Introduction to Fatigue Design April 09

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April 2009/ John Wægter

Note 1.1

Introduction to fatigue design

General.............................................................................................................................................2 The S-N curve..................................................................................................................................2Fatigue crack propagation................................................................................................................3Definition of basic S-N curves.........................................................................................................6 Tubular joints...................................................................................................................................9Influence of the parent material .....................................................................................................11Influence of plate thickness............................................................................................................11Influence of residual stresses.........................................................................................................13Cumulative damage........................................................................................................................14References......................................................................................................................................17

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Introduction to fatigue design

General

Fatigue may be defined as a mechanism of failure based on theformation and growth of cracksunder the action of repeated stresses. Normally, small cracks will not cause failure, but if the designis insufficient in relation to fatigue, the cracks may propagate to such an extent that failure of theconsidered detail occurs.

Fatigue is associated with stresses that vary with time often in a repeated manner. There are manypossible sources of time varying stresses e.g.

• Fluctuating live loads

• Acceleration forces in moving structures• Pressure changes• Temperature fluctuations

• Mechanical vibrations• Environmental loading (wind, waves and current)

Fatigue failure is therefore likely to occur in many types of structures, but in the context of thiscourse the main emphasis will be on welded offshore steel structures. Further, only fatigue failurefrom many stress cycles (high cycle fatigue) will be considered.

Fatigue is the most common cause of structural failures, it is frequently claimed that at least 80 % of

all structural failures are associated with fatigue. It should however be noticed, that in most caseswhere fatigue was identified as responsible for a structural failure, it was not because the fatiguedesign was inadequate, but rather because no attempt was made to design against fatigue and giveattention to proper joint selection and detailing.

With todays design approaches it is possible to perform a fatigue design that can almost eliminatethe risk of failure. In theory it can be based on fracture mechanics techniques, and significantadvances have been made in this field. But as such calculations inevitably must be based on certainassumptions (e.g. initial crack/flaw size) they are mainly used for comparative analyses where theresults finally can be calibrated to test results.

Most fatigue design in the industry is therefore closely based on fatigue strength data (S-Ncurves)that have been obtained experimentally.

The S-N curve

Fatigue cracks initiate at some form of stress concentration. The stress concentration may be causedby either a gross change of shape in the structure or by a local change of shape e.g. a bolt hole. Inwelded structures fatigue cracks typically initiate at a weld toe or from a defect at the root of theweld. For these reasons fatigue cracks are very commonly associated with joints.

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Data on fatigue strength are usually presented in the form of S-Ncurves, which show, for aparticular type of specimen, the number of stress cycles (N) required to cause failure under a givenrepeated constant stress range (S). S is the difference between the maximum and the minimum stressin the cycle.

To produce such anS-Ncurve fatigue tests are carried out at different stress ranges on a number of identical specimens representing the detail under consideration. The results are plotted on the basisof logSversus logN, and a typical relationship is shown in Figure 1.

Figure 1 Typical S-Ncurve for welded joints

It can be seen that, as the applied stress range is decreased, the number of cycles required to causefailure increases until ultimately a stress is reached below which failure does not occur. Thislimiting stress is referred to asthe fatigue limit (the endurance limit).

The two branches of theS-Ncurve represent an important different behaviour. For stress rangesabove the fatigue limit, cracks initiate and propagate to final rupture, whilebelow the fatigue limitcrack are initiated, but they do not propagate. In both cases the existence of cracks suggests thatfracture mechanics may be a useful tool for analysing the behaviour.

Fatigue crack propagation

For welded joints it has been found that crack propagation is the essential parameter for the fatiguelife. The rate of propagation depends on the actual stress or strain condition at the crack tip at anymoment, which in linear fracture mechanics (LFM) can be expressed in terms of the stress intensityfactor,K .

Conventionally, studies of fatigue crack propagation aim to relate the rate of propagation to therange of the stress intensity factor, or max minK K K Δ = − , where maxK and minK are the values of K

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corresponding to the upper and lower values of the stress cycle, and . This approach is

logical under wholly tensile loading, but it may not be so obviously correct when the stress cyclealternates between tension and compression, since at some stage the crack will close and at leastpart of the compressive part of the cycle will have no effect on the crack.

maxS minS

The first investigators to suggest that the rate of fatigue crack propagation was related to K Δ wereParis and Erdogan. They found that the rate of propagation depended on the stress conditionadjacent to the tip of the crack, and that it could be expressed in terms of K Δ . They expressed therate of propagation as

( )mda

C K dn

= Δ (1)

whereC andmare material constants,a is the crack size andn is the number of stress cycles.

It has since been shown that this relationship is only valid for a certain range of K Δ . If the equation

forda

dnis plotted against the value of K Δ on a log–log basis, a curve of the shape shown in Figure

2 is obtained, where the Paris law is a straight line in the middle region.

Figure 2 Typical relationship betweenda

dnand K Δ

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The lower value of K Δ represents a threshold value, thK Δ , that must be exceeded before any

propagation can occur at all. The existence of this threshold value explains why some cracks do notpropagate under fatigue loading.K C represents a critical value of K Δ triggering fast failure.

For steels it has been found, Ref. /1/, that there is a simple empirical relationship betweenmandC,in the Paris law region which is

41.315 10

895.4mC

−⋅= (2)

forda

dnin mm/cycle and K Δ in Nmm-3/2. Furthermore, for normal weldable structural steels

virtually all the results lie in the rangem=2.4 – 3.6 with an average value of 3.0.

Given that the values of mandC are known and that an initial crack size is assumed, it is possible tointegrate Equation (1) to calculate the expected fatigue life. Although this in principle is simple, ithas not been possible to determine values of mandC that relate to the full interval of K Δ for agiven steel quality in a particular structure.

In general using linear fracture mechanics, the values of K Δ for a crack deptha, at the toe of theweld subjected to stress range σ Δ can be written as

( )K F a aσ π Δ = ⋅ Δ (3)

whereF is a function of the crack size and the joint geometry. As previously indicated, the stressintensity factorK describes the stress field at the crack tip as indicated in Figure 3.

Figure 3 Stress field around crack tip

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Combining Equations (1) and (3) gives

( )( )mda

C F a adn

σ π = ⋅ Δ (4)

which on integration gives

( )2

( )

mf

i

am

ma

daC

F a aσ π N= ⋅ Δ ⋅ ⋅

⋅∫ (5)

whereN is the number of cycles for the crack to grow from an initial size to a final sizeia f a .

Examination of Equation (5) shows, that for a given type of joint (i.e. a known value of F(a)) and

constant values of andia f a , the value of the integral is a constant, so that Equation (5) can bewritten as

( ) constant =m

N aσ Δ ⋅ = (6a)

log log logN a m σ = − ⋅ Δ (6b)

which is the equation of anS-Ncurve. The constanta in Equation (6) above should not be confusedwith the crack sizeaused in fracture mechanics.

The implication of Equation (6) is that if the fatigue life of a specimen consists wholly of crackpropagation, then theS-Ncurve would be a straight line with slope -1/mwhen plotted on the basisof log σ Δ versus logN.

Many welded joints effectively fall into this category, since it has been found that fatigue cracksnormally initiate at small, slag intrusions at the weld toe. It should, however be emphasised that notall fatigue cracks initiate from the weld toe.

WhenF, C andmare known, Equation (5) generally provides the key equation for fatigue

calculations since it allows the calculation of any one of the four parameters ,ia f a , σ Δ andN

given the three others.

Defini tion of basic S-N curves

It is not possible to determineS-Ncurves for welded joints for all joint geometries entirely based onfracture mechanics. This would require knowledge of the stress intensity factor for each type of joint together with details of the assumed pre-existing defect (flaw) and the weld shape in thevicinity of those defects. Such information does not, in general exist. It is therefore inevitable thatthe basicS-N curves for design must be based on experimental results obtained under constantamplitude loading.

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The fatigue strength S of a welded joint depends on the stress concentration it produces. It is thedifference between the degrees of stress concentration that distinguish the good solution from thebad one. Considering the joints shown in Figure 4, it can be seen that the butt weld will produce lessdisturbance of the stress flow than any of the other solutions. The stress concentration at the weld

toe will be lower than in any of the last three solutions based on fillet welds, thus the butt weldsolution has the highest fatigue strength. It is important to notice thatfeatures that may increase thestatic strength of a welded joint, such as the presence of butt weld overfill or welded attachments orstiffeners inevitably reduce the fatigue strength.

Figure 4 Various methods of welded joints and lines of stress flow

A summary of some typical fatigue strengths (at 2 x 106 cycles) for welded joints together with an

indication of the point of origin of fatigue cracks in them is shown in Table 1. While the strengthquoted in different design standards vary somewhat, they are usually not very different from thevalues given in the table which is based upon analysis of a large mass of experimental data.

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Table 1 Fatigue strengths and cracking patterns in simple welded joints

For these simple joints, the design stress range given ( localσ ) is the nominal value as determined in

beam stress analysis, away from the stress concentration. When calculating the allowable designstress range, the stress concentration effect of the joint itself is thus included in the quoted fatigue

strengths. However, if e.g. a hole exists adjacent to the joint, the extra stress concentration from the

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hole (SCF) must be taken into consideration when determining the local stresses( local nominalSCFσ σ = ⋅ ), see Figure 5.

Figure 5 Determination of local stresses

The fatigue strength in Table 1 refers to simple joints, typically 13 mm thick, tested in air. In some

cases these allowable stress ranges may need to be modified

• If there is a larger eccentricity than assumed in the test• If the plate is thicker• If the joint is placed in a corrosive environment

Similarly, if the point of crack initiation is affected by a large geometric discontinuity in thestructure, e.g. a cut-out in a plate as shown in Figure 5, it will be necessary to consider this stressconcentration when determining the design stress as already mentioned above.

Based on the above it can be concluded, thatthe general rule in designing against fatigue is to keep

all stress flow paths smooth by eliminating as many stress concentrations as possible.

Tubular joints

In more complicated joints, such as tubular joints as used in the offshore industry, it is not possibleto relate the fatigue strength directly to nominal stresses (beam stresses) as for the simple casesconsidered in Table 1. However, a reasonable correlation with fatigue test data can be obtained byincluding the stress concentration of the joint itself in the stresses considered, using a special designS-Ncurve and stresses in terms of thehot spot stress range.

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A tubular joint consists of a chord and one or more braces, see Figure 6. The joint is analysed byconsidering the hot spot stress range in both the brace and in the chord based onthe nominal stressranges in the braceand appropriatestress concentration factors(SCFs).

Nominal stresses arise due to the tubes behaving as beam-columns, and may be calculated by frameanalysis of the structure. However, the total stress picture in a tubular joint is generally caused bythree main influences

1. The basic structural response of the joint to the applied loads in terms of beam stresses(nominal stresses)

2. The stress concentration caused by the actual shape of the tubular joint (geometric stresses)

3. Highly localised stresses caused by the shape of the weld in part of the tube and the weldnear the brace-chord intersection (notch stresses)

Figure 6 Determination of hot spot stresses in a tubular joint

To capture the different types of stresses different types of FE analysis are required. As alreadymentioned nominal stresses may be determined by frame analysis, geometric stresses can be foundusing shell elements while the determination of local stresses in the vicinity of the weld requiressolid elements.

Under cyclic loadingfatigue cracks will appear first in the point or region of highest stress range. This point is calledthe hot spot, and the stress in the hot spot isthe hot spot stress.

The hot spot stress incorporates the effects of the overall tube geometry (brace and chord). Itdisregards the stress concentration from the weld itself, but this stress concentration is included in

the appropriate S-N curve to be used for tubular joint design.

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The determination of hot spot stresses can be done very reliably using FE analysis, or for simple joints where parametric formulae exist, based on nominal stresses and (parametric) stressconcentration factors. For unusual and complex details an experimental determination of the fatigue

strength is sometimes carried out.

Influence of the parent material

For unwelded components the fatigue strength increases with material tensile strength. For example,plain polished steel specimens, with or without notches, display an increase of the fatigue strengthwith the tensile strength. However, this is not the case with welded joints, in which fatigue failureoccurs by crack propagation from initial flaws. For such welded joints the fatigue strength isindependent of the static strength, see Figure 7.

Figure 7 Fatigue strength versus static strength of parent material

An element in the explanation of this difference is that for the unwelded component, a fatigue crackfirst has to be initiated and then propagate, while in the welded joint it is only necessary for the

crack to propagate.

Influence of plate thickness

It has been found experimentally that the fatigue strength of test specimens is size-dependent. Thisis true for both unwelded and welded specimens, see Figure 8.

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Figure 8 Size effect on fatigue specimens

For the same limiting stress a ‘thin’ plate experiences a steeper stress gradient than a ‘thick’ plateunder pure bending. Such a steep stress gradient has been found to be less damaging than a small orno gradient. It has also been found that a combination of bending stresses and axial stresses is lessdamaging than a pure axial stress of the same value.

In practical design it has always been assumed that the use of fatigue strengths obtained under axialloading would be safe, even if conservative, under bending stresses. In consequence, fatigue designrules for welded joints have traditionally been based upon results obtained under axial loading.

With the issue of the British Department of Energy Guidance Notes (1984), Ref. /3/, it became

usual to quantify a thickness correction factor. They based their recommendations on newer testnormalised to a thickness of 32 mm as shown in Figure 9. Based on these results a thicknesscorrection factor was introduced of the form

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BB

tS S

t

⎛ ⎞= ⎜ ⎟

⎝ ⎠(7)

whereS is the fatigue strength of the joint under consideration, t is the thickness,SB is the fatiguestrength of the joint using the basicS-Ncurve andt

B

BB is the thickness corresponding to the basicS-N curve. Newer design codes often use the termtref instead of tB.B

The practical implication of Figure 9 being, that thicker plates have lower fatigue strength than thinplates.

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Figure 9 Thickness corrections, summary of tests

For design purposestB =32 mm should be used for tubular joints, whiletB BB =22 mm should be used

for other types of joints. The choice of reference thickness (tB =32 mm) for tubular joints wasconsistent with the data actually used for tubular joints, while for other types of joints the chosenvalue of t

B

BB =22 mm was less obvious since the results were bases on specimen of approximately12.7 mm on average. - Newer design codes represent a further development of the above basicconsiderations in relation to thickness correction.

Influence of residual stresses

The majority of welded structures are not stress relieved, so that it is realistic to assume that hightensile residual stresses of yield stress magnitude will exist in some places, both in weld metal andin the adjacent parent material. Since residual stresses are not normally modelled in the structuralanalysis, this means that the stress range determined coexist with an unknown level of residualstresses. In other words, the actual stress range may be determined as the nominal stress rangefound in the structural analysis, but the mean stress level would be changed in the presence of residual stresses.

In Figure 10 a test specimen with a non-load carrying attachment has been tested under a pulsatingtension (R =0), one series being as-welded and one series being stress relieved.

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partly tensile/partly compressive

pulsating tension

Figure 10 Influence of stress ratio and stress relief on fatigue strength

It can be seen that the fatigue strength is increased when the specimen is stress relieved, althoughnot so much. In addition a stress relieved specimen with partly tensile and partly compressivestresses (R =-1) was tested. The increase in allowable stress range is significant, and the resultsconfirms that if a joint contains low residual stresses and is subjected to partly compressive stresses,then the fatigue strength is increased.

As can be seen in Figure 10, for stress relieved joints there is a significant increase in fatiguestrength when subjected to partially compressive stresses as opposed to pulsating tension. It wouldtherefore theoretically be safe to increase the allowable stress range for such joints, and Gurney,Ref. /1/, gives the following advice: “A possible method of doing that would be to design for astress range equal to the tensile component of stress, plus 60 % of the compressive component.”However, it is emphasised thata necessary assumption for using the increased fatigue strength in

practical design work is that the residual stresses have been reduced effectively.

Based on these considerations, the conclusion for practical design work normally is thatthe fatiguestrength of welded joints should be considered independent of the mean stress and be based on thestress range alone.

Cumulative damage

Most fatigue tests have been carried out under constant amplitude loading and design rules forwelded joints have been based on these tests, see Figure 11.

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popular testing area

Figure 11 Definition of terms forS-N testing

The parameters that characterize the loading is the stress amplitudeSa or stress rangeSr, and themean stressSm.

minmax SSSr −= (8)

min

max

SR

S= (9)

1

2 1r

m

SS

R

R

+= ⋅

−(10)

In real life many structures experience loads of variable amplitude. For such structures the fatiguedesign is normally carried out based onMiner’s linear damage rule, which states that the fatiguedamageD can be found as

1 2

1 2

......... 1.0i i

ii i

n n n nD

N N N N= + + = ≤∑ (11)

wheren1, n2 etc. are the numbers of stress cycles of stress rangeS1, S2 etc. expected during the lifeof the structure, andN1, N2 etc. are the corresponding numbers of cycles to failure under constantamplitude loading at those stress ranges, see Figure 12. - The values of N1, N2 etc. should be derived

from the relevant designS-Ncurve for the joint in question.

The application of Miner’s rule requires knowledge of the expected loading/response historybecausea fatigue design is to be based on the long term distribution of stress ranges for the jointconsidered.

In some instances the loading assumed for design purposes will be specified in the relevantapplication standard/design basis. In cases where such information is not available, the designer hasto make reasonable (conservative) assumptions as to the stress range history, e.g. based oninformation from similar structures, or from loading/response readings obtained from continuousmonitoring.

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Figure 12 Miner’s rule

When long term stress histories are known, the corresponding long term stress range history may be

established using an appropriate standard cycle counting technique such as Rainflow counting. Thestress history can then be expressed in terms of coexisting stress ranges and number of stress cycles.

As may be noted, Miner’s rule does not account for the sequence of the stress ranges, only for thenumber of stress ranges. However, experimental data suggest that the sequence of stress rangesdoes actually influence the fatigue strength of a welded joint. At present no simple and reliable remedy isavailable to account for this sequence effect. For design purposed a pragmatic correction is typicallycarried out by reducing the limiting damage to a value less than unity in combination with amodifiedS-Ncurve.

The modifiedS-Ncurve is normally modified to account for the stresses lower than the constantamplitude ‘fatigue limit’ because in a variable loading history the higher stress ranges tend topropagate small defects and fatigue cracks to such an extent, that they will become capable of beingpropagated further by even the lowest stress ranges.

Nevertheless, the lower stress ranges can usually be assumed to be less damaging so that in manystandards it is assumed that theS-Ncurve is bent from am=3 to a shallower slopem=5 at 107 cycles.

Experimental data suggest that the designS-Ncurve should be determined as the mean minus twostandard deviations, and this is used by most design standards i.e.

σ Δ⋅−⋅−= log2loglog msaN (12)

wheres is the standard deviation of logN.

The designS-Ncurve is therefore often written as

log log logN a m σ = − ⋅ Δ (13)

where saa ⋅−= 2loglog .

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References

Ref. /1/ Gurney, T. R.: Chapter 5.4 Fatigue Design. Constructional Steel Design. AnInternational Guide. Elsevier Applied Science, 1992.

Ref. /2/ Fatigue Handbook. Offshore Steel Structures. Edited by A. Almar-Næss. Tapir 1985.

Ref. /3/ Department of Energy. Offshore Installations: Guidance on design, construction andcertification. London: HMSO, 1984.

Ref. /4/ DNV-RP-C203: Fatigue Design of Offshore Steel Structures. October 2008. DetNorske Veritas.

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Introduction to Fatigue Design April 09