a lack of solutions available for predicting crack aspect ratio evolution.This has been identi"ed [19] to represent the greatest hindrance to good predictionsof remaining life of cracked components. Di!erent researchers have used di!erentapproaches to incorporate the e!ect of crack aspect ratio into stress intensity factormodels used for fatigue crack growth prediction. One approach highlighted byBrennan [20] is based on the use of a root mean square (RMS) or average stressintensity factor for the transverse and longitudinal directions of crack growth. Cruseand Besuner [21] proposed this approach and it has been used by Dedhia and Harris[22] in the analyses of fatigue cracks in pipes. The use of this method for the

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 579

Fig. 11. Comparison of crack shape evolution curves for > and ¹-joints.

prediction of > factors in welded joints is outside the scope of work presented in thispaper and will not be discussed in any further detail. However, a new semi-empiricalmodel which accounts for the e!ect of crack aspect ratio is required.

6. Further discussion and conclusions

Fig. 13 shows the > factor curves predicted by various existing > factor models.Those shown in Fig. 13 include the TPM, AVS, modi"ed AVS and the adapted #atplate solution based on the Newman and Raju equations. As shown in Fig. 13, theexperimental > factors obtained for this study are all below those predicted by theabove equations.

The importance of crack shape evolution in the accurate prediction of crack growthin cracked components was highlighted in the previous section. The TPM, AVS andmodi"ed AVS models do not account for this e!ect. It is therefore possible that theiraccuracy in predicting > factors in tubular welded joints will depend on whether thecrack shape evolution in the welded joint of interest is representative of thoseoriginally used to derive the respective equations. The main reason for this is that bothjoint geometry and mode of loading in#uence crack shape evolution.

The prediction of crack aspect ratio has been identi"ed to represent a major sourceof uncertainty in the fatigue crack growth prediction. This is mainly as a result of thelarge scatter on crack aspect ratio obtainable from experimental data at the currentstate of the art in fatigue testing.

The derivation of accurate stress intensity factor solutions is imperative if fatiguecrack growth prediction models are to be reliable. However, there are other important

580 L.S. Etube et al. / Marine Structures 12 (1999) 565}583

Fig. 12. Comparison of Myers' and Monahan's solution with >-joint data.

factors which are often ignored, partly due to the lack of su$cient data and partly dueto the inherent di$culty which is often encountered in reducing the level of uncertain-ty to a reasonable level. One of these factors is the e!ect of variable amplitude loadingand the associated sequence e!ects.

Fatigue crack growth prediction under variable amplitude loading conditions isstill in its infancy. The reason for this is that there is a lack of suitable models, whichaccount for all the relevant e!ects unique to variable amplitude loading conditions.

The results obtained from the variable amplitude fatigue tests conducted for thisstudy have been analysed using existing fracture mechanics models. It is apparent thatone of the main di$culties in trying to quantify fatigue crack growth under variableamplitude loading conditions is the large number of variables involved which almostalways operate together to in#uence fatigue crack growth at any one time. Some ofthese variables include material properties determined by the alloying elementspresent, the nature of the corrosive environment determined mainly by its chemicalcomposition and other additional factors. Crack shape evolution has also beenidenti"ed as an important parameter, which in#uences the stress intensity factor. Atpresent, empirical and semi-empirical fatigue crack growth models for the analysis offatigue crack growth under variable amplitude conditions are limited. A methodo-logy, which accounts for the statistical distribution of crack aspect ratio data undervariable amplitude loading conditions, has been developed for the prediction of> factors in tubular welded joints. This is currently being prepared for publication.Most of the existing models do not allow for interaction e!ects to be accounted for.Those that attempt to model interaction e!ects are based on a cycle by cycle analysis

L.S. Etube et al. / Marine Structures 12 (1999) 565}583 581

Fig. 13. Comparison of > factors from di!erent models with >-joint data.

with emphasis on single overloads or underloads. This often makes their use onengineering structures impractical.

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[1] Gri$th AA. Philos Trans R Soc London 1920;A221:163 (republished with additional commentary inTrans Am Soc Metals 1968;61:871).

[2] Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate. Trans ASMEJ Appl Mech 1957;E24:361.

[3] BS PD 6493. Guidance on methods for assessing the acceptability of #aws in welded structures.London: British Standards Institution, 1991.

[4] Technical Software Consultants Ltd. ACFM crack microguage* model U10, user manual. MiltonKeynes, April 1991.

[5] Etube LS, Myers P, Brennan FP, Dover WD, Stacey A. Constant and variable amplitude corrosionfatigue performance of a high strength jack-up steel. International O!shore and Polar EngineeringConference, Montreal, Canada, May 1998.

[6] Etube LS. Variable amplitude corrosion fatigue and fracture mechanics of high strength weldablehigh strength steels. PhD Thesis, Department of Mechanical Engineering, University College Lon-don, September 1998.

[7] Billingham J, Spurier J, Healy J, Kilgallon P. Corrosion fatigue fracture mechanics of jack-up steels.Project Report, University of Cran"eld, 1998.

[8] Balladon P, Coudert E. TPG 500 structural assessment. Rapport Technique 95072 C, September1995.

[9] Holdbrook SJ, Dover WD. The stress intensity factors for a deep surface crack in a "nite plate. EngFracture Mech 1979;12:347}64.

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[10] Dover WD, Dharmavasan S. Fatigue fracture mechanics analysis of ¹ and> joints. Paper OTC 4404of O!shore Technology Conference, TX, 1982.

[11] Austin JA. The role of corrosion fatigue crack growth mechanisms in predicting fatigue life of o!shoretubular joints. PhD Thesis, Department of Mechanical Engineering, University College London,October 1994.

[12] Kam JCP, Topp DA, Dover WD. Fracture mechanics modelling and structural integrity of weldedtubular joints in fatigue. Proceedings of the Sixth International O!shore Mechanics and ArcticEngineering Symposium. ASME 1987;3:395}402.

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