Lecture 12.1: Basic Introduction to FatigueOBJECTIVE/SCOPE:To
summarize the main factors affecting fatigue strength, as opposed
to static strength, of welded joints and to illustrate the method
of carrying out a fatigue check.PREREQUISITESLecture 11.1.2:
Introduction to Connection DesignLecture 11.2.1: Generalities on
Welded ConnectionsRELATED LECTURESLecture 12.12: Determination of
Stress Intensity FactorsLecture 12.13: Fracture Mechanics Applied
to FatigueSUMMARYThis lecture gives an explanation of the mechanism
of fatigue and the influence of welding on that mechanism. It
summaries the primary factors affecting fatigue strength and
introduces S-N Curves. The classification of fatigue details is
presented and important details reviewed. The calculation of stress
range is summarised. The principal types of fatigue loading and the
bases for their calculation are presented with an introduction to
cycle counting and damage calculations for mixed amplitude
loading.NOTATIONa design weld strength parameterDsRstress
rangeDsDnon-propagating stress, i.e. the constant amplitude stress
range below which cracks will not growN endurance number of
cycles.1. INTRODUCTION1.1 Nature of FatigueFatigue is the mechanism
whereby cracks grow in a structure. Growth only occurs under
fluctuating stress. Final failure generally occurs in regions of
tensile stress when the reduced cross-section becomes insufficient
to carry the peak load without rupture. Whilst the loading on the
structure is stationary the crack does not grow under normal
service temperatures. Many structures, such as building frames, do
not experience sufficient fluctuating stress to give rise to
fatigue problems. Others do, such as bridges, cranes, and offshore
structures, where the live loading is a higher proportion of the
total load.1.2 How Welds FatigueIn welded steel structures, fatigue
cracks will almost certainly start to grow from welds, rather than
other details, because: Most welding processes leave minute
metallurgical discontinuities from which cracks may grow. As a
result, the initiation period, which is normally needed to start a
crack in plain wrought material, is either very short or
no-existent. Cracks therefore spend most of their life propagating,
i.e. getting longer. Most structural welds have a rough profile.
Sharp changes of direction generally occur at the toes of butt
welds and at the toes and roots of fillet welds, see Figure 1.
These points cause local stress concentrations of the type shown in
Figure 2. Small discontinuities close to these points will
therefore react as though they are in a more highly stressed member
and grow faster.
1.3 Crack Growth HistoryThe study of fracture mechanisms shows
that the growth rate of a crack is proportional to the square root
of its length, given the same stress fluctuation and degree of
stress concentration. For this reason fatigue cracks spend most of
their life as very small cracks which are hard to detect. Only in
the last stages of life does the crack start to cause a significant
loss of cross-section area, as shown in Figure 3. This behaviour
poses problems for in-service inspection of structures.
2. FATIGUE STRENGTH2.1 Definition of Fatigue Strength and
Fatigue LifeThe fatigue strength of a welded component is defined
as the stress range (R) which fluctuating at constant amplitude,
causes failure of the component after a specified number of cycles
(N). The stress range is the difference between the maximum and
minimum points in the cycle, see Figure 4. The number of cycles to
failure is known as the endurance or fatigue life.
2.2 Primary Factors Affecting Fatigue LifeFor practical design
purposes there are two main factors which affect the fatigue life
of a detail, namely: The stress range (R) at the location of crack
initiation. There are special rules for calculating this range. The
fatigue strength of the detail. This strength is primarily a
function of the geometry and is defined by the parameter 'a', which
varies from joint to joint.The fatigue life (N), or endurance, in
number of cycles to failure can be calculated from the
expression:
where m is a constant, which for most welded details is equal to
3. Predictions of life are therefore particularly sensitive to
accuracy of stress prediction.2.3 S-N CurveThe expression linking N
andRmcan be plotted on a logarithmic scale as a straight line,
Equation (2), and is referred to as an S-N curve. An example is
shown in Figure 5. The relationship holds for a wide range of
endurance. It is limited at the low endurance end by static failure
when the ultimate material strength is exceeded. At endurances
exceeding about 5-10 million cycles the stress ranges are generally
too small to permit propagation under constant amplitude loading.
This limit is called the non-propagating stress (D). Below this
stress range cracks will not grow.
For design purposes it is usual to use design S-N curves which
give fatigue strengths about 25% below the mean failure values, as
shown in Figure 5. 'a' is used to define these lines.2.4 Effect of
Mean StressIn non-welded details the endurance is reduced as the
mean stress becomes more tensile. In welded details the endurance
is not usually reduced in those circumstances. This behaviour
occurs because the weld shrinkage stresses (or residual stresses),
which are locked into the weld regions at fabrication, often attain
tensile yield. The crack cannot distinguish between applied and
residual stress. Thus, for the purposes of design, the S-N curve
always assumes the worst, i.e. that the maximum stress in the cycle
is at yield point in tension. It is particularly important to
appreciate this point as it means that fatigue cracks can grow in
parts of members which are nominally 'in compression'.2.5 Effect of
Mechanical StrengthThe rate of crack growth is not significantly
affected by variations in proof stress or ultimate tensile strength
within the range of low alloy steels used for general structural
purposes. These properties only affect the initiation period,
which, being negligible in welds, results in little influence on
fatigue life. This behaviour contrasts with the fatigue of
non-welded details where increased mechanical strength generally
results in improved fatigue strength, as shown in Figure 6.
3. CLASSIFICATION OF DETAILS3.1 Detail ClassesThe fatigue
strength parameter (K2) of different welded details varies
according to the severity of the stress concentration effect. As
there are a wide variety of detail in common use, details with
similar K2values are grouped together into a single detail class
and given a single K2value.This data has been obtained from
constant amplitude fatigue tests on simple specimens containing
different welded detail types. For the most commonly used details,
it has been found convenient to divide the results into fourteen
main classes. The classes are:
As shown in Figure 7, these classes can be plotted as a family
of S-N curves. The difference in stress range between neighbouring
curves is usually between 15 and 20%.
The above table has been taken from Eurocode 3 [1]. It does not
include S-N data for unstiffened hollow tubular joints.3.2 Detail
TypesThere are usually a number of detail types within each class.
Each type has a very specific description which defines the
geometry both microscopically and macroscopically. The main
features that affect the detail type, and hence its classification,
are: Form of the member:e.g. plate, rolled section, reinforcing
bar. Location of anticipated crack initiation:The location must be
defined with respect to the direction of stress fluctuation. A
given structural joint may contain more than one potential
initiation site, in which case the joint may fall into two or more
detail types. Leading dimensions:e.g. weld shape, size of
component, proximity of edges, abruptness of change of
cross-section. Fabrication requirements:e.g. type of weld process,
any grinding smooth of particular parts of the joint. Inspection
requirements:Special inspection procedures may be required on
higher class details to ensure that detrimental welding defects are
not present.It should be noted that if fatigue is critical in the
design, the extra controls on fabrication incurred by the last two
requirements may increase the total cost significantly above that
for purely static strength.Examples of different types of welded
detail and their classes are shown in Eurocode 3: Part 1.1 [1].3.3
Commonly Used Detail TypesFigure 8 shows some of the most important
details to look out for in welded steelwork.
They are: Load carrying fillet welds and partial penetration
butt welds. These details are category 36 for failure starting at
the root and propagating through the throat. Welded attachments on
edges. They are category 45. Note that the attachment weld may not
be transferring any stress. Failure is from the weld toe into the
member. Ends of long flat plates, e.g. cover plates are category
50. Most short attachments in the stress direction are category 80
or 71 as long as they are not at an edge. Transverse full
penetration butt welds can range from category 12,5 to 36 depending
on how they are made. Long continuous welds on site welded
structures are found to be category 100.It should be borne in mind
that most potential fatigue sites on welded structures are found to
be category 80 or below.4. STRESS PARAMETERS FOR FATIGUE4.1 Stress
AreaThe stress areas are essentially similar to those used for
static design. For a crack starting at a weld toe, the
cross-section of the member through which propagation occurs is
used. For a crack starting at the root, and propagating through the
weld throat, the minimum throat area is used, as shown in Figure
8a.4.2 Calculation of Stress RangeThe force fluctuation in the
structure must be calculated elastically. No plastic redistribution
is permitted.The stress on the critical cross-section is the
principal stress at the position of the weld toe (in the case of
weld toes cracks). Simple elastic theory is used assuming plane
sections remain plane, see Figure 9. The effect of the local stress
concentration caused by the weld profile is ignored as this is
already catered for by the parameter 'd' which determines the weld
class.
In the case of throat failures, the vector sum of the stresses
on the weld throat at the position of highest vector stress along
the weld is used, as in static design.Exceptions to these rules
occur in the case of unstiffened joints between slender members
such as tubes. In this case the stress parameter is the Hot Spot
Stress. This stress is calculated at the point of expected crack
initiation, taking into account the true elastic deformation in the
joint, i.e. not assuming plane sections to remain plane.4.3 Effects
of Geometrical Stress Concentrations and Other EffectsWhere a
member has large changes in cross-section, e.g. at access holes,
there will be regions of stress concentration due to the change of
geometry. In static design the stresses are based on the net area
as plastic redistribution will normally reduce these peaks at
ultimate load. With fatigue this is not so, and if there is a
welded detail in the area of the geometrical stress raiser the true
stress must be used, as shown in Figure 10.
4.4 Secondary EffectsSimilarly any secondary effects, such as
those due to joint fixity in latticed structures, and shear lag and
other distortional effects in slender beams, are allowed for in
calculating the stresses.5. LOADINGS FOR FATIGUE5.1 Types of
LoadingExamples of structures and the loads which can cause fatigue
are:Bridges: Commercial vehicles, goods trainsCranes: Lifting,
rolling and inertial loadsOffshore structures: WavesSlender
chimneys: Wind gustingThe designer's objective is to anticipate the
sequence of service loading throughout the structure's life. The
magnitude of the peak load, which is vital for static design
purposes, is generally of little concern as it only represents one
cycle in millions. For example, highway bridge girders may
experience 100 million significant cycles in their lifetime. The
sequence is important because it affects the stress range,
particularly if the structure is loaded by more than one
independent load system.For convenience, loadings are usually
simplified into a load spectrum, which defines a series of bands of
constant load levels, and the number of times that each band is
experienced, as shown in Figure 11.
Slender structures, with natural frequencies low enough to
respond to the loading frequency, may suffer dynamic magnification
of stress. This magnification can shorten the life considerably.A
useful source of information on fatigue loading is Eurocode 1
[2].5.2 Cycle CountingIn practice most stress histories in real
structures are of the variable amplitude type, shown in Figure 12,
as opposed to the constant amplitude shown in Figure 4. Such
histories pose a problem in defining the number and amplitude of
the cycles.
The first step is to break the sequence into a stress spectrum
as shown in Figure 12 using a cycle counting method. There are
various methods in use. The two most used are the Rainflow Method
and the Reservoir Method. The latter, which is easy to use by hand
for short stress histories, is described inLecture 12.2. The former
is more convenient for analysing long stress histories using a
computer.6. CALCULATION OF DAMAGEUnder variable amplitude loading
the life is estimated by calculation of the total damage done by
each cycle in the stress spectrum. In practice the spectrum is
simplified into a manageable number of bands, as shown in Figure
13.
The damage done by each band in the spectrum is defined aswhere
n is the required number of cycles in the band during the design
life and N is the endurance under that stress range, see Figure
14.
If failure is to be prevented before the end of the specified
design life, the Palmgren-Miner's Rule must be compiled with. This
rule states that the damage done by all bands together must not
exceed unity, i.e.:
It should be noted that, when variable amplitude loading occurs,
the bands in the spectrum withvalues less thanDmay still cause
damage. Damage occurs because the larger amplitude cycles may start
to propagate the crack. Once it starts to grow lower cycles become
effective. In this case, the horizontal constant amplitude fatigue
limitDshown in Figure 5, is replaced by a sloping line with a log
gradient of.7. CONCLUDING SUMMARY Fatigue and static failure
(whether by rupture or buckling) are dependent on very different
factors, namely:- Fatigue depends on the whole service loading
sequence (not one extreme load event).- Fatigue of welds is not
improved by better mechanical properties.- Fatigue is very
sensitive to the geometry of details.- Fatigue requires more
accurate prediction of elastic stress.- Fatigue makes more demands
on workmanship and inspection. It is therefore important to check
early in the design whether fatigue is likely to be critical.
Acceptable margins of safety against static collapse cannot be
relied upon to give adequate safety against fatigue. Areas with a
high live/dead stress ratio and low category 36 details should be
checked first. The check must cover any welded attachment to a
member, however insignificant, and not just the main structural
connections. Note that this check should include welded additions
to the structure in service. If fatigue is critical, then the
choice of details will be limited. Simplicity of detail and
smoothness of stress path should be sought. Be prepared for fatigue
critical structures to cost more.8. REFERENCES[1] Eurocode 3:
"Design of Steel Structures": ENV1993-1-1: Part 1.1: General Rules
and Rules for Buildings, CEN 1992.[2] Eurocode 1: "Basis of Design
and Actions on Structures", CEN (in preparation).9. ADDITIONAL
READING1. Maddox, S.J. "Fatigue Strength of Welded Structures",
Cambridge, Abington Publishing, 1991.2. Gurney, T. R., "Fatigue of
Welded Structures", 2nd ed., Cambridge University Press, 1991.3.
Narayanan, R. (ed), "Structures Subjected to Repeated Loading",
London, Elsevier Applied Science, 1991.
Lecture 12.2: Advanced Introduction toFatigueOBJECTIVE/SCOPE:To
introduce the main concepts and definitions regarding the fatigue
process and to identify the main factors that influence the fatigue
performance of materials, components and
structures.PREREQUISITESLecture 12.1: Basic Introduction to
FatigueRELATED LECTURESSUMMARYThe physical process of the
initiation of fatigue cracks in smooth and notched test specimens
under the influence of repeated loads is described and the
relevance 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 are described.1. INTRODUCTIONFatigue is
commonly referred to as a process in which damage is accumulated in
a material undergoing fluctuating loading, eventually resulting in
failure even if 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 as metallic alloys, polymers
and composites, eg. concrete and fibre reinforced plastics.
Although the phenomenological details of the process may differ
from one material to another the following definition given by ASTM
[1] encompasses fatigue failures in all materials:Fatigue - the
process ofprogressive localisedpermanent structural change
occurring in a material subjected to conditions that produce
fluctuating stresses 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 a crack 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 most case 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
the components, with only slightly lower stresses, no fatigue
damage may occur and these parts thus have an infinite fatigue
life. The initiation process usually results in a number of
micro-cracks that may grow more or less independently until one
crack becomes dominant through a coalescence process at the
microcracks start to interact. Under steady fatigue loading this
crack grows slowly, but starts to accelerate when the reduction of
the cross-section increases the local 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
the fatigue process can in many cases be related to distinctive
features of the fracture surface of components that have failed
under fluctuating loads, the presence of these features can
therefore be used to identify fatigue as the probable cause of
failure.2. CHARACTERISTICS OF FATIGUE FRACTURE SURFACESTypical
fracture surfaces in mechanical components that were subjected to
fatigue loads are shown in Slide 1. One characteristic feature of
the surface morphology which is evident in both macrographs is the
flat, smooth region of the surface exhibiting beach marks (also
called clamshell marks). This part represents the portion of the
fracture surface over which the crack grew in a stable, slow mode.
The rougher regions, showing evidence of large plastic deformation,
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 other cases 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
concentration fatigue 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. While the presence of any
defects at the origin may indicate the cause of the fatigue
failure, the crack propagation area may yield some information
regarding the magnitude 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 fracture regions 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 a high maximum load, whereas a small area
indicates that the load was lower at fracture. Similarly, for a
fixed maximum stress, the relative area corresponding to slow 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 a changing corrosive environment. Beach
marks are therefore not observed in the surfaces of fatigue
specimens tested under constant amplitude loading conditions
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 clear that 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 microscope at
magnifications between 1,000x and 30,000x may reveal characteristic
surface ripples called fatigue striations, see Slide 2. Although
somewhat similar in appearance, these lines are not the beach marks
described above as one beach mark may contain thousands of
striations. During constant amplitude fatigue loading 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 these conditions the crack does
not advance simultaneously along the crack front, growth occurring
instead only along some portions during a few cycles, then arrests
while growth occurs along other segments. Striations as shown in
Figure 3 are not seen if the crack grows by other mechanisms such
as microvoid coalescence or, in brittle materials, microclevage. In
structural steels the crack can propagate by all three mechanism,
and striations may be difficult to observe. Slide 3 shows 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 PROCESSFrom the description of the
characteristics of fatigue fracture surfaces, three stages in the
fatigue process may be identified:Stage I: Crack initiationStage
II: Propagation of one dominant crackStage III: Final
fractureFatigue cracking in metals is always associated with the
accumulation of irreversible plastic strain. The crack process
which is discussed in the following applies 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 the amount 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 45to the tensile stress direction. The results
of such deformation is atomic planes sliding relative to each
other, resulting in a roughening of the surface in slip bands.
During further cycling 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). The name originated from
the observation in early studies of fatigue that slip band would
reappear - "persist" - at the same location after a thin surface
layer was removed 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 process
small cracks develop in the intrusions. These microcracks grow
along slip planes, ie. a shear stress driven process. Growth in the
shear mode, called stage I crack growth extends over a few grains.
During continued cycling the microcracks in different grains
coalesce resulting in one or a few dominating cracks. The stress
field associated with the dominating crack cause further growth
under the primary action of maximum principal stress; this is
called stage II growth. The crack path is now essentially
perpendicular to the tensile stress axis. Crack advancement is,
however, still influenced by the crystallographic orientation of
the grains 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 become weaker 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 fatigue life, 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
higher fatigue 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 resistant to slip deformation, the
interface between the matrix and inclusion may be relatively weak,
and cracks will start here if decohesion occurs at the inclusion
surface, 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 rather imprecise 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 the crack compared with microstructural dimensions such as
grain size invalidates a fracture mechanics based analysis of this
growth phase. Instead, local stresses and 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 LOADINGThe simplest form of stress
spectrum to which a structural element may be subjected is a
sinusoidal or constant amplitude stress-time history with a
constant mean 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 most fatigue tests. The
following six parameters are used to define a constant amplitude
stress cycle:
Smax = maximum stress in the cycleSmin = minimum stress in the
cycleSm= mean stress in the cycle = (Smax+ Smin)/2Sa= stress
amplitude = (SmaxSmin)/2AS = stress range = Smax- Smin= 2SaR =
stress ratio = Smin/SmaxThe stress cycle is uniquely defined by any
two of these quantities, except combinations of stress range and
stress amplitude. Various stress patterns are shown in 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 of the 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 with
constant 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 high
temperatures 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 stress cycles 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 is called 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 main load with superimposed smaller excursions
that could be caused by eg. second order vibrations or by
electronic noise in the load acquisition system. In case of true
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 damage accumulation. 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
stress cycles 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 DATAThe total fatigue life in terms of cycles to
failure can be expressed as:Nt = Ni+ Np (1)where Niand Npare 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
assessment of 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 CurvesFatigue data for components whose lives consist of
an initiation phase followed by crack propagation are usually
presented in the form of S-N curves, where applied 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 105to 107cycle 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 105cycles.
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 same stress level, as
illustrated in Figure 9. Plotting the data for a given stress level
along a logarithmic endurance axis gives a distribution which can
be approximated by the Gaussian (or normal) distribution, hence
endurance data are said to have a log normal distribution.
Alternatively the Weibull distribution may 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 testing at 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 any desired 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
schematically in 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 asK is reduced towards a limit or
threshold,Kth, 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 useful design data.5.2 Fatigue TestingThe basis for any
design methodology aimed at preventing fatigue failures is data
characterising the fatigue strength of components and structures.
Fatigue testing 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 service load spectrum of
that structure. However, life estimates are required before the
design is finalised or details of the loading history are known.
Additionally, each structure will experience a particular load
history that is unique for that structure, so many simplifications
and assumptions need to be made regarding the test stress 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 the data. 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
components and structures requires additional knowledge of
influencing factors related to the geometry, size, surface
condition and corrosive environment. S-N tests of components are
also normally standardised tests that make life predictions more
accurate compared with the three other tests because the
uncertainties regarding the influence of notches and surface
conditions are reduced. Service loading or variable amplitude
testing normally requires a knowledge of the response 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
hydraulic grips that facilitate the insertion and removal of
specimens. A schematic diagram of such a testing machine is shown
in Figure 12. These machines are capable of 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 DataAmong the first systematic
fatigue investigations reported in the literature are those set up
and conducted by the German railway engineer, August Whler, 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 Whler's original data are shown
in Figure 13. These data are presented in what is now well known as
Whler or S-N diagrams. Such diagrams are still commonly used in the
presentation of fatigue data, although the stress axis is often on
a logarithmic scale in contrast to Whler's linear stress axis.
Basquin's equation is often fitted to test data, it has the
form:SaNb = constant (2)where Sais 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 in Figure 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 LIFEThe difference in
fatigue behaviour of full scale machine or structural components as
compared with small laboratory specimens of the same material is
sometimes striking. In the majority of cases the real life
component exhibits a considerably poorer fatigue performance than
the laboratory specimen although the computed stresses are the
same. This difference in fatigue response can be examined in a
systematic manner by evaluating the various factors that influence
fatigue strength. Qualitative and quantitative assessments of these
effects are presented in the following paragraphs.6.1 Material
EffectsEffect of static strength on basic S-N dataFor 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 106to 107cycles (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 Soto the ultimate tensile strength Suis 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 strength materials. 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
are quickly 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
therefore almost independent of the tensile strength. An important
example is welded joints which always contain small crack-like
defects from which crack start growing after a very short
initiation period. Consequently the fatigue design stresses in
current design rules for welded joints are independent of the
ultimate tensile strength.
Crack Growth DataFatigue crack growth rates seem to be much less
dependent on static strength properties than crack initiation, at
least within a given alloy system. In a comparison 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 that
grouping the steels according to microstructure would minimise
scatter. His data for ferritic-pearlitic, matensitic and austenitic
are shown in Figure 17. Also shown 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 alloy systems 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 an aluminium plate. Thus, a rough assessment of the
fatigue strength of an aluminium component whose life is dominated
by crack growth can be obtained by dividing the fatigue strength of
a similarly shaped steel component by three.
6.2 Mean Stress EffectsIn 1870 Whler identified the stress
amplitude as the primary loading variable in fatigue testing;
however, the static or mean stress also affects fatigue life as
shown 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 = constant gives 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/ Sois
plotted against Sm/ Su. Sois the fatigue strength at a given life
under fully reversed (Sm= 0,R = -1) conditions. Suis the ultimate
tensile strength. The data points thus represent combinations of
Saand Smgiving 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 the following
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 of the data, while the Soderberg
line is a relatively conservative lower bound that is sometimes
used in design. These expressions should be used with care in
design 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 EffectsFatigue 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 the overall
deformation, cracks may start growing here and eventually result in
fracture of the part. It is therefore necessary to calculate the
local stress and relate this 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 Ktfactor. 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 cycle region, 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 strength reduction in fatigues
is less than that predicted by the stress concentration factor.
Instead the fatigue notch factor Kfis used to evaluate the effect
of notches in fatigue. Kfis defined as the unnotched to notched
fatigue strength, obtained in fatigue tests:
Kf = (8)From Figure 22 it is evident that Kfvaries with fatigue
life, however, Kfis commonly defined as the ratio between the
fatigue limits. With this definition Kfis less than Kt, the stress
increase due to the notch is therefore not fully effective in
fatigue. The difference between Kfand Ktarise 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 at the 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 stressed region 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 Kfis related to
Kt. 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
wheremax,effis the effective maximum stress, see Figure 23. This
definition of Kfprovides a scale for q that ranges from zero to
unity. When q = 0, Kf= Kt= 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 annealed copper,
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 values of 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 incorporates a notch
size effect.
The fatigue notch factor applies to the high cycle range, at
shorter lives Kfapproaches 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 applied only
to the alternating part of the stress cycle and not to the mean
stress. For brittle materials, however, Kfshould be applied to the
mean stress as well.6.4 Size EffectsAlthough 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. KfKt, 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 apparent
limit.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
optimum conditions 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 will
experience 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 with other 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 weld toe 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 is determined by the welding process and is therefore
essentially constant for different size joints. The t/r ratio
therefore increases and also the local stress when the 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 the experimentally determined size effects for these
components are among the strongest recorded. An example of size
effects for welded joints is shown in Figure 25. The solid line
represents current design practice, according to eg. Eurocode 3 and
the UK Department of Energy Guidance Notes. The equation for this
line is 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 thickness correction 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
concentration joints 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 the following size
reduction factor has been proposed to account for the larger stress
gradient found in notched specimens [8].n = 0.10 + 0.15 log Kt
(11)6.5 Effects of Surface FinishAlmost all fatigue cracks nucleate
at the surface since slip occurs easier here than in the interior.
Additionally, simple fracture mechanics considerations show that
surface defects and notches are much more damaging than internal
defects of similar size. The physical condition and stress
situation at the surface is therefore of prime importance for the
fatigue performance. One of the important variables influencing the
fatigue strength, the surface finish, commonly characterised by Ru,
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 standard specimens. The surface reduction factor
Cris the defined as the ratio between the two fatigue limits. Since
steels become increasingly more notch sensitive with higher
strength, the surface factor Crdecreases with increasing tensile
strength, Su.
6.6 Residual Stress EffectsResidual stresses or internal
stresses are produced when a region of a part is strained beyond
the elastic limit while other regions are elastically deformed.
When the force or deformation causing the deformation are removed,
the elastically deformed material springs back and impose residual
stresses in the plastically deformed material. Yielding can be
caused by thermal expansion as well as by external force. The
residual stresses are of the opposite sign to the initially applied
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 during fabrication. These
stresses are of yield stress magnitude and tensile and compressive
stresses must always balance each other, as indicated in Figure 26.
The high 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 deformation occurs. 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 resistant to fatigue loading than cut threads. Shot peening
and hammer peening have been shown to be highly effective methods
for increasing the fatigue strength of welded joints.Thermal
processes produce a hardened surface layer with a high compressive
stress, often of yield stress magnitude. The high hardness also
produces a wear resistant 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 stress increases life.
There is, however, an important difference which relates to the
stability of residual stresses. While an externally imposed mean
stress, eg. stress caused 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 spectrum that cause
local yielding at stress concentrations.6.8 Effects of
CorrosionCorrosion 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 introduces notch-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 is obtained when
corrosion and fatigue cycling act simultaneously. The strongest
effect of corrosion is observed for unnotched specimens, the
fatigue strength reduction 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 are deposited either by galvanic or
electrolytic deposition or by spraying. The preferred method for
marine structures, however, is cathodic protection which is
obtained by the use of sacrificial anodes or, more infrequently, by
impressed current. The use of cathodic protection normally restores
the high cycle fatigue strength 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 on
life.7. CYCLE COUNTING PROCEDURE FOR VARIABLE AMPLITUDE LOADINGIn
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 of loading 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 what is
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 this difficulty. The methods most commonly
adopted for use in connection with Codes and Standards are the
'reservoir' or the 'rainflow' method.
7.1 The Reservoir MethodThe 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 kind has been
obtained from strain gauges attached to the structure at the detail
under consideration or has been estimated by computer simulation.
It is important that the results analysed should be representative
of long term behaviour. To analyse these results, a representative
period is chosen so that the peak stress level repeats 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 to form 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 this trough T1but remains
tapped in adjacent troughs separated by intermediate peaks as shown
in Figure 30b. The draining of the first trough T1corresponds to
one cycle of stress range Stas 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 T2as 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 from the 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) it is ensured that the
greatest possible stress range is counted first, and this procedure
is repeated sequentially so that the highest ranges are identified
as the random 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 SmN=constant. The reservoir method
procedure does ensure that practical combinations of minima and
maxima are considered 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
as shown 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 of stress from zero to maximum and
back.
7.2 The 'Rainflow' Counting MethodThe alternative 'rainflow'
cycle counting procedure is illustrated in Figure 32a for the same
stress time history of Figure 29. This is essentially the same
picture turned 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 by the 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 than the 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 of water 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 roof completely. 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 to follow 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 by the 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.
The complete rainflow diagram for the stress pattern of Figure 29
is shown in Figure 32d. This procedure when correctly applied also
counts the highest stress range cycles first and ensures that only
practical combinations of minima and maxima within a sequence are
considered. The rainflow method is somewhat more difficult 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 1This
design example is based on the stress cycle history of Figure 29 as
analysed above for stress cycle counting purposes. Firstly the
stress history represents a relatively short time period, and has
to be extrapolated to represent the total required life. Obviously
the first requirement is to ascertain the required 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
by the 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 of long
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 and falling to low values in the middle of the
night. Furthermore there is possibility that the heaviest loads may
not have occurred during the sampling time considered. 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 inLecture 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 106cycles at stress range
90N/mm2with slope - 1/3 continued down to a stress level of
66N/mm2at design life 5 x 106cycles, with a change in slope to -1/5
on down to a stress range of 36N/mm2which is the fatigue limit at
10 million 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 be constructed:
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/mm2fell in the intermediate range between 36 and 66N/mm2and the
available life N was calculated using the changed slope of the S-N
curve for this region. The stress range of 30N/mm2is below the cut
off for the S90 classification and does not contribute to the
fatigue damage.7.3 Exceedance Diagram MethodsA convenient way of
summarising the fatigue loading applied to structures is by the use
of exceedance diagrams. These diagrams present a summary of the
magnitude 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 stress levels. 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 loading of 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
exceedances so 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 the requirement 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 be
treated as an equivalent histogram with cycles as follows:
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/mm2and the stress ranges of 20N/mm2would 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 ofSmN 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 1010for stress ranges of 80N/mm2and above, and 1.621 x
1015for the 40 and 60N/mm2stress 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 SmN=constant as follows:+ = 0.298From 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/mm2contributing to the S5N part of the
design curve.7.4 Block LoadingBlock 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 n1heavy 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 n2medium lorries which cause a stress range S2, and
n3cars which cause a stress range S3at 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 of cycles 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 the appropriate
classification for the geometric detail being considered and to
identify the appropriate S-N design curve. For present purposes,
let us assume that the 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 S1at the detail concerned, the
available design life would be N1cycles 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 Stis n1/N1.
Similarly if the only fatigue loading applied to the bridge was the
crossing of the medium lorries with stress range S2the available
design life would be N2and the fatigue damage caused would be
n2/N2. For the passage of the cars at stress range S3the available
design life if this was the only loading would be N3and 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-Miner law
of linear damage, or more simply as Miner's law and is summarised
as follows:
+ + + .... + = 1 (11)7.5 Frequency and Spectrum AspectsIt 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 be significant
effects of frequency on the crack growth mechanism leading to
increased crack growth rates, shorter lives and reduction or
elimination of the fatigue limit. In particular it is necessary in
fatigue testing of materials where environmental conditions may be
important to carry out the testing at the same frequency as 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 a conventional 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 the natural resonance
frequencies of the structure are well separated from the
frequencies of applied loading which may occur. Even so the
structure may respond with frequencies of stress fluctuation which
are a combination of the applied loading frequency and its own
natural vibration frequencies. Furthermore since the magnitude of
the loading may also vary with time it is necessary to consider
both time domain and frequency domain aspects. Figure 35 shows a
typical frequency domain response for stress fluctuations at a
particular location in an offshore structure. This diagram gives
information on number of times different stress 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 are those
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 the result 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 cycle counting 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 top of 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, local and global
geometry, defects and material properties combine to give a worst
case situation. The crack thus nucleates at a local peak spot, and
may cause failure 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 to details 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 loading condition and
associated number of cycles this loading should be applied and
stresses determined. The stresses should be analysed for range of
variation in principal stress or of direct stress aligned
perpendicular or parallel to the geometric detail as defined in
Eurocode 3. Treatments for shear stresses are given in Eurocode 3.
The stress ranges should be multiplied by appropriate partial
factors, and for variable amplitude loading either combined
together to give 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 design S-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 results are obtained.9.
REFERENCES1. 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. rjaster 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.
Lecture 12.3: Effect of Workmanship onFatigue Strength of
Longitudinal andTransverse WeldsOBJECTIVE/SCOPEIdentification of
factors influencing the fatigue strength of welded joints and of
the consequences for design, fabrication and
inspection.PREREQUISITESLecture 12.1: Basic Introduction to
FatigueLecture 12.6: Fatigue Behaviour of Bolted ConnectionsRELATED
LECTURESLecture 3.4: Welding ProcessesLecture 3.6: Inspection/QA
AssuranceSUMMARYThe data on fatigue strength given in Eurocode 3
[1] are briefly reviewed. The strengths of longitudinal and
transverse welds are related to the quality of workmanship. The
need for inspection and the limitations of non-destructive testing
are examined. The implications for economic design, detailing and
specification are set out.1. INTRODUCTIONAny joint in a structure
or in any part of it is a potential point of weakness, both in
static strength and in fatigue.For fatigue the potential weakness
is evident from the fatigue strength data given in Eurocode 3 [1]
(Figure 1). There the perfect plate is in detail category 160,
which is the fatigue strength at 2.106cycles, whilst the joint
detail with the worst geometry and hence stress concentration, is
in category 36.
In a welded joint potential sites for initiation of a fatigue
crack are:1. In the parent metal of either part joined, adjacent
to:(i) the end of the weld(ii) a weld toe(iii) a change of
direction of the weld.2. In the weld metal itself, starting
from:(i) the weld root(ii) the weld surface(iii) an internal
flaw.Even one type of joint, the longitudinal fillet or butt weld,
can fall into any one of four categories, from 140 to 100,
depending on workmanship, see Figure 2.
Transverse butt welds can have an even wider range of strengths
(Figure 2) - from category 125 to category 36, 7 categories in all.
If one excludes butt welds made from one side only, with and
without backing strips, i.e. detail categories 71, 50 and 36, four
categories are left for "good" butt welds. Here the category
depends on both weld geometry and workmanship.Other welds
(transverse fillets, welds to attachments, etc.) also show wide
variations in strength depending on geometry and workmanship.It is
important to note that a number of other (usually accidental)
results of poor workmanship can reduce the performance of a detail
to below what its category would indicate:(a) weld spatter(b)
accidental arc strikes(c) unauthorised attachments(d) corrosion
pitting(e) weld flaws, particularly in transverse butt welds(f)
poor fit-up(g) eccentricity and misalignment.Most of these are
largely unquantifiable and must be controlled by adequate
inspection and repair.It is the purpose of this lecture to describe
in greater detail welded joints and the matters to be considered by
the designer before deciding the fatigue strength that will be used
in calculations.2. LONGITUDINAL WELDSThe highest category for
longitudinal welds, 140, applies only where there are "no
significant flaws". This implies automatic welding, no stop/start
positions, no slag inclusions or blow holes - near perfection
"demonstrated by specialist inspection".The next category down,
125, requires automatic welding and expert repair, followed by
inspection, of any accidental stop/start positions. Leaving
stop/start positions brings the category of longitudinal fillet
welds down to 112 and that of longitudinal butt welds down to
100.Manual fillet or butt welds and one-sided butt welds are all in
category 100, as are "repaired" welds.There is experimental
evidence that small slag inclusions can bring the strength of a
longitudinal fillet weld down to category 90.A lower limit to the
strength of defective longitudinal fillet welds is probably that of
an intermittent weld, category 80, or even the end of such a weld
at a cope hole, category 71.3. TRANSVERSE BUTT WELDSTransverse butt
welds can reach category 125 when "high quality welding" is
obtained and proved to have been achieved by later inspection.
Amongst other requirements, the proposed welding standard limits
solid inclusions in such welds to a width of 2mm and a length of
6mm, thus acknowledging the importance of internal defects.Lower
quality welds fall into category 112 provided the welds are ground
flush. Otherwise they are in category 90, or 80 for splices in
rolled sections or girders. Here the category depends on the weld
profile and the likely quality of workmanship; internal defects are
not mentioned.In fact, internal defects have at least as great an
influence on the fatigue strength of transverse butt welds as does
the weld profile.Another factor which affects the strength of
splices in girders, and which is not mentioned explicitly in the
description of the detail categories, is the order in which the
welds are made. This can affect the level of residual stress.The
test results shown in Figure 3 illustrate these points. They are
all results of tests on transverse butt welds shown against the
grid of lines representing the fatigue strengths given in Eurocode
3 [1] for the various detail categories. The short thick lines
represent test results on small plate specimens, 40mm wide and 10mm
thick. All other points represent results from tests of complete
beams.
The results range from category 112 for the plate specimens down
to category 63 or so for a butt joint in a rolled I beam.The
reasons for this spread of results are partly weld quality and
partly residual stresses caused by different welding procedures.3.1
Effect of Internal Defects.It is likely that the plate specimens
were reasonably free from internal defects. The butt welds in the
plate girder flanges, shown as large circles, contained various
small defects, in the range of 3mm2to 30mm2from which the cracks
leading to failure originated. Allowing for the fact that the plate
girder flanges were 35mm thick, all results would fit into category
112. So would the results from tests on small girders with 25mm
thick flanges, shown as triangles.The results shown as small dots
were obtained for a butt weld between a rolled and a built-up I
section. The failure was due to a large "lack of fusion" defect in
the 30mm thick flange directly above the web to which it was joined
by 24mm radii. The defect had an area of about 80mm2and is sketched
in Figure 4 and was attributed to faulty weld preparation. It must
be pointed out, however, that it was the work of experienced
fabricators, who clearly had not appreciated the difficulty of
achieving full penetration at this point.
Even allowing for size effect, one would put this result into
category 63. There is no information on the strength of such welds
in Eurocode 3 - they should not be used. A British Standard,
BS5400: Part 10 puts such welds in a class which corresponds, as
regards strength, to category 63 [2]. This classification fits the
test results.These few test results suffice to indicate that
internal weld defects occur and that they have a decisive influence
on the fatigue strength of a welded joint.To determine this effect
quantitatively, a fracture mechanics study has been undertaken,
based on fatigue test results from butt welds containing known
defects. The results were used to obtain basic fracture mechanics
data. These results showed the scatter typical of all fatigue test
results. A lower bound of the values was then used to calculate the
fatigue strength of butt welds of various thicknesses containing
defects of various sizes. The defect size was expressed as an area;
a reasonable approximation which avoided the need to give two
dimensions for every defect and to investigate various shapes.The
figures shown in Figure 5 are approximate and were obtained by
interpolation, and some extrapolation, from the results produced by
the investigation.
The agreement between these figures and the few large beam
results is quite good.It will be noted that near surface defects
cause a greater loss of fatigue strength than deep ones and that a
12mm2defect, such as suggested in the draft welding standard [1],
would bring a butt weld strength down to category 100, or even 90
if near the surface.It is clear, therefore, that high fatigue
strength in a butt weld requires nearly perfect welds.3.2 Effect of
Welding ProcedureThe results in Figure 3 showing the effect of
different welding procedures and, hence, residual stress are the
two groups of squares.They were obtained from tests on butt joints
through rolled I sections. The higher set, full squares, were from
specimens in which the flange butt welds were made before the web
butt welds. The lower set, open squares, were obtained from
specimens in which the reverse procedure had been used - web butt
weld first, then flange butt welds so that their contraction was
resisted by the web.One set fits category 100, the other category
80 - a considerable loss of strength through using the wrong weld
sequence.These results do not stand alone; similar ones have been
obtained in the United States and they are confirmed by the results
shown by the circles on the figure.These results were obtained from
plate girder specimens with 35mm thick flanges. The full circles
show results from specimens in which only the flange plates were
butt welded, and that before they were welded to the webs. Allowing
for size effect, they fit category 112 or, possibly, 125. The open
circles are results from butt welds right through similar plate
girders. They are little, if any, worse than those shown by full
circles.However, the welding procedure, shown in Figure 6, was
designed to minimise residual stresses in the flange welds.
Initially, the webs were not welded to the flange for some 110mm
either side of the joint, the flange butt welds were made first,
then the web butt weld and, finally, the web was welded to the
flanges. This weld also served to close the small slot which had
been left under the flange butt weld to allow radiography of these
welds. Cope holes were neither needed nor provided.
It is clear from these results that butt joints right through a
girder can have the same fatigue strength as a butt weld through a
plate provided that the right welding procedure is specified by the
designer and followed in the fabrication shops. Otherwise there is
a loss of fatigue strength of the order of 25%.The results were
obtained from tests on plate girders, but the conclusions have a
wider application. They apply, for example, to joints in portal
frames at or near the corners and any situation where there is a
risk of high restraint of butt welds.There is some evidence that
similar considerations apply to welding attachments to girders. In
one test, plates welded to the compression flange of a plate girder
caused early cracks, as was expected. When similar attachments were
welded to the flange plate before it was welded to the girder, no
cracks were observed at about double the endurance; again an
improvement of about 25% in the fatigue strength.Given the effect
welding procedure can have on the fatigue strength of a joint, it
must be considered at the design stage and be specified; it cannot
be left to the fabricator.4. OTHER WELDS4.1 GeneralSo far
discussion has been limited to those types of weld (