Advanced Introduction to Fatigue

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ESDEP WG 12

FATIGUE

Lecture 12.2: Advanced Introduction to

Fatigue

OBJECTIVE/SCOPE:

To introduce the main concepts and definitions regarding the fatigue process and to identify the main factors that influence the fatigue performance ofmaterials, components and structures.

PREREQUISITES

Lecture 12.1: Basic Introduction to Fatigue

RELATED LECTURES

SUMMARY

The physical process of the initiation of fatigue cracks in smooth and notched test specimens under the influence of repeated loads is described and therelevance of this process for the fatigue of real structures is discussed.

The basis of different stress cycle counting procedures is explained for variable amplitude loading. Exceedance diagram and frequency spectrum effects aredescribed.

1. INTRODUCTION

Fatigue is commonly referred to as a process in which damage is accumulated in a material undergoing fluctuating loading, eventually resulting in failure evenif the maximum load is well below the elastic limit of the material. Fatigue is a process of local strength reduction that occurs in engineering materials such asmetallic alloys, polymers and composites, eg. concrete and fibre reinforced plastics. Although the phenomenological details of the process may differ from onematerial to another the following definition given by ASTM [1] encompasses fatigue failures in all materials:

Fatigue - the process of progressive localised permanent structural change occurring in a material subjected to conditions that produce fluctuatingstresses and strains at some point or points and that may culminate in cracks or complete fracture after a sufficient number of fluctuations.

The important features of the process relevant to fatigue in metallic materials are indicated by the underlined words in the definition above. Fatigue is a progressive process in which the damage develops slowly in the early stages and accelerates very quickly towards the end. Thus the first stage consists of acrack initiation phase, which for smooth and mildly notched parts that are subjected to small loads cycles may occupy more than 90 percent of the life. In mostcase cases the initiation process is confined to a small area, usually of high local stress, where the damage accumulates during stressing. In adjacent parts of thecomponents, with only slightly lower stresses, no fatigue damage may occur and these parts thus have an infinite fatigue life. The initiation process usuallyresults in a number of micro-cracks that may grow more or less independently until one crack becomes dominant through a coalescence process at themicrocracks start to interact. Under steady fatigue loading this crack grows slowly, but starts to accelerate when the reduction of the cross-section increases thelocal stress field near the crack front. Final failure occur as an unstable fracture when the remaining area is too small to support the load. These stages in thefatigue process can in many cases be related to distinctive features of the fracture surface of components that have failed under fluctuating loads, the presenceof these features can therefore be used to identify fatigue as the probable cause of failure.

2. CHARACTERISTICS OF FATIGUE FRACTURE SURFACES

Typical fracture surfaces in mechanical components that were subjected to fatigue loads are shown in Slide 1. One characteristic feature of the surfacemorphology which is evident in both macrographs is the flat, smooth region of the surface exhibiting beach marks (also called clamshell marks). This partrepresents the portion of the fracture surface over which the crack grew in a stable, slow mode. The rougher regions, showing evidence of large plasticdeformation, is the final fracture area through which the crack progressed in an unstable mode. The beach marks may form concentric rings that point toward

the areas of initiation. The origin of the fatigue crack may be more or less distinct. In some cases a defect may be identified as the origin of the crack, in othercases there is no apparent reason why the crack should start at a particular point in a fracture surface. If the critical section is at a high stress concentrationfatigue initiation may occur at many points, in contrast to the case of unnotched parts where the crack usually grows from one point only see Figure 1. Whilethe presence of any defects at the origin may indicate the cause of the fatigue failure, the crack propagation area may yield some information regarding themagnitude of the fatigue loads and also about the variation in the loading pattern. Firstly, the relative magnitude of the areas of slow-growth and final fractureregions give an indication of the maximum stresses and the fracture toughness of the material. Thus, a large final fracture area for a given material indicates ahigh maximum load, whereas a small area indicates that the load was lower at fracture. Similarly, for a fixed maximum stress, the relative area corresponding toslow crack growth increases with the fracture toughness of the material (or with the tensile strength if the final fracture is a fully ductile overload fracture).



Slide 1 : Typical fatigue failures in steel components.

Beach marks are formed when the crack grows intermittently and at different rates during random variations in the loading pattern under the influence of achanging corrosive environment. Beach marks are therefore not observed in the surfaces of fatigue specimens tested under constant amplitude loadingconditions without any start-stop periods. The average crack growth is of the order of a few millimetres per million cycles in high cycle fatigue, and it is clearthat the distance between bands in the beach marks are not a measure of the rate of crack advance per load cycle. However, examination by electron microscopeat magnifications between 1,000x and 30,000x may reveal characteristic surface ripples called fatigue striations, see Slide 2. Although somewhat similar inappearance, these lines are not the beach marks described above as one beach mark may contain thousands of striations. During constant amplitude fatigueloading at relatively high growth rates in ductile material such as stainless steels and aluminium alloys the striation spacing represents the crack advancement

per load cycle. However, in low stress, high cycle fatigue where the striation spacing is less than one atomic spacing (- 2.5 x 10-8m) per cycle. Under theseconditions the crack does not advance simultaneously along the crack front, growth occurring instead only along some portions during a few cycles, then arrestswhile growth occurs along other segments. Striations as shown in Figure 3 are not seen if the crack grows by other mechanisms such as microvoid coalescenceor, in brittle materials, microclevage. In structural steels the crack can propagate by all three mechanism, and striations may be difficult to observe. Slide 3shows an example of beach marks and striations in the fracture originating at a large defect in a welded C-Mn steel with a yield strength of about 360Mpa.



Slide 2 : Striations in an aluminium alloy.

Slide 3 : Fatigue failures in the Alexander L Kielland platform.

3. NATURE OF THE FATIGUE PROCESS

From the description of the characteristics of fatigue fracture surfaces, three stages in the fatigue process may be identified:

Stage I: Crack initiation

Stage II: Propagation of one dominant crack

Stage III: Final fracture

Fatigue cracking in metals is always associated with the accumulation of irreversible plastic strain. The crack process which is discussed in the followingapplies to smooth specimens made of ductile materials.

In high cycle fatigue the maximum stress in cyclic loads that eventually cause fatigue failure may be well below the elastic limit of the material, and large scale plastic deformation does not occur. However, at a free surface plastic strains may accumulate as a result of dislocation movements. Dislocations are line defects



in the lattice structure which can move and multiply under the action of shear stresses, leaving a permanent deformation. Dislocation mobility and hence theamount of deformation (or slip) is greater at a free surface than in the interior of crystalline materials due to lack of constraint from grain boundaries. Grains in polycrystalline structural metals are individually oriented in a random manner. Each grain, however, has an ordered atomic structure giving rise to directional properties. Deformation for example, takes place on crystallographic planes of easy slip along which dislocations can move more easily than other planes.

Since slip is controlled primarily by shear stress, slip deformation takes place along crystallographic planes that are orientated close to 45° to the tensile stressdirection. The results of such deformation is atomic planes sliding relative to each other, resulting in a roughening of the surface in slip bands. During furthercycling slip band deformation is intensified at the surface and extending into the interior of the grain, resulting in so-called persistent slip bands, (PSB's). Thename originated from the observation in early studies of fatigue that slip band would reappear - "persist" - at the same location after a thin surface layer wasremoved by elastopolishing. The accumulation of local plastic flow result in surface ridges and troughs called extrusions and intrusions, respectively, Figure 2.The cohesion between the layers in slip band is weakened by oxidation of fresh surfaces and hardening of the strained material. At some point in this processsmall cracks develop in the intrusions. These microcracks grow along slip planes, ie. a shear stress driven process. Growth in the shear mode, called stage Icrack growth extends over a few grains. During continued cycling the microcracks in different grains coalesce resulting in one or a few dominating cracks. Thestress field associated with the dominating crack cause further growth under the primary action of maximum principal stress; this is called stage II growth. Thecrack path is now essentially perpendicular to the tensile stress axis. Crack advancement is, however, still influenced by the crystallographic orientation of thegrains and the crack grows in a zigzag path along slip planes and cleavage planes from grain to grain, see Figure 3. Most fatigue cracks advance across grain boundaries as indicated in Figure 3, ie. in a transcrystalline mode. However, at high temperatures or in a corrosive environment, grain boundaries may becomeweaker than the grain matrix, resulting in intercrystalline crack growth. The fracture surface created by stage II crack growth are in ductile metals characterised by striations whose density and width can be related to the applied stress level.

Since crack nucleation is related to the magnitude of stress, any stress concentration in the form of external or internal surface flaws can marked reduce fatiguelife, in particular when the initiation phase occupies a significant portion of the total life. Thus a part with a smooth, polished surface generally has a higherfatigue strength than one with a rough surface. Crack initiation can also be facilitated by inclusions, which act as internal stress raisers. In ductile materials slip band deformations at inclusions are higher than elsewhere and fatigue cracks may initiate here unless other stress raisers dominate.

In high strength materials, notably steels and aluminium alloys, a different initiation mechanism is often observed. In such materials, which are highly resistantto slip deformation, the interface between the matrix and inclusion may be relatively weak, and cracks will start here if decohesion occurs at the inclusionsurface, aided by the increased stress/strain field around the inclusion. Slide 4 shows small fatigue cracks originating at inclusions in a high strength steel.Alternatively, a hard brittle inclusion may break and a fatigue crack may initiate at the edges of the cleavage fracture.

Slide 4 : Fatigue crack initiation at an inclusion in a high strength steel alloy.

From the discussion above it is evidently not possible to make a clear distinction between crack nucleation and stage I growth. "Crack initiation" is thus a ratherimprecise term used to describe a series of events leading to stage II crack. Although the initiation stage includes some crack growth, the small scale of thecrack compared with microstructural dimensions such as grain size invalidates a fracture mechanics based analysis of this growth phase. Instead, local stressesand strains are commonly related to material constants in prediction models used to estimate the length of stage I. The material constants are normally obtained

from tests on smooth specimens subjected to stress or strain controlled cycling.

4. FATIGUE LOADING

The simplest form of stress spectrum to which a structural element may be subjected is a sinusoidal or constant amplitude stress-time history with a constantmean load, as illustrated in Figure 4. Since this is a loading pattern which is easily defined and simple to reproduce in the laboratory it forms the basis for mostfatigue tests. The following six parameters are used to define a constant amplitude stress cycle:



Smax = maximum stress in the cycle

Smin = minimum stress in the cycle

Sm = mean stress in the cycle = (Smax + Smin)/2

Sa = stress amplitude = (Smax Smin)/2

AS = stress range = Smax - Smin = 2Sa

R = stress ratio = Smin/Smax

The stress cycle is uniquely defined by any two of these quantities, except combinations of stress range and stress amplitude. Various stress patterns are shownin Figure 5, with definitions in accordance with ISO [2] terminology.

The stress range is the primary parameter influencing fatigue life, with mean stress as a secondary parameter. The stress ratio is often used as an indication ofthe influence of mean loads, but the effect of a constant mean load is not the same as for a constant mean stress. The difference between S-N curves withconstant mean stress or constant R-ratio is discussed in the section on fatigue testing.

The test frequency is needed to define a stress history, but in the fatigue of metallic materials the frequency is not an important parameter, except at hightemperatures when creep interacts with fatigue, or when corrosion influences fatigue life. In both cases a lower test frequency results in a shorter life.

Typical stress-time histories obtained from real structures are one shown in Figure 6. The sequence in Figure 6a has a constant mean stress, individual stresscycles are easily identifiable, and it necessary to evaluate this stress history in terms of stress range only. The more "random" stress variations in Figure 10b iscalled a broad band process because the power density function (a plot of energy vs. frequency) spans a wide frequency range, in contrast to the one in Figure

6a which contains essentially one frequency. The difference is illustrated in Figure 7. The load history in Figure 6 can be interpreted as a variation of the mainload with superimposed smaller excursions that could be caused by eg. second order vibrations or by electronic noise in the load acquisition system. In case oftrue mean load variations not only the range but also the mean of each cycle needs to be recorded in order to estimate the influence of mean load on the damageaccumulation. In both cases it is necessary to eliminate the smaller cycles since they may be below the fatigue limit and therefore cause no fatigue damage, or because they do not represent real load cycles. Thus a more complicated evaluation procedure is required for identifying and counting individual major stresscycles and their associated mean stresses. Counting methods such as the range pair, rainflow and the reservoir methods are designed to achieve this. These procedures are described in paragraph 7.



5. FATIGUE LIFE DATA

The total fatigue life in terms of cycles to failure can be expressed as:



Nt = Ni + N p (1)

where Ni and N p are number of cycles spent in the initiation and propagation stages, respectively. As noted, the two stages are distinctly different in nature and

different material parameters control their length. The life of unnotched components, for example, is dominated by crack initiation. In sharply notched parts,however, or in parts containing crack-life defects, eg. welded joints, the crack growth stage dominates and crack propagation data may be used in an assessmentof fatigue life using fracture mechanics analysis. Therefore different test methods are necessary to assess the fatigue properties of these types of components.

5.1 Fatigue Strength Curves

Fatigue data for components whose lives consist of an initiation phase followed by crack propagation are usually presented in the form of S-N curves, whereapplied stress S is plotted against total cycles to failure, N (= Nt). As the stress decreases, the life in cycles to failure increases, as illustrated in Figure 8. The S-

N curves for ferrous and titanium alloys exhibit a limiting stress below which failure does not occur; this is called the fatigue or the endurance limit. The branch point or "knee" of the curve lies normally in the 105 to 107 cycle range. In aluminium and other nonferrous alloys there is no stress asymptote and a finite

fatigue life exists at any stress level. All materials, however, exhibit a relatively flat curve in the high-cycle region, ie. at lives longer than about 105 cycles.

A characteristic feature of fatigue tests is the large scatter in fatigue strength data, this is particularly evident when a number of specimens are tested at the samestress level, as illustrated in Figure 9. Plotting the data for a given stress level along a logarithmic endurance axis gives a distribution which can beapproximated by the Gaussian (or normal) distribution, hence endurance data are said to have a log normal distribution. Alternatively the Weibull distributionmay be used, but the choice is not important since about 200 specimens, tested at the same stress level, are required to make a statistically significant distinction between the two distributions. This number is about one order of magnitude larger than the quantity of specimens that typically are available for fatigue testingat one stress level.



Assuming the life distribution to be log normal, the associated mean life curve and the standard deviation can be used to define a design S-N curve for anydesired probability of failure.

When the crack propagation stage dominates fatigue life, design data may be obtained from crack growth curves, an example of which is shown schematicallyin Figure 10. The stress intensity factor K uniquely describes the stress field near the crack tip, and is therefore used in the design against unstable fracture.

Likewise, the range of the stress intensity factor, ∆K, may be expected to govern fatigue crack growth. The validity of this assumption was first proved by Paris[3], and later verified by many other researchers. The crack growth curve, which has a sigmoidal shape, spans three regions as indicated schematically in Figure

10. In Region I the crack growth rate drops off asymptotically as ∆K is reduced towards a limit or threshold, ∆K th, below which no crack growth takes place.

Life fatigue endurance data, crack growth data show considerable scatter and test results must be evaluated by statistical methods in order to derive usefuldesign data.

5.2 Fatigue Testing

The basis for any design methodology aimed at preventing fatigue failures is data characterising the fatigue strength of components and structures. Fatiguetesting is therefore essential for the fatigue design process. The ideal fatigue test may be defined as a test in which an actual structure is subjected to the serviceload spectrum of that structure. However, life estimates are required before the design is finalised or details of the loading history are known. Additionally, eachstructure will experience a particular load history that is unique for that structure, so many simplifications and assumptions need to be made regarding the teststress sequence which is going to represent the many types of service histories that can occur in practice.

Fatigue testing is therefore performed in several ways, depending on the stage the design or production of the structure has reached or the intended use of thedata. The following four main types of tests can be identified:

1. Stress-life testing of small specimens.2. Strain-life testing of small specimens.3. Crack growth testing.4. S-N tests of components.5. Prototype testing for design validation.

The first three tests are idealised tests that produce information on the material response. The use of the results from these tests in life prediction of componentsand structures requires additional knowledge of influencing factors related to the geometry, size, surface condition and corrosive environment. S-N tests ofcomponents are also normally standardised tests that make life predictions more accurate compared with the three other tests because the uncertaintiesregarding the influence of notches and surface conditions are reduced. Service loading or variable amplitude testing normally requires a knowledge of theresponse of the actual structure to the loading environment, and is therefore normally used only for prototype or component testing at a late stage in the production process.

Rotating bending machines were used in the past to generate large amounts of test data in a relatively inexpensive way. Two types are shown schematically in

Figure 11. The computer-controlled closed loop testing machines are widely used in all modern fatigue testing laboratories. Most are equipped with hydraulicgrips that facilitate the insertion and removal of specimens. A schematic diagram of such a testing machine is shown in Figure 12. These machines are capableof a precise control of almost any type of stress-time, strain-time or load pattern and are therefore replacing other types of testing machines.



5.3 Presentation of Fatigue Test Data

Among the first systematic fatigue investigations reported in the literature are those set up and conducted by the German railway engineer, August Wöhler, between 1852 and 1870. He performed tests on full scale railway axles and also small scale bending, axial and torsion tests on several types of materials.



Typical examples of Wöhler's original data are shown in Figure 13. These data are presented in what is now well known as Wöhler or S-N diagrams. Suchdiagrams are still commonly used in the presentation of fatigue data, although the stress axis is often on a logarithmic scale in contrast to Wöhler's linear stressaxis. Basquin's equation is often fitted to test data, it has the form:

Sa N b = constant (2)

where Sa is the stress amplitude, and b is the slope. When both axes have logarithmic scales, Basquin's equation becomes a straight line.

Other types of diagrams are used, for instance to demonstrate the influence of mean stress; examples are the Smith or Haigh diagrams which are shown inFigure 14. Low cycle fatigue data are almost universally plotted in strain vs. life diagrams since strain is a more meaningful and more easily measurable parameter than stress when the stress exceeds the elastic limit.

6. PRIMARY FACTORS AFFECTING FATIGUE LIFE

The difference in fatigue behaviour of full scale machine or structural components as compared with small laboratory specimens of the same material issometimes striking. In the majority of cases the real life component exhibits a considerably poorer fatigue performance than the laboratory specimen althoughthe computed stresses are the same. This difference in fatigue response can be examined in a systematic manner by evaluating the various factors that influencefatigue strength. Qualitative and quantitative assessments of these effects are presented in the following paragraphs.

6.1 Material Effects

Effect of static strength on basic S-N data



For small unnotched, polished specimens tested in rotating bending or fully reversed axial loading there is a strong correlation between the high-cycle fatigue

strengths at 106 to 107 cycles (or fatigue limit) So, and the ultimate tensile strength Su. For many steel materials the fatigue limit (amplitude) is approximately

50% of the tensile strength, ie. So = 0.5 Su. The ratio of the alternating fatigue strength So to the ultimate tensile strength Su is called the fatigue ratio. The

relationship between the fatigue limit and the ultimate tensile strength is shown in Figure 15 for carbon and alloy steels. The majority of data are grouped between the lines corresponding to fatigue ratios of 0.6 and 0.35. Another feature is that the fatigue strength does not increase significantly for Su>1400 Mpa.

Other relationships between fatigue strength and static strength properties based on statistical analysis of test data may be found in the literature.

For real life components, the effects of notches, surface roughness and corrosion reduce the fatigue strength, the effects being strongest for the higher strengthmaterials. The variation in fatigue strength with the tensile strength is illustrated in Figure 16. The data in Figure 16 are consistent with the fact that cracks arequickly initiated in components that are sharply notched or subjected to severe corrosion. The fatigue life then consists almost entirely of crack growth. Crack

growth is very little influenced by the static strength of the material, as illustrated in Figure 16, and the fatigue lives of sharply notched parts are thereforealmost independent of the tensile strength. An important example is welded joints which always contain small crack-like defects from which crack startgrowing after a very short initiation period. Consequently the fatigue design stresses in current design rules for welded joints are independent of the ultimatetensile strength.

Crack Growth Data

Fatigue crack growth rates seem to be much less dependent on static strength properties than crack initiation, at least within a given alloy system. In acomparison of crack growth data for many different types of steel, with yield strengths from 250 to about 2000 Mpa levels of steel, Barsom [4] found thatgrouping the steels according to microstructure would minimise scatter. His data for ferritic-pearlitic, matensitic and austenitic are shown in Figure 17. Alsoshown in the same diagram is a common scatter band which indicates a relatively small difference in crack growth behaviour between the three classes of steel.



While data for aluminium alloys show a larger scatter than for steels, it is still possible to define a common scatter band. Recognising that different alloysystems seem to have their characteristic crack growth curves, attempts have been made to correlate crack growth data on the basis of the following expression

= C (3)

An implication of Equation 3 is that at equal crack growth rates, a crack in a steel plate can sustain three times higher stress than the same crack in analuminium plate. Thus, a rough assessment of the fatigue strength of an aluminium component whose life is dominated by crack growth can be obtained bydividing the fatigue strength of a similarly shaped steel component by three.

6.2 Mean Stress Effects

In 1870 Wöhler identified the stress amplitude as the primary loading variable in fatigue testing; however, the static or mean stress also affects fatigue life asshown schematically in Figure 10. In general, a tensile mean stress reduces fatigue life while a compressive mean stress increases life. Mean stress effects are presented either by the mean stress itself as a parameter or the stress ratio, R. Although the two are interrelated through:

Sm = Sa (4)

the effects on life are not the same, ie. testing with a constant value of R does not have the same effect on life as a constant value of Sm, the difference is shown

schematically in Figure 18.



As indicated in Figure 19a, testing at a constant R value means that the mean stress decreases when the stress range is reduced, therefore testing at R = constantgives a better S-N curve than the Sm = constant curve, as indicated in Figure 19b. It should also be noted that when the same data set is plotted in an S-N

diagram with R = constant or with Sm = constant the two S-N curves appear to be different, as shown in Figure 20.

The effect of mean stress on the fatigue strength is commonly presented in Haigh diagrams as shown in Figure 21, where Sa / So is plotted against Sm / Su. So is

the fatigue strength at a given life under fully reversed (Sm = 0,R = -1) conditions. Su is the ultimate tensile strength. The data points thus represent

combinations of Sa and Sm giving that life. The results were obtained for small unnotched specimens, tested at various tensile mean stresses. The straight lines



are the modified Goodman and the Soderberg lines, and the curved line is the Gerber parabola. These are empirical relationships that are represented by thefollowing equations:

Modified Goodman Sa/So + Sm/Su = 1 (5)

Gerber Sa/ So + (Sm/ Su)2 = 1 (6)

Soderberg Sa/ So + Sm/ Sy = 1 (7)

The Gerber curves gives a reasonably good fit to the data, but some points fall below the line, ie. on the unsafe side. The Goodman line represents a lower ofthe data, while the Soderberg line is a relatively conservative lower bound that is sometimes used in design. These expressions should be used with care indesign of actual components since the effects of notches, surface condition, size and environment are not accounted for. Also stress interaction effect due to

mean load variation during spectrum loading might modify the mean stress effects given in the three equations.

6.3 Notch Effects

Fatigue is a weakest link process which depends on the local stress in a small area. While the higher strain at a notch makes no significant contribution to theoverall deformation, cracks may start growing here and eventually result in fracture of the part. It is therefore necessary to calculate the local stress and relatethis to the fatigue behaviour of the notched component. A first approximation is to use the S-N curve for unnotched specimens and reduce the stress by the K t

factor. An example of this approach is shown in Figure 22 for a sharply notched steel specimen. The predicted curve fits reasonably well in the high cycleregion, but at shorter lives the calculated curve is far too conservative. The tendency shown in Figure 22 is in fact a general one, namely that the actual strengthreduction in fatigues is less than that predicted by the stress concentration factor. Instead the fatigue notch factor K f is used to evaluate the effect of notches in

fatigue. K f is defined as the unnotched to notched fatigue strength, obtained in fatigue tests:



K f = (8)

From Figure 22 it is evident that K f varies with fatigue life, however, K f is commonly defined as the ratio between the fatigue limits. With this definition K f is

less than K t, the stress increase due to the notch is therefore not fully effective in fatigue. The difference between K f and K t arise from several sources. Firstly,

the material in the notch may be subject to cyclic softening during fatigue loading and the local stress is reduced. Secondly, the material in the small region atthe bottom of the notch experiences a support effect caused by the constraint from the surrounding material so that the average strain in the critical region is

less than that indicated by the elastic stress concentration factor. Finally, there is a statistical variability effect arising from the fact that the highly stressedregion at the notch root is small, so there is a smaller probability of finding a weak spot.

The notch sensitivity q is a measure of how the material in the notch responds to fatigue cycling, ie. how K f is related to K t. q is defined as the ratio of effective

stress increase in fatigue due to the notch, to the theoretical stress increase given by the elastic stress concentration factor. Thus, with reference to Figure 21

q = (σmax,eff - σn)/(σmax - σn) = (K f σn - σn)/(K tσn - σn) = (K f - 1)/(K t - 1) (9)

where σmax,eff is the effective maximum stress, see Figure 23. This definition of K f provides a scale for q that ranges from zero to unity. When q = 0, K f = K t =

1 and the material is fully insensitive to notches, ie. a notch does not lower the fatigue strength. For extremely ductile, low strength materials such as annealedcopper, q approaches 0. Also materials with large defects, eg. grey cast iron with graphite flakes have values of q close to 0. Hard brittle materials have valuesof q close to unity. In general q is found to be a function of both material and the notch root radius. The concept of notch sensitivity therefore also incorporatesa notch size effect.



The fatigue notch factor applies to the high cycle range, at shorter lives K f approaches unity as the S-N curves for notched and unnotched specimens converge

and coincide at N = 1/4 (tensile test). In experimental investigations involving ductile materials it was found that the fatigue notch factor need to be appliedonly to the alternating part of the stress cycle and not to the mean stress. For brittle materials, however, K f should be applied to the mean stress as well.

6.4 Size Effects

Although a size effect is implicit in the fatigue notch factor approach, a size reduction factor is normally employed in when designing against fatigue. The need

for this additional size correlation arises from the fact that the notch size effect saturates at notch root radii larger than about 3-4mm, ie. K f → K t, while it is

well known from tests on full scale components, also unnotched ones, that the fatigue strength continues to drop off with increasing size, without any apparentlimit.

The size effect in fatigue is generally ascribed to the following sources:

A statistical size effect, which is an inherent feature of the fatigue process the nature of fatigue crack initiation which is a weakest link process where a

crack initiates when variables such as internal and external stresses, geometry, defect size and number, and material properties combine to give optimumconditions for crack nucleation and growth. Increasing size therefore produces a higher probability of a weak location.

A technological size effect, which is due to the different material processing route and different fabrication processes experienced by large and small parts. Different surface conditions and residual stresses are important aspects of this type of size effect.

A geometrical size also called the stress gradient effect. This effect is due to the lower stress gradient present in a thick section compared with a thin one,see Figure 24. If a defect, in the form of a surface scratch or a weld defect, has the same depth in the thin and thick parts, the defect in the thick part willexperience a higher stress than the one in the thin part, due to the difference in stress gradient, as indicated in Figure 24.

A stress increase effect, due to incomplete geometric scaling of the micro-geometry of the notch. This takes place if the notch radius is not scaled up withother dimensions.



Examples of components for which the latter effect is important are welded joints and threaded fasteners. The critical locations for crack initiation are the weldtoe and the thread root, respectively. In both cases the local stress is a function of the ratio of thickness (diameter) to the notch radius. In welds the toe radius isdetermined by the welding process and is therefore essentially constant for different size joints. The t/r ratio therefore increases and also the local stress whenthe plate is made thicker, with r remaining constant. A similar situation exists for bolts, due to the fact that the thread root radius is scaled to the thread pitch,

rather than the diameter for standard (eg. ISO) threads. Since the pitch increases much slower than the diameter the result is an increase in the notch stress with bolt size. For bolts as well as welded joints the increased notch acuity effect comes in addition to the notch size effect discussed earlier, the result is that theexperimentally determined size effects for these components are among the strongest recorded. An example of size effects for welded joints is shown in Figure25. The solid line represents current design practice, according to eg. Eurocode 3 and the UK Department of Energy Guidance Notes. The equation for this lineis given by:

= (10)



The exponent n, the slope of the lines in Figure 25a, is the size correction exponent.

The experimental data points indicate that the thickness correction with n = 1/4 is on the unsafe side in some cases. As indicated in Figure 25a thicknesscorrection exponent of n = 1/3 instead of the current value of 1/4 gives a better fit to the data in Figure 25a. For unwelded plates and low stress concentrationoints in Figure 25b a value of n = 1/5 seems appropriate [7].

There is experimental evidence that indicate a relationship between the stress gradient and the size effect. Based on an analysis of experimental data similar thefollowing size reduction factor has been proposed to account for the larger stress gradient found in notched specimens [8].

n = 0.10 + 0.15 log K t (11)

6.5 Effects of Surface Finish

Almost all fatigue cracks nucleate at the surface since slip occurs easier here than in the interior. Additionally, simple fracture mechanics considerations showthat surface defects and notches are much more damaging than internal defects of similar size. The physical condition and stress situation at the surface istherefore of prime importance for the fatigue performance. One of the important variables influencing the fatigue strength, the surface finish, commonlycharacterised by R u, the average surface roughness which is the mean distance between peaks and troughs over a specified measuring distance. The effect of

surface finish is determined by comparing the fatigue limit of specimens with a given surface finish with the fatigue limit of highly polished standardspecimens. The surface reduction factor Cr is the defined as the ratio between the two fatigue limits. Since steels become increasingly more notch sensitive with

higher strength, the surface factor Cr decreases with increasing tensile strength, Su.

6.6 Residual Stress Effects

Residual stresses or internal stresses are produced when a region of a part is strained beyond the elastic limit while other regions are elastically deformed. Whenthe force or deformation causing the deformation are removed, the elastically deformed material springs back and impose residual stresses in the plasticallydeformed material. Yielding can be caused by thermal expansion as well as by external force. The residual stresses are of the opposite sign to the initiallyapplied stress. Therefore, if a notched member is loaded in tension until yielding occurs, the notch root will experience a compressive stress after unloading.Welding stresses which are locked in when the weld metal contracts during cooling are an example of highly damaging stresses that cannot be avoided duringfabrication. These stresses are of yield stress magnitude and tensile and compressive stresses must always balance each other, as indicated in Figure 26. Thehigh tensile welding stresses contribute to a large extent to the poor fatigue performance of welded joints.



Stresses can be introduced by mechanical methods, for example by simply loading the part the same way service loading acts until local plastic deformationoccurs. Local surface deformation a such as shot peening or rolling are other mechanical methods frequently used in industrial applications. Cold rolling is the preferred method to improve the fatigue strength by axi-symmetric parts such as axles and crankshafts. Bolt threads formed by rolling are much more resistantto fatigue loading than cut threads. Shot peening and hammer peening have been shown to be highly effective methods for increasing the fatigue strength ofwelded joints.

Thermal processes produce a hardened surface layer with a high compressive stress, often of yield stress magnitude. The high hardness also produces a wearresistant surface; in many cases this may be the primary reason for performing the hardness treatment. Surface hardening can be accomplished by carburising,nitriding or induction hardening.

Since the magnitude of internal stresses is related to the yield stress their effect on fatigue performance is stronger the higher strength of the material.Improving the fatigue life of components or structures by introducing residual stresses is therefore normally only cost effective for higher strength materials.

Residual stresses have a similar influence on fatigue life as externally imposed mean stresses, ie. a tensile stress reduces fatigue life while a compressive stressincreases life. There is, however, an important difference which relates to the stability of residual stresses. While an externally imposed mean stress, eg. stresscaused by dead weight always acts (as long as the load is present), residual stress may relax with time, especially if there are high peaks in the load spectrumthat cause local yielding at stress concentrations.

6.8 Effects of Corrosion

Corrosion in fresh or salt water can have a very detrimental effect on the fatigue strength of engineering materials. Even distilled water may reduce the high-

cycle fatigue strength to less than two thirds of its value in dry air.

Figure 27 schematically shows typical S-N curves for the effect of corrosion on unnotched steel specimens. Precorrosion, prior to fatigue testing introducesnotch-like pits that act as stress raisers. The synergistic nature of corrosion fatigue is illustrated in the figure by the drastic lower fatigue strength which isobtained when corrosion and fatigue cycling act simultaneously. The strongest effect of corrosion is observed for unnotched specimens, the fatigue strengthreduction is much less for notched specimens, as shown in Figure 28.



Protection against corrosion can successfully be achieved by surface coatings, either by paint systems or through the use of metal coatings. Metal coating aredeposited either by galvanic or electrolytic deposition or by spraying. The preferred method for marine structures, however, is cathodic protection which isobtained by the use of sacrificial anodes or, more infrequently, by impressed current. The use of cathodic protection normally restores the high cycle fatiguestrength of welded structural steels to its in-air value, while at higher stresses hydrogen embrittlement effects may reduce the fatigue life by a factor of 3 to 4 onlife.

7. CYCLE COUNTING PROCEDURE FOR VARIABLE AMPLITUDE LOADING

In practice the pattern of the stress history with time at any particular detail is likely to be irregular and may indeed be random. A more realistic pattern ofloading would involve a sequence of loads of different magnitude producing a stress history perhaps as shown in Figure 29. The problem now arises as to whatis meant by a cycle and what is the corresponding stress range. A number of alternative methods of stress cycle counting have been proposed to overcome thisdifficulty. The methods most commonly adopted for use in connection with Codes and Standards are the 'reservoir' or the 'rainflow' method.



7.1 The Reservoir Method

The basis of the reservoir method is shown in Figure 30 using the stress time history as Figure 29. it should be assumed that a stress time history of this kindhas been obtained from strain gauges attached to the structure at the detail under consideration or has been estimated by computer simulation. It is importantthat the results analysed should be representative of long term behaviour. To analyse these results, a representative period is chosen so that the peak stress levelrepeats itself and a line is drawn to join the two peaks as shown in Figure 30a. The region between these two peaks is then regarded as being filled with water toform a reservoir. The procedure is then to take the lowest trough position and imaging that one opens a tap to drain the reservoir. Water drains out from thistrough T1 but remains tapped in adjacent troughs separated by intermediate peaks as shown in Figure 30b. The draining of the first trough T1 corresponds to

one cycle of stress range St as shown, and the remaining level of water is now lowered to the level of the next highest peak. A tap is now opened at the next

lowest trough T2 as shown in Figure 30c and the water allowed to drain out. The height of the water released by this operation corresponds to one cycle of

stress range S2. This procedure is continued sequentially through each next lowest trough, gradually building up a series of numbers of cycles of different stress

ranges. It is also essential to allow for the one cycle from zero to peak stress. For the particular stress time history shown in Figure 29 the results obtained fromthe sample time period taken would be:



1 cycle at 120N/mm2, 1 at 100N/mm2, 4 at 80N/mm2, 6 at 60N/mm2, 10 at 30N/mm2.

The important principle of the above procedure is the recognition that by taking the difference between the lowest and highest stress levels (trough and peak) itis ensured that the greatest possible stress range is counted first, and this procedure is repeated sequentially so that the highest ranges are identified as therandom fluctuations take place. In the assessment of the effects of the different cycles the greatest damage is caused by the higher stress ranges since the design

curves follow a relationship of the kind Sm N=constant. The reservoir method procedure does ensure that practical combinations of minima and maxima areconsidered together whereas this is not always the case in other stress cycle coating procedures.

An alternative way of carrying out the reservoir cycle counting method is to turn the diagram upside down and use the complementary part of the diagram asshown in Figure 31. This version of the reservoir method gives identical results to the normal method but has the advantage of including the major cycle ofstress from zero to maximum and back.

7.2 The 'Rainflow' Counting Method

The alternative 'rainflow' cycle counting procedure is illustrated in Figure 32a for the same stress time history of Figure 29. This is essentially the same pictureturned onto its side as shown in Figure 32a. Water (rain) is allowed to fall from the top onto the pattern considered as a roof structure and the paths followed bythe rain are followed. However it is important that a number of standard rules are followed and the procedure is rather more complex and subject to error thanthe reservoir method. For each leg of the roof an imaginary flow of water is introduced at its highest point as shown by the dots in Figure 32b. The flow ofwater is followed for the outermost starting point first, allowing the water to drop onto any parts of the roof below and continue to drain until it falls off the roofcompletely. The width from the stress level at which the water started until it left the roof represents the magnitude of one cycle of stress. It is necessary tofollow the flow paths from each starting point sequentially, moving progressively in from the points which are furthest out. If however the flow reaches a position where water has drained from a previous flow, it is terminated at that point as shown in Figure 32c for the flow starting from position 3 terminated bythe previous flow position 1. The stress range for a cycle terminated in this way is limited to the width between the starting point and the termination point. Thecomplete rainflow diagram for the stress pattern of Figure 29 is shown in Figure 32d. This procedure when correctly applied also counts the highest stress rangecycles first and ensures that only practical combinations of minima and maxima within a sequence are considered. The rainflow method is somewhat moredifficult to apply correctly than the reservoir method and it is recommended that both for teaching and for design purposes the reservoir method should be used.The results for the stress ranges from the rainflow method applied to the stress history from Figure 29 are identical to those from the reservoir method ie.



1 cycle at 120N/mm2, 1 at 100N/mm2, 4 at 80N/mm2, 6 at 60N/mm2, 10 at 30N/mm2.







There are two other cycle counting methods, the 'range pair counting' method and the 'mean crossing level' which are sometimes used although they tend not to be specified in Codes.

Example 1 This design example is based on the stress cycle history of Figure 29 as analysed above for stress cycle counting purposes. Firstly the stress historyrepresents a relatively short time period, and has to be extrapolated to represent the total required life. Obviously the first requirement is to ascertain therequired design life, and to multiply the numbers of cycles of each stress range determined as above by the ratio of the design life to the period represented bythe sample time record taken. For example, if the design life was 20 years, and the sample time period was 6 hours, the numbers of cycles should be multiplied by 20 x 365 x 4 = 29200. Caution should be exercised with such an extrapolation however, as to whether such a short length time sample is representative oflong term behaviour. For example in the case of a bridge structure the traffic flows are likely to vary at different times of day, peaking at rush hour times andfalling to low values in the middle of the night. Furthermore there is possibility that the heaviest loads may not have occurred during the sampling timeconsidered. Problems of extrapolation from samples to full data are common in the statistical world and statistical procedures may be necessary to ensure that potential differences in scaling up the data are allowed for. To a large extent this depends on the absolute size of the sample taken.

To check whether the design is satisfactory for any particular detail, it is necessary to decide on the appropriate design S-N design curve for the detail. The basis of doing this for Eurocode 3 will be explained in Lecture 12.9. For present purposes it will be assumed that the stress history of Figure 29 analysed above

applies to a detail for which the design S-N curve is S90, for which the design life is 2 x 106 cycles at stress range 90N/mm2 with slope - 1/3 continued down toa stress level of 66N/mm2 at design life 5 x 106 cycles, with a change in slope to -1/5 on down to a stress range of 36N/mm2 which is the fatigue limit at 10million cycles. For a twenty year design life assuming the stress history of Figure 29 is representative of 6 hours typical loading the following table can beconstructed:

Stress range

N/mm2

Cycles applied

n

Available cycles

N

n

N

120 29200 843750 0.0346

100 29200 1.458 x 106 0.0200

80 116800 2.848 x 106 0.0410

60 175200 8.053 x 106 0.0218

30 292000 below cut off 0

Σ n/N = 0.1174



For these assumptions the loading is acceptable for the detail and life required. Indeed the 'Damage Sum' value of 0.1174 based on a 20 year design life

indicates the available design life is 20/0.1174 = 170 years. For this particular case the stress range of 60N/mm2 fell in the intermediate range between 36 and

66N/mm2 and the available life N was calculated using the changed slope of the S-N curve for this region. The stress range of 30N/mm2 is below the cut off forthe S90 classification and does not contribute to the fatigue damage.

7.3 Exceedance Diagram Methods

A convenient way of summarising the fatigue loading applied to structures is by the use of exceedance diagrams. These diagrams present a summary of themagnitude of a particular event against the number of times this magnitude is exceeded. Whilst in principle this presentation can be applied to a wide variety of phenomena for the purposes of fatigue analyses the appropriate form is a graph of log (number of times exceeded) against the occurrence of different stresslevels. An example is shown in Figure 33. This might represent the stresses caused at a particular location in a bridge by traffic passing over r by wave loadingof an offshore structure. A typical feature of natural phenomena of this kind is that the number of exceedances increases as the stress level decreases. The form

of the exceedance diagram for natural phenomena of this kind is often close to linear as shown. It is important to note that the diagram represents exceedancesso that any particular point on the graph includes all of the numbers of cycles of stress range above that value. For use in fatigue analysis using Miner's law therequirement is a summary of the numbers of cycles of each stress level occurring. Thus the loading represented by the exceedance diagram of Figure 33 can betreated as an equivalent histogram with cycles as follows:

Stress range

N/mm2

No. of times exceeded Cycles occurring

180 1 1

160 10 9

140 100 90

120 1000 900

100 10000 9000

80 100000 90000

60 1000000 900000

40 10000000 9000000

20 100000000 90000000



Some of the stress ranges will be found to be below the fatigue limit and hence will not contribute to the Miners law damage sum. For example for the detail

considered in Example 1 above, the cut off limit was 36N/mm2 and the stress ranges of 20N/mm2 would not contribute to the fatigue damage. The stress ranges

above this level will contribute however and their effects must be included. This is done by finding the value of Σ Sm N separately for the remaining stress

levels above and below the change in slope of the S-N curve, and for the figures given above this will be found to be 5.692 x 1010 for stress ranges of 80N/mm2

and above, and 1.621 x 1015 for the 40 and 60N/mm2 stress ranges. For an S90 detail with the spectrum of loading shown above, the fatigue damage from each

part of the S-N curve has to be calculated based on the appropriate value of Sm N=constant as follows:

+ = 0.298

From these figures the damage sum factor calculated as 0.298 is acceptable. Detailed examination of the figures leading up to this result would indicate that the

majority of the damage calculated occurs at the lowest stress ranges of 40 and 60N/mm2 contributing to the S5 N part of the design curve.

7.4 Block Loading

Block loading is a particular case of an exceedance diagram.

Consider the particular case of a one lane bridge structure on which the loading is idealised as falling into three categories. Suppose that there are n1 heavy

lorries travelling across the bridge during its lifetime, and that at a particular welded detail each lorry causes a stress range S1. In addition there are n2 medium

lorries which cause a stress range S2, and n3 cars which cause a stress range S3 at the same welded detail as they cross the bridge. To assess the combined effect

of the different stress ranges all being applied in some form of sequence the procedure adopted is to assume that the damage caused by each individual group ofcycles of a given stress range is the same as would be caused under constant amplitude loading at that stress range. It is necessary first to decide on theappropriate classification for the geometric detail being considered and to identify the appropriate S-N design curve. For present purposes, let us assume thatthe design curve is as shown in Figure 34. If the only fatigue loading applied to the bridge was the crossing of the heavy lorries with stress range S1 at the detail

concerned, the available design life would be N1

cycles as shown in Figure 34. In fact the number of cycles applied at this stress range is n1

. It is assumed that

the fatigue damage caused at stress range St is n1/N1. Similarly if the only fatigue loading applied to the bridge was the crossing of the medium lorries with

stress range S2 the available design life would be N2 and the fatigue damage caused would be n2/N2. For the passage of the cars at stress range S3 the available

design life if this was the only loading would be N3 and the fatigue damage caused would be n3/N3. When all three loadings occur together the assumption for

design purposes is that the total fatigue damage is the sum of that occurring at each individual stress range independently. This is known as the Palmgren-Minerlaw of linear damage, or more simply as Miner's law and is summarised as follows:

+ + + .... + = 1 (11)

7.5 Frequency and Spectrum Aspects

It is not uncommon for loading to occur at more than one frequency. It is generally considered that for non aggressive environmental conditions, eg. steel in air,there is little or no effect of frequency on constant amplitude fatigue behaviour. In aggressive conditions however, eg. steel in seawater, there may besignificant effects of frequency on the crack growth mechanism leading to increased crack growth rates, shorter lives and reduction or elimination of the fatiguelimit. In particular it is necessary in fatigue testing of materials where environmental conditions may be important to carry out the testing at the same frequencyas that of the service loading. An example of this is the effect of wave loading on offshore structures where a typical frequency of waves is about 0.16Hz.Clearly this has major implications on the time required for testing since to accumulate one million cycles at 0.16Hz would take about 70 days whereas aconventional test in air at say 16Hz would reach the same life in less than 1 day. With any structure the response of the structure to dynamic loading depends on



the frequency or rate of the applied loading and on the vibration characteristics of the structure itself. It is most important for the designer to ensure that thenatural resonance frequencies of the structure are well separated from the frequencies of applied loading which may occur. Even so the structure may respondwith frequencies of stress fluctuation which are a combination of the applied loading frequency and its own natural vibration frequencies. Furthermore since themagnitude of the loading may also vary with time it is necessary to consider both time domain and frequency domain aspects. Figure 35 shows a typicalfrequency domain response for stress fluctuations at a particular location in an offshore structure. This diagram gives information on number of times differentstress levels are exceeded as well as the frequency data. The peaks at about 0.16Hz correspond to the applied loading whereas the higher frequency peaks arethose due to the vibration response of the structure.

With variable amplitude fatigue loading of this kind there are additional complexities with regard to frequency effects to be considered. Where the stressing

occurs close to or at a single frequency the condition is known as 'narrow band' and when there are a range of different frequencies involved it is known as'broad band'. If the frequency domain response of Figure 35 is converted back into the time domain response in which the data was originally recorded theresult would look like Figure 36. Clearly some assumptions must have been in the conversion of one diagram into the other and in this case it is that stress cyclecounting has been carried out by the reservoir method. In Figure 36 however, it is clear that because the higher frequency stress cycles are superimposed on topof the lower frequency cycles, some of the higher frequency cycles occur at higher mean stress or stress ratio.

8. CONCLUDING SUMMARY

In this lecture it has been shown that fatigue is a weakest link process of a statistical nature in which a crack will initiate at a location where stress, localand global geometry, defects and material properties combine to give a worst case situation. The crack thus nucleates at a local peak spot, and may causefailure of the structure, even if the rest of the structure has a high fatigue resistance. Good fatigue design practice is therefore based on close attention todetails that increases the stress locally and therefore are potentially initiation sites for fatigue cracks.

A positive aspect of the local nature of the fatigue process is that only a relatively small area of highly stressed material need to be improved in order to



increase the load carrying capacity of the structure when fatigue is the limiting design criterion. Another general conclusion is that increasing the size of a structure generally leads to a lower strength with respect to brittle fracture as well as fatigue.

Size effects must therefore be properly accounted for. The larger number of factors influencing fatigue strength makes the combined effects of these factors very difficult to predict. The safest way to obtain

design data is therefore still to perform fatigue tests on prototype components with realistic environmental conditions. A normal structural design analysis must be carried out for the maximum design loads and for a series of intermediate loads with known number of

occurrences in the design life to give stress results at typical details. Alternatively if the application Code gives an equivalent constant amplitude loadingcondition and associated number of cycles this loading should be applied and stresses determined. The stresses should be analysed for range of variationin principal stress or of direct stress aligned perpendicular or parallel to the geometric detail as defined in Eurocode 3. Treatments for shear stresses aregiven in Eurocode 3. The stress ranges should be multiplied by appropriate partial factors, and for variable amplitude loading either combined together togive an equivalent constant amplitude stress range and number of cycles or used to sum up fatigue damage.

The correct detail classification must be identified for typical critical details and the applied fatigue damage for the design life checked against the designS-N curve for the detail concerned. If the design is not satisfactory either the stress ranges must be reduced or the detail changed until satisfactory resultsare obtained.

9. REFERENCES

1. Metals Handbook, ASM 1985.2. ISO Standard, 373 - 1964.3. P.C. Paris and F. Erdogan, "A Critical Analysis of Crack Propagation Laws", Trans, ASME, Vol. 85, No. 4, 1963.4. J.M. Barsom, "Fatigue Crack Propagation", Trans, ASME, SEr. B, No.4, 1971.5. H. Neuber, "Kerbspannungslehre", Springer, 1958.6. R.E. Peterson, "Stress Concentration Factors", John Wiley & Sons, 1974.7. O. Ørjasæter et al, "Effect of Plate Thickness on the Fatigue Properties of a Low Carbon Micro-Alloyed Steel", Proc. 3rd Int. ECSC Conf. on Steel in

Marine Structures (SIMS'87), Delft, 15-18 June 1987.8. P. J. Haagensen, "Size Effects in Fatigue of Non-Welded Components", Proc. 9th Int, Conf. on Offshore Mechanics and Arctic Engineering, (OMAE),

Houston, Texas, 18-23 February 1990.

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