Journal of Constructional Steel Research 62 (2006) 1199–1209www.elsevier.com/locate/jcsr

Fatigue reliability of welded steel structures

M.K. Chryssanthopoulos∗, T.D. Righiniotis

School of Engineering, University of Surrey, Guildford, Surrey, UK

Abstract

In general, two different approaches to the formulation of the fatigue limit state are considered, the first based on S–N lines in combinationwith Miner’s damage accumulation rule, and the second based on fracture mechanics crack growth models and failure criteria. Often, the twoapproaches are used sequentially, with S–N being used at the design or preliminary assessment stage and fracture mechanics for more refinedremaining life or inspection and repair estimates. However, it is essential to link the results, and the decisions made, at the design and assessmentstages, and it is therefore important to develop compatible methodologies for using these two approaches in tandem. In doing so, it is essentialto understand and quantify different uncertainty sources and how they might affect the robustness of the results obtained, and the subsequentdecisions made about the structure. The objective of this paper is to highlight parts of recent research at the University of Surrey on the fatigueassessment of steel bridges. The work includes the development of a probabilistic fracture mechanics methodology for the prediction of fatiguereliability, using up-to-date crack growth and fracture assessment criteria and incorporating information on inspection and subsequent managementactions.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Crack growth; Fatigue; Fracture mechanics; Inspection; Steel bridges; Structural reliability

1. Introduction

Over the last twenty years or so, probabilistic methods forthe assessment of fatigue reliability have attracted significantattention. In the civil engineering field, much of the research,from the mid-80s onwards, was directed towards applicationsin offshore structures, in particular tubular joints subjectto stochastic loading. This effort was aided considerablyby progress in experimental techniques associated with themeasurement of crack growth data under laboratory conditions,and the development of technology aimed at measuring cracksin actual structures.

At the same time, advances in probabilistic methods,especially the concerted effort in developing structuralreliability methods for damage accumulation problems undertime-varying loads, made it possible to cast fatigue assessmentproblems in a reliability format. As a result, by the late 80s, notonly had a large number of publications appeared addressing

∗ Corresponding address: Department of Civil Engineering, School ofEngineering, University of Surrey, GU2 7XH, Guildford, Surrey, UK. Tel.: +441483 686 632; fax: +44 1483 450 984.

E-mail address: mkchry@surrey.ac.uk (M.K. Chryssanthopoulos).

0143-974X/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2006.06.007

particular resistance and load modelling issues but also thefirst papers dealing with a complete methodology for fatiguereliability evaluation were being produced, including updatingfollowing inspection and repair [1–5]. In the following years,this approach was adopted by offshore operators for estimatingremaining fatigue life, and for determining inspection plans, ofageing structures [6–8].

In the past decade, considerable interest has arisen inadapting and implementing these techniques for applicationsin metallic bridges, thus focusing on fatigue details found ingirders and plated structures subjected to traffic loading [9–13].The characteristics of bridge live loading being substantiallydifferent from wave loading on offshore structures, led torevised formulations for the reliability problem. In parallel withthese developments, improved methods for fatigue and fractureassessment were actively being pursued for other structures,e.g. nuclear plants [14] and ships [15].

An additional factor, contributing to an increased interestin fatigue design and assessment, has been the flurry ofactivity associated with the development of a new generationof structural codes, both at national and international level;for example, a new European standard for fatigue design [16]has been prepared, whilst other documents have been updated

1200 M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209

and extended in order to incorporate recent developments infracture mechanics assessment methods [17]. The underlyingphilosophy in these regulatory bodies, as well as amongstowners and operators, is increasingly focusing on the needto introduce probabilistic concepts for fatigue life prediction.Thus, the development of probabilistic fatigue models, and theirtesting and validation through examples and case studies, couldconsiderably enhance available guidance documents. In thisrespect, the effort of the Joint Committee of Structural Safetyin developing the Probabilistic Model Code [18] is of particularnote.

In general, two different approaches to the formulation ofthe fatigue limit state may be considered, the first based onS–N curves in combination with Miner’s damage accumulationrule, and the second based on fracture mechanics crack growthmodels and associated failure criteria. Often, as can be seenin some of the references cited above, the two approaches areused sequentially, with S–N being used at the ‘design’ stageand fracture mechanics at the ‘assessment’ stage, in other wordsfor new and existing structures respectively. In the former case,the purpose of a fatigue analysis is to determine a design life,associated with a target reliability, whereas in the latter theobjective is to determine inspection intervals or time to repair,once more linked to target reliabilities.

It is often desirable to link the results, and hence thedecisions, at the ‘design’ and ‘assessment’ stages. Thus, it isimportant to develop compatible methodologies for using thesetwo approaches in tandem. Although the majority of engineersworking in the above mentioned industries are more familiarwith the S–N , rather than the fracture mechanics, approach forfatigue analysis, the latter is increasingly gaining ground asfitness-for-purpose criteria are becoming popular with ownersand regulators.

The fatigue process may be regarded as comprising threestages; crack initiation, crack propagation and final failure. Afatigue analysis based on S–N curves, the latter being derivedfrom standard fatigue tests, typically lumps all three stagesinto one, though the definition of failure within this context isnot always clear. On the other hand, a fatigue analysis basedon fracture mechanics is concerned primarily with the secondstage, though it can be extended to include final failure throughthe introduction of appropriate limit state criteria related tofracture resistance (which could be expressed in terms of acritical crack size). The extent to which a fracture mechanicsapproach can provide comparable information on fatigue lifewith that derived from S–N curves will depend, in part, onthe number of load cycles expended during the initiation stage.A common assumption in fatigue analysis of welded jointsis that the initiation stage is negligibly small compared tothe propagation stage. This is because fatigue cracks developfrom small defects introduced right from the outset in areas ofstress concentration. The weld toe is considered as a criticalarea in which such cracks are often to be found. Clearly, thisassumption needs to be evaluated for the specific prevailingconditions.

The objective of this paper is to highlight the key factorsthat need to be considered in fatigue reliability analysis, and

to present a case study, pertaining to welded bridge details, inwhich the proposed procedures are implemented and utilisedin support of decision making. As will become evident, muchwork has been, and is still being, carried out in this areastemming from different industrial sectors and applications.Thus, it is considered essential to sift through and processinformation from experiments and field observations, as wellas to integrate and consolidate the procedures to be followed infatigue reliability analysis.

2. S–N approach

An S–N curve is a relation between the stress range underconstant amplitude loading and the number of stress cycles tofailure. The standard S–N curve can be expressed in the formof:

N Sm= A (1)

where N is the number of stress cycles to failure at aconstant amplitude stress range S, A and m are the materialparameters. Sometimes a model with two segments is used,having parameters A1 and m1, A2 and m2. The stress rangelevel at which the two curves intersect is defined by A1Sm1

0 =

A2Sm20 .

Many steels subjected to pure constant amplitude loadingin inert environments exhibit a fatigue limit, i.e. a stress levelbelow which fatigue failure appears to never occur. However, itis generally accepted that even infrequent overloads (i.e. stresscycles that exceed the fatigue limit value) may lead to fatiguedamage even though the vast majority of stress cycles are belowthe fatigue limit. In essence, a fatigue limit no longer exists andevery stress cycle is treated as damaging, as determined fromthe S–N curve(s) [19].

There are many sources of uncertainty in the fatigue processand its analysis. Wirsching [20] has produced an itemisedlist, which includes the fatigue process itself, the extrapolationfrom laboratory test specimens and procedures to details inreal structures, the loading conditions, the local environment(temperature, presence of water/humidity etc.), the dynamiceffects, as well as the stress analysis methods used to obtainestimates of local stress ranges from globally applied forces anddisplacements.

The uncertainty associated with S–N curves is typically as-sessed from laboratory tests on nominally identical specimensunder constant amplitude loading. In a typical fatigue test, thestress level (i.e. the independent variable) is specified and thecycles to failure (i.e. the dependent variable) are recorded. Un-der these conditions, it has been observed that the distributionof ln N for a fixed value of S exhibits a variation which is in-dependent of S (at least until test results at very low stress lev-els are considered), whereas the mean value of ln N varies lin-early with ln S. In general, the lognormal distribution providesa better fit to N than other candidate distributions such as theWeibull distribution, though there is no apparent physical ormathematical reason for this [20]. Assuming then that in Eq.(1) the parameter m is deterministic and that the uncertainty islumped into the second parameter A, it is easily shown that if

M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209 1201

the variable N follows a lognormal distribution with a coeffi-cient of variation equal to CN then A is also lognormal with acoefficient of variation CA = CN . More general treatment ofthe uncertainty in ln S– ln N lines has been undertaken [21] al-lowing for joint probabilistic modelling of the material param-eters m and A, but in many practical applications the simplerapproach outlined above has been adopted.

It should be emphasised that statistical analysis of standardfatigue tests is a challenging task, as it requires carefulconsideration of many factors, such as the merging of data fromdifferent sources, influence of different failure criteria adoptedby different laboratories (e.g. ‘visible’ or through-thicknesscrack), influence of specimen preparation as well as the contrastbetween laboratory and field conditions, etc. In recent years,effort has been directed towards the development of the newEurocode for fatigue [16]. The above, and many other, factorswere assessed in producing S–N lines for different fatiguedetails [22].

The stress ranges in a fatigue analysis using the S–Napproach should be obtained bearing in mind the locationof the detail within the structure as well as the basis of theS–N curves used. For example, chord to brace connections inoffshore tubular joints are assessed using the so-called T-curve,for which the ‘hot-spot’ stress range has to be determined.The ‘hot-spot’ stress is related to the nominal stress in thebrace through empirical stress concentration factors, usuallydeveloped from limited parametric studies using experimentsor finite element analyses of typical geometries. On the otherhand, for many other fatigue details the corresponding S–Nlines take into account the local stress concentrations created bythe joints themselves or by the weld profile [23]. The relevantstress ranges to be used could then be determined from thenominal stresses present in the vicinity of the joint. However, ifthe detail is situated in a region of stress concentration resultingfrom the overall shape of the structure, this must be accountedfor in estimating the nominal stress. The different approaches,adopted in fatigue assessment of welded joints, have beenreviewed by Fricke [24].

As far as probabilistic modelling is concerned, the importantmessage from the above is that modelling uncertainties areinvolved both in the method for calculating nominal stressesin the vicinity of a fatigue sensitive detail and in theestimation of stress concentration factors due to gross andlocal discontinuities, or due to deviations from an ideal shape.The mean value and coefficient of variation adopted for suchmodel uncertainty variables would vary depending on thetype of global analysis procedures (e.g. finite element vs.simplified theory), on the characteristics of the local geometryand of the detail itself, as well as on the factors included inthe derivation of the relevant S–N curve. Typically, effectsassociated with residual stresses, weld profiling, environment(e.g. air vs. water), through thickness stress variation (fora reference plate thickness) and fabrication tolerances areincluded in the derivation of any particular S–N curve, whereaseffects associated with post-weld treatment (e.g. grinding),difference in plate thickness (from a reference value), globaldiscontinuities etc. are accounted for through modification

factors. However, as can be evidenced from the literature, it isdifficult to establish rigid rules regarding these issues; it is thusessential in any analysis, whether deterministic or probabilistic,to be aware of the idealisations and simplifications adopted,and the implications for uncertainty modelling. Straub [25] hasreviewed studies pertaining to marine structures under waveloads and has tabulated uncertainty levels for a factor, whichacts as a multiplier on calculated stress ranges.

In general, structures experience variable loading duringtheir life. Thus, fatigue loading is characterised by the numberof stress cycles and the magnitude of stress range for eachcycle. Moreover, fatigue damage is quantified in terms of thePalmgren-Miner rule. According to this rule, each load cyclecauses fatigue damage proportional to the inverse of the fatiguelife at that stress range amplitude. Therefore, letting Si bethe stress range of the i th cycle, damage may be defined inaccordance with the Palmgren-Miner rule as

Dn =

n∑i=1

1N (Si )

(2)

where N (Si ) is the number of cycles to failure at stress level Si .It can readily be shown that by introducing a simple linear S–Ncurve, see Eq. (1), Eq. (2) can be written as

Dn =1A

n∑i=1

Smi =

1AψL(t) (3)

where ψL(t) may be referred to as the ‘fatigue loading’. In adeterministic format the Palmgren-Miner rule simply states thatdamage increments, expressed as life fractions, are additive andindependent of sequencing, and that fatigue failure is expectedwhen such fractions sum to unity (i.e. Dn = 1). Experienceshows that this sum at failure is subject to considerableuncertainty; for example, analyses undertaken in relation tooffshore structures have revealed median values between 0.8and 1.2, and even wider variations are reported for bridgedetails. Wirsching [20] has suggested that, in the absence ofspecific data for the problem in hand, a lognormal distributionwith a median value of unity and a coefficient of variation of0.30 may be adopted for the damage at failure.

On the other hand, ψL(t) may be described througha probability function, whose nature depends on thecharacteristics of the underlying loading process. For somemodels the expectation of the mth moment of the arbitrarypoint-in-time distribution of the stress range is sufficient. Ingeneral, the derivation of the appropriate probabilistic modelfor ψL(t) is application specific.

3. Fracture mechanics approach

3.1. Crack growth relationships

Crack growth is a process not fully understood at the atomiclevel but for which macroscopic observations have been made,and models have been developed. Crack growth can occur undercyclic loading, however, in a hostile chemical environment, itcan even take place in the presence of statically applied loads.

1202 M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209

The former is known as fatigue while the latter is described asenvironmentally assisted crack growth. Under cyclic loading ina hostile environment, both types may occur in combination.

Engineering analysis of crack growth can be undertakenusing relationships between the stress intensity factor (SIF),which characterises the stress conditions at the crack tip, andthe associated crack growth rates. The underpinning (linearelastic fracture mechanics—LEFM) theory, which is used todetermine SIFs for cracked bodies is based on stress analysis ofan isotropic and homogeneous solid containing an ideally sharpcrack. For cracks of the order of a few mm or greater, micro-structural material (e.g. grain boundaries) and geometrical(e.g. crack kinks) features are in effect assumed to occur onsuch a small scale that only average behaviour needs to beconsidered [23]. If, however, the crack is sufficiently small (ofthe order of a few microns), it can interact with the micro-structure and special treatment might be needed. Here, thedistinction between small and short cracks is relevant. Forthe former, all the crack dimensions are similar or smallerthan the dimension of the greatest micro-structural feature,whereas for the latter, one crack dimension is large comparedto the microstructure. There is considerable research interestin the behaviour and modelling of small and short cracks [23,26], which, from an engineering point of view is focused ondetermining conditions under which such cracks may growand reach the level at which LEFM becomes applicable. Asmentioned in the introduction, this can also be viewed in termsof the three stages of fatigue life of engineering components andstructures. The ensuing discussion focuses on the modelling ofcrack propagation stage, assuming that LEFM is applicable;initial defect distributions are considered in the case studiespresented.

The most widely used fatigue crack growth model basedon LEFM, commonly known as Paris’s Law, was proposed byParis and Erdogan [27] and can be written as:

da

dn= C(1K )m (4)

where a is the crack depth, da/dn is the instantaneous (median)crack propagation rate and 1K is the alternating SIF at thecrack tip, generated from an applied nominal stress range; Cand m are material constants which can be determined fromexperiments on simple specimens in which a small initial crackis introduced and then propagated through the application ofconstant amplitude cyclic loading. Clearly, if the above modelis correct, then a plot of ln(da/dn) against ln(1K )would resultin a straight line. As shown in many experimental studies ona wide range of metals, this is the case for the middle rangeof growth rates, e.g. typically for a range between 10−5 and10−3 mm/cycle. At higher rates, the model underestimatesactual rates, whereas for lower values it overestimates them.Thus, fatigue crack growth over a wide range of crack growthrates spanning several orders of magnitude invariably results inthe well-known ‘sigmoidal’ curve, the middle portion of whichmay be described accurately by Eq. (4) (see Fig. 1).

Before more detailed models that attempt to capture theentire crack growth behaviour are presented, it is helpful to

Fig. 1. Schematic representation of a typical crack growth curve and itsapproximations.

summarise the treatment of uncertainty in relation to Paristype models. This is believed to arise from inhomogeneitiesin the metal, the type of crack measurement method, themeasurement procedures (e.g. whether da/dn is determinedby averaging over predetermined intervals in n or a), and thestatistical procedures adopted for parameter estimation. Thevariability can also be viewed as having a component presentwithin a particular test and a component manifested whenresults from several nominally identical tests are combined.The approaches that have been developed to deal with randomfatigue crack propagation include: treatment of C and m asrandom variables (individually or as a correlated pair) [2,21];formulation as a first-passage problem focusing particularly onthe effect of averaging that is necessarily present in measuringcrack increments [28]; Markov process description [29]; andthe introduction of a lognormal random process as a multiplierof the right-hand side of the Paris law [30,31].

Unfortunately, comparisons between the different ap-proaches on realistic structural details are virtually absent fromthe literature. However, most of the engineering applicationsin offshore and bridge structures have been undertaken with thesimplest of the approaches mentioned above, i.e. treatment of Cand m as random variables. One notable exception is the fatigueanalysis of welded miter gates in river locks, which has beencarried out using the lognormal random process approach [31].

In the JCSS Probabilistic Model Code [18], the suggestedapproach is to treat m as a deterministic parameter and tomodel C as a lognormal random variable. It is emphasisedthat this should be viewed as an operational model for whicha probabilistic fatigue analysis under different load processescan be undertaken using standard reliability techniques, e.g.FORM/SORM or Monte Carlo simulation.

As mentioned previously, the Paris model is only strictlyvalid in the central region of crack growth (Region II,see Fig. 1), although in fatigue assessments of engineering

M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209 1203

components it is sometimes extended to the near-thresholdregion (Region I in Fig. 1). In many studies the effect of a cut-off or threshold value,1Kthr, below which the crack is assumedto be non-propagating is also included. Thus, Eq. (4) is replacedby

da

dn= C(1K )m for 1K > 1Kthr. (5)

This equation forms the basis for the JCSS commonlyadopted probabilistic crack growth model, with m assumeddeterministic and C and1Kthr treated as random variables [18].Eq. (5) is conservative and often quite acceptable, particularlyin benign environments, such as inert gases or vacuum.However, more refined descriptions may be necessary when alarge part of the fatigue damage is due to a very large numberof low-amplitude stress cycles, or the environmental responseof the material deviates significantly from linear behaviour.

As highlighted by King et al. [32], both these factorsbecome important in the fatigue assessment of offshore details.Furthermore, the presence of a very large number of low-amplitude stress cycles is a key feature of the traffic loadon highway bridges [12,33,34]. A re-appraisal of availabledata for fatigue crack growth rates of steels in air andseawater environments [32,35] has led to the development andcharacterisation of a bi-linear relationship to represent crackgrowth in the near threshold region (Region I) as well as in thetraditional intermediate (Paris) region (Region II). Eq. (5) maythus be replaced by

da

dN= C1(1K )m1 for 1Kthr < 1K ≤ 1Ktr

da

dN= C2(1K )m2 for 1Ktr < 1K .

(6)

The above bi-linear crack growth model is also included in theJCSS document [18], with the information on variability of C1

and C2 taken from the work by King et al. [32]. It is worthnoting that the coefficient of variation of C1 is significantlyhigher than the corresponding value for C2 (or C in the case ofthe simpler linear model). This reflects the higher uncertaintyinvolved in both the actual crack growth process, as well as itsmeasurement, when the rates are small. Finally, it should beemphasised that the mean values adopted for the C parametersshould properly reflect the influence of the environmentalconditions (e.g. air vs. seawater), and the stress ratio R (definedas the ratio between minimum and maximum applied stresses),as there are important differences in mean values depending onthe prevailing conditions.

For either Eq. (5) or (6), the variability associated with thethreshold value for the stress intensity factor range, 1Kthr,should be included. As reported by King et al. [32] thecoefficient of variation for 1Kthr is quite high; furthermore, itsmean value appears to be dependent on the R ratio. A lognormaldistribution with a CoV of 0.4 [36] has been proposed. Note thatin the case of freely corroding steels 1Kthr tends to zero.

3.2. Stress intensity factor

The discussion has so far focused on crack growth rates.However, the power law relationship between crack growth rate(da/dn = damage) and crack driving force (1K = loading)implied by Eq. (4) or any other alternative crack growth model(see for example Eq. (6)) suggests that a closer examination ofK or 1K is warranted. Within the context of LEFM, the stressintensity factor K for a given applied ‘far field’ stress Sa isgiven as:

K = SaY Mk√πa (7)

where Y is the “geometry factor”. The Y -factor depends on thegeneric geometry of the cracked joint (i.e. excluding any localstress concentration effects), the nature of load distribution(e.g. bending or membrane) and the crack size normalised withrespect to other relevant dimensions (e.g. thickness). The factorMk accounts for the effect of a crack being in the immediatevicinity of a stress concentration, such as a weld toe, a notchor a hole. It therefore depends on the local geometry and is afunction of crack size and loading. Solutions for Y -factors areavailable for a number of standard load cases (tension, bending,torsion) and generic geometries in handbooks (for exampleRefs. [37–39]). Solutions for Mk may be found in the literature[40,41] or can be developed through FE analysis.

The stress Sa should be determined from the applied loadingin the vicinity of the crack (but not influenced by it). Thethrough-thickness distribution of this stress is often assumedto consist of ‘membrane’ and ‘bending’ components and ischaracterised in terms of the ratio of the bending stress to thetotal stress, called ‘degree-of-bending’ [42–44]. Eq. (7), may bere-cast as follows

K = (SamYm Mkm + SabYb Mkb)√πa (8)

where the subscripts ‘m’ and ‘b’ refer to membrane and bendingcomponents respectively.

In addition to the stresses due to the applied loading, residualstresses Sres may also be present, e.g. in the case of weldedjoints. The distribution of residual stresses is usually obtainedfrom experimental measurements on typical joint and welddetails; the uncertainty associated with such distributions islarge both in terms of physical and statistical components.Note that the presence of residual stresses should also be takeninto account in determining the stress ratio R, which in turndetermines the appropriate values of C and m parameters inParis type models.

3.3. Crack growth modelling

For welded joints it is often observed that micro-cracksinitiate from surface-breaking defects at the toe of the weld.These micro-cracks tend to coalesce to form a single, dominantfatigue crack of roughly semi-elliptical shape. Hence, semi-elliptical cracks in plated structures are of interest in manypractical applications (see Fig. 2). In this case, two crackdimensions, the depth a and the half-length at the surface c,

1204 M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209

Fig. 2. Butt welded plate containing a semi-elliptical toe crack.

become relevant both of which are functions of the fatigueloading process.

For each principal direction of crack growth, a Paris typeexpression may be formulated thus

da

dn= C(1K A)

m for 1K A > 1Kthr

dc

dn= C(1K B)

m for 1K B > 1Kthr

(9)

where the first expression relates to point A and growth in thedepth direction, whereas the second expression relates to pointB and growth in the length direction (Fig. 2). For each of thesetwo points, the stress intensity factor range, 1K A and 1K B , isgiven by

1K A = SYA Mk A√πa/Q

1K B = SYB Mk B√πc/Q

(10)

where S is the applied stress range, YA, YB are geometryfactors, Mk A,Mk B are stress magnification factors for pointsA and B respectively, and Q is the elliptic shape factor whichmay be approximated by [44]

Q = 1 + 1.464(a

c

)1.65. (11)

Once Eqs. (10) and (11) are substituted into (9), a pair ofcoupled differential equations is obtained. With the exception ofthe material parameters (C and m) and the applied stress rangeS, all other terms are a function of the crack size (a, c), whichclearly changes during the fatigue loading process.

Under variable loading and considering the uncertaintiesassociated with it, some further modifications may beintroduced to Eq. (9). The first pertains to the effect ofthe threshold value, 1Kthr, which essentially acts as a filterdistinguishing between damaging and non-damaging cyclesin the loading process. A second factor is related to crackclosure [45], which deals with the possibility that a crack maynot propagate (i.e. remain ‘closed’) due to local compressivestresses at the crack tip, even though the applied stresses aretensile. As a result, Eq. (9) can be re-written as

da

dn= C(1KeffA)

m G(a)

dc

dn= C(1KeffB)

m G(c)(12)

where G(a) and G(c) are ‘threshold correction factors’, givenby [46]

G(a) =E[Sm

]∞

Sthr(a)

E[Sm]∞

0(13)

and

G(c) =E[Sm

]∞

Sthr(c)

E[Sm]∞

0(14)

where Sthr is the stress range for either crack dimension (a or c)corresponding to 1Kthr, which can be obtained from Eq. (7)or (8). The values of G(a) and G(c) are in the range fromzero to unity, with the latter corresponding to the case whereall cycles contribute to fatigue damage. 1KeffA and 1KeffBare ‘effective stress intensity factor ranges’, which account forresidual compressive stresses and stress ratio effects [25,45].They can be defined through ‘effective stress intensity factorratios’ U (a) and U (c), according to [4]

1KeffA = U (a)1K A (15)

and

1KeffB = U (c)1K B . (16)

However, as pointed out by Straub [25], although moresophisticated crack growth relationships may lead to a bettercomparison between predicted and measured values, theyalso lead to difficulties with regard to calibration and modeluncertainty characterisation, due to the additional parametersthat are introduced. There is considerable debate in the fatigueand fracture community regarding the significance of crackclosure effects for different applications.

Integration of the first differential equation (12) froman initial crack size a0 to a crack size a(t) after time tcorresponding to a number of stress cycles n results in

1C

∫ a(t)

a0

dx

GaU ma Y m

a Mmka(πx/Qa)m/2

=

n∑i=1

Smi = ψL(t) (17)

For brevity, the subscript ‘a’ is used here to denotethe functional dependence of the variable on crack size.Comparison of Eq. (17) with Eq. (3) reveals the similarity ofthe two approaches adopted for fatigue damage accumulationproblems, i.e. S–N and fracture mechanics.

In general, the above integral is evaluated through anincremental numerical procedure, which involves sub-divisionof the interval between a(t) and a0. At each step, followingthe evaluation of Eq. (17), the second crack dimension c iscomputed as well, using the relationship that can be obtainedby combining the two differential equations, i.e.

dc

da=

G(c)

G(a)

(1KeffB

1KeffA

)m

. (18)

When the life of a fatigue sensitive detail is required, theupper bound of the integral in Eq. (17) may be taken asthe critical value ac determined through the application of afracture criterion or some other requirement, e.g. a maximumpermissible crack size.

M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209 1205

Fig. 3. Buried elliptical crack located in a plate of thickness t (The direction ofthe applied loading is perpendicular to the indicated plane).

3.4. Initial defect size

As can be seen from Eq. (17), the fracture mechanicsapproach considers the growth of a crack from an initial size toa critical size. The initial crack size distribution is one of the keyinputs in determining fatigue lives through the LEFM methodspresented in the preceding section. It should be emphasisedthat the application of LEFM implies that the stress intensityfactor concept is meaningful in describing crack growth [47].As reported by Straub [25], recent studies seem to indicatethat 0.1 mm is a reasonable lower bound for the applicationof LEFM in common metallic structures.

In engineering structures, many possible mechanisms leadto the presence of cracks, arising from material processing ormanufacturing factors. Cracks may be classified as buried orsurface, and for analysis purposes such cracks are generallyidealised as elliptical, semi-elliptical or quarter-elliptical. Fig. 3shows schematically an elliptical crack that lies below thesurface, and is characterised through the dimensions, a, c ands. Surface defects (i.e. s = 0) are usually more dangerous thanembedded defects, particularly in welded structures, because(a) they are associated with higher Y factors, (b) they are oftenlocated at stress concentrations, which further increase Y and(c) they are oriented normal to the principal stress. Moreover,their presence is caused by limitations in workmanship andtheir initial size tends to be larger than that of embeddedcracks for the normal range of component thickness. Intentionalor unintentional lack of penetration in butt welds form anexception to this last observation, however, experience supportsthe view that many fatigue cracks in welded structures tend togrow from an initial surface defect [2,48].

Meaningful statistics on crack frequency, size and locationare difficult to obtain, due to measuring and sampling issues.That said, in the past twenty-five years there have beenseveral studies aimed at improving our understanding andproviding quantitative information on relevant crack parametersfor different detail geometries used in a range of structuralapplications (ships, offshore platforms, nuclear etc.). Cracksize distributions are typically modelled by exponential orlognormal distributions, whereas the occurrence rate is assumedto be Poisson distributed. Information on crack aspect ratios(a/c) is very limited, though there have been some attemptsto determine the sensitivity of fatigue reliability towards thisparameter under simplified conditions (e.g. c and a are eitherindependent or fully correlated). A review of published data oninitial defect distributions can be found in [2,25,49]. Straub [25]also addresses the issue of small crack growth, based on the

work of Lassen [50], in order to distinguish between crackinitiation and crack propagation under LEFM conditions. Asmentioned earlier, in fatigue analysis of welded joints it iscommonly assumed that the initiation stage is negligibly smallcompared to the propagation stage, and hence the methodspresented in the previous section can be adopted for anevaluation of the entire fatigue life.

4. Limit state functions

4.1. S–N approach

For the S–N approach, the limit state function may be givenby

g(X , t) = 1− D = 1−ψL(t)

A

(Tp

TB

)ξBm

scf Bmglob (19)

where X is the vector of random variables, t is any point intime, ξ is the thickness correction factor [51], 1 is Miner’sdamage sum at failure, Tp is the actual thickness of the plateand TB is the reference plate thickness. In Eq. (19), Bscf andBglob are model uncertainties associated with global and localstress analysis. In the work of Shetty and Baker [3–5] pertainingto welded tubular joints for offshore applications, a lognormaldistribution is proposed with a mean of unity and a coefficientof variation of 0.20 and 0.10 for Bscf and Bglob respectively.Straub [25] reports higher CoV values from published literaturepertaining to ships, offshore structures and FPSOs, and meanvalues in the range from 0.7 to 1.0.

4.2. Fracture mechanics approach

Using the fracture mechanics approach, the limit statefunction can be formulated as

g(X , t) = ac − a(t) (20)

where ac is a limiting crack depth (for example plate thickness)and a(t) is the crack depth after a service exposure of timet . Starting from an initial crack depth of a0, the crack deptha(t) after time t can be calculated using the crack growthpropagation models highlighted above. In this case, a(t) is afunction of random variables such as initial defect size, fatiguematerial properties, uncertainties in service loading, modeluncertainties, etc. (see Eq. (17)).

Alternatively, in terms of a fatigue resistance function anda fatigue loading function, the limit state function can also beexpressed as

g(X , t) =1C

∫ ac

a0

dx

GaU ma Bm

sifYma Mm

ka(πx/Qa)m/2

− Bmglob Bm

scfψL(t) (21)

where Bsif is a model uncertainty associated with the estimationof the stress intensity factor; in general, this will be influencedby the methods used for determining both Y and Mk factors.At present, it is suggested to model Bsif as a lognormal variablewith a mean of unity and coefficient of variation in the rangefrom 0.07 to 0.20. Lower variability may be appropriate if the

1206 M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209

factors are computed using finite element models and/or weightfunction techniques.

4.3. Fatigue crack growth approach with fracture resistancemodel

The maximum crack size, ac, that can be sustained by thecomponent will, in general, depend on the material’s resistanceto fracture i.e. its fracture toughness Kmat. The limit statefunction with respect to fracture can be defined as

g(X , t) = f (Kr , Lr ) (22)

where Kr is the fracture ratio, Lr is a measure of the proximityto plastic collapse and f (.) is an appropriate interactioncriterion, as for example given in BS7910 [17]. In this approachfailure occurs when a stress cycle occurs that causes the residualload bearing capacity of the cross section to be exceeded. Theresidual load bearing capacity depends on the actual crack sizeand the interaction between plastic resistance and the brittlefracture. The quantities Kr and Lr are defined as follows [17]:

Kr =Ks + Kres

Kmat+ ρ; Lr =

Sref

Sy(23)

where Kmat is the material fracture toughness, Ks is the stressintensity factor for maximum applied stress, Kres is the stressintensity factor for residual stresses, Sref(=SBglob Bscf) is thenet section stress (function of the crack size), Sy is the yieldstress and ρ is a factor, which accounts for the interactionbetween primary (loading) and secondary (residual) stresses.

The crack size a may be determined from the fatiguecrack growth models described above. Different residual stressprofiles for various cracked geometries and restraint conditionsmay be assumed but their use implies that Kres would needto be evaluated using either finite element techniques orthe weight function method. A simplified (and conservative)method would be to approximate the residual stress field viaa linear stress field subtended from the surface and crack tipstress values. The corresponding Kres can then be obtainedby superposition of the tensile and bending solutions for thegeometry in question. Further details of this approximation aregiven by Tada et al. [37]. If the bending solution is not known,in which case the previously mentioned procedure cannot beapplied, the residual stress field may be assumed to be uniform.This approach will, in general, yield very conservative resultsfor deep cracks and less conservative results for shallow cracks.

Clearly, if this approach is followed, the randomness in thematerial fracture toughness needs to be considered. Severalmodels for fracture toughness have been developed. Forexample, two [11,52,53], three [54] and four [55]-parameterWeibull distributions have been used to describe fracturetoughness.

5. Case studies

To highlight some of the issues discussed previously, anumber of case studies are presented in this section. The fatiguereliability problem is here considered in terms of the FMapproach as this is applied to a butt-welded bridge detail (seeFig. 2).

Fig. 4. Comparison between the BS 5400 [58] mean and 97.7% probability ofsurvival lines for class E and their FM for a butt weld counterparts. Also shownin the figure are the results presented in [56] and [57].

5.1. S–N based FM model calibration

As was previously discussed, one of the greatest uncertain-ties related to the fatigue process arises from the initial cracksize distribution. This distribution will be a function of the man-ufacturing process and the type of welded detail in question.Since models describing the fatigue crack growth process (dis-tributions and their parameters for C,m, Kmat etc.) are fairlywell established (for example [17,32,52]), it is expedient touse the appropriate S–N curve, or alternatively published S–Ndata, to calibrate the FM model in terms of the initial defectsize. This approach has been used in the past in the offshoreindustry [6]. Note that for FM modelling as discussed in Sec-tion 3.3, whereby a crack possesses both depth and length, thiscalibration would have to be carried out in terms of both a0 andc0, or, alternatively, for a0 and (a/c)0.

The limit state function (Eq. (21)) may be readily modifiedto cater for the constant amplitude load case and the statistics ofthe fatigue life at different stress ranges obtained from MonteCarlo simulation. Stress intensity factors for the butt-weldeddetail are presented in [53]. Crack growth was modelled basedon the bi-linear model discussed in Section 3.1 (see Eq. (6))coupled with a fracture resistance model (see Section 4.3).Distribution types and their parameters used for C, Kmat etc.as well as the relevant detail dimensions may be found in [12,53]. Fig. 4 depicts the results of this type of analysis for a butt-welded plate. Also shown in this figure are the results reportedin [56] and the scatter band reported in [57].

The FM results (mean and mean − 2 std.dev.) presentedin Fig. 4 were generated by assuming a0 and (a/c)0 log-normally distributed with means 0.2, 0.01 and coefficients ofvariation (CoV) of 0.2 and 0.2, respectively. Although thesevalues are not unique and other distribution types with differentmean–CoV combinations for a0 and (a/c)0 could have equallybeen used to yield similar results, the proposed values comparewell with published results in the literature (see [53] for furtherdiscussion).

It has to be noted that in the absence of specific data oninitial defect sizes pertinent to the welded detail in question,

M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209 1207

Fig. 5. Basic reliability curve for a butt-welded detail.

S–N based calibration of the FM model discussed here formsa crucial step in a fatigue reliability analysis. Naturally, goodcorrelation between the experimentally based S–N data andthe FM results increases confidence when considering variableamplitude loading.

5.2. Fatigue reliability based on inspections

Calibration of the FM model allows the treatment of thevariable amplitude case to proceed. Here the butt-welded detailis assumed to be located at the mid-span of a 10 m bridgegirder. The loading characteristics were obtained by traversingthe BS 5400 Pt 10 Table 11 of vehicles [58] over the spanand obtaining the relevant bending moment history assumingan annual vehicle flow of 106. Bending moment ranges werederived using the rainflow method and converted into stressranges on the assumption that the detail’s S–N design fatiguelife is 120 years. The associated basic reliability curve, obtainedusing the limit state function of Eq. (20) is shown in Fig. 5. Ascan be seen in this figure, the detail’s failure probability P fat year 120 of continuous operation is approximately 1.5 ×

10−2. This compares well with the S–N anticipated failureprobability of 2.3 × 10−2, which is not surprising in view ofthe results presented in Fig. 4. A similar curve could have beenobtained using an S–N based probabilistic methodology (seeSection 4.1). However, the great utility of the FM approach isdemonstrated when inspections and/or other types of invasiveaction are considered. To illustrate this point it is here assumedthat following the design and construction of the bridge, theowner has decided to impose a more stringent requirement onthe detail’s reliability, namely P f = 10−3. As can be seenin Fig. 5, this requirement is violated at year 36 of operation.Clearly, an inspection is warranted at that time, which may ormay not lead to crack detection.

Modifications to the basic reliability problem to cater for aseries of n inspections are introduced by considering a subsetof the fatigue crack sizes a(t) to reflect the outcome of theseinspections. Accordingly, the limit state function (Eq. (20))becomes

g(X , t) = ac −

[a(t)

∣∣∣∣∣ n⋂i=1

ai (Ti )

](24)

Fig. 6. Reliability curve for a butt-welded detail following multiple inspections(no detection).

where the bar indicates a conditional event, Ti is the timeat which the inspection takes place and ai is the previouslymentioned subspace, which is defined as a(Ti ) ≥ ad in the caseof crack detection and a(Ti ) < ad otherwise. The parameter ad

is the detectable crack size, which is randomised through theinspection method’s probability of detection (PoD).

Using the modified limit state function of Eq. (24) andassuming inspections using the ACFM method whose PoDcurve is given by [59]

PoD(ad) = 1 −1

1 + e2.2 (ad − 0.7)1.9(25)

leads to the results shown in Fig. 6. The scenario consideredin Fig. 6 involves multiple inspections at the times at whichthe reliability is violated (years 36, 47, 62 and 80) under theassumption of no detection during these four inspections. Notethat the first branch of the ‘saw-tooth’ curve is the initial partof the curve shown in Fig. 5 and that each segment of the curverepresents a separate analysis. Fig. 6 demonstrates that in orderto maintain a P f = 10−3 for this detail, using ACFM andassuming that at no stage cracks are detected, a minimum offour inspections would need to be carried out throughout thedetail’s life.

However, it has to be emphasised that different inspectionscenarios could have been considered if for example the bridgeowner had decided that inspections at those specific times wereunsuitable. It also has to be noted that the results of this type ofreliability analysis are rather sensitive to the inspection methodused as well as the inspection outcome. This last point ishighlighted in Fig. 7, whereby the fourth inspection at year80 results in crack detection. Since the failure probabilitiesbecome unacceptably high leading to a failure probability of0.26 at year 120, a decision has to be made regarding thedetail’s operation. Methods that can be used by the bridgeowner in order to maintain the desirable reliability includerepair and/or imposition of load restrictions. The way some ofthese techniques may be quantified within an FM-based fatiguereliability analysis is discussed in the next case study.

1208 M.K. Chryssanthopoulos, T.D. Righiniotis / Journal of Constructional Steel Research 62 (2006) 1199–1209

Fig. 7. Reliability curve for a butt-welded detail following multiple inspections(no detection–detection).

5.3. Fatigue reliability based on inspections and invasiveactions

Once again, for this case study the butt-welded detail isselected. Here, however, it is assumed that, following the initialinspection at year 36 a crack is detected. The results of thisanalysis are shown in Fig. 8 in the form of triangles. As can beseen, the detail’s fatigue reliability becomes unacceptably lowalmost immediately following crack detection (Pf = 2.1×10−2

at year 36). It is here assumed that in the light of these resultsthe bridge owner decides to repair the crack by re-weldingthe toe. This type of operation is assumed here to restore thecrack depth to its as-welded state, in other words, a1(T1 =

36) = a0. Subsequent analysis results in the curve denoted inFig. 8 as ‘Detection & Re-welding’, which leads to a reliabilityviolation at year 64. Thus, an inspection would have to beundertaken at that time. The remaining two curves reflect thetwo possible outcomes of this inspection i.e. detection and nodetection at year 64. Note that in this case, even the lattercurve leads to violation of the reliability criterion at around year104 of operation. At that point the bridge owner may considerthe option of imposing load restrictions on the bridge and/orpreventative repair even though a crack is not detected. Theimposition of load restrictions and a different type of repair,namely weld toe grinding, were considered for a different typeof welded detail in [13].

6. Concluding remarks

The application of structural reliability techniques tofatigue related problems in welded steel structures hasoccupied intensively the engineering community for the pasttwenty years. An appropriate probabilistic framework, andassociated methodology, has by now been established andused in the context of assessment, inspection and repair.However, it is clear, from reviewing the literature, thatconsiderable differences can still be found on key assumptionsregarding fatigue and fracture behaviour, the parameters of theprobabilistic models and the idealised processes adopted forrepresenting random loading. It is of paramount importanceto examine carefully for each application, be it an offshorenode, a ship’s deck or a bridge girder detail, the factors that

Fig. 8. Reliability curve for a butt-welded detail following an initial crackdetection and repair.

play an important role in fatigue and fracture behaviour and tofocus attention on those which are of particular relevance tothe case in hand. Fatigue problems, and decisions, are multi-faceted and it is unwise to opt for a ‘one-size fits all’ approach.Procedures have been identified, tools exist, required data areslowly becoming available, and case studies can be consulted,but the need for well thought out benchmarking of analyticalresults against experimental databases, careful probabilisticmodelling and clear understanding of the limitations associatedwith any approach should not be underestimated.

Acknowledgments

The first author would like to express his sincere thanksto Prof. Ton Vrouwenvelder, TNO/TU Delft with whom hehas had the opportunity to collaborate on probabilistic fatiguemodels within the Joint Committee of Structural Safety (JCSS).The case studies presented in this paper have been producedin the course of a project sponsored by the Highways Agency,in co-operation with Flint and Neill and TWI. The opinionsexpressed in the paper are those of the authors and do notnecessarily reflect those of the sponsors.

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