Fatigue Analysis - lite
Andrei Lozzi 2012
1.1
The problem generated by variable loads. There is almost no mechanical
component that in practice is subjected to a truly steady load, that is experiences a steady stress,
over its service life. Hence static strengths, such as Su , Sus , Sy & Sys etc, of the material from which
a part is made from, are not immediately applicable in calculating the likelihood of failure under
variable loads. We need to be able to estimate the strength of the material, at any specific location
of a part, which is subjected to some variable stress.
Now, variable can mean infinity of load patterns, so we
resort to presuming that any variable load can be divided
into its Fourier components. We thus arrive at the
representation of any variable load as a combination of a
steady load with a range of sinusoidal load elements, of
different frequencies and amplitudes, added to it.
In real-world situations this can usually be further
simplified to the steady load with just one, that is the most
demanding sinusoidal element, added to it. Hence we may
restate our objective as this: we want to be able to closely
estimating the degree of safety, at any particular location
in a component, where a stress variation as shown in Fig 1
exists. We will leave you to consider on how to deal with
random loads or multiple significant Fourier load
components.
Fig 1
1.2 Tests that determine strengths under oscillatory loads – the fatigue strength. We will
continue making simplifying assumptions by presuming that we can separate the steady or mean
stress: σm from the alternating stress: σa. We will compare σm to the steady or static strength - Su of
the part’s material, and the amplitude of the oscillatory component: σa to the fatigue strength - Sf.
We will see later that there are limitations to these strengths if steady and alternating stresses
coexist. We will first consider some of the means of determining the fatigue strength, which we
will see can have a multitude of meanings.
1.2.1
Moore rotating beam test. Below at left in Fig 2 a) is shows a 3D representation of
a Moore test piece and at right, Fig 2 b) a side view or 2D drawing of a similar piece. These pieces
are usually prepared, machined, stress relieved and polished so that they are essentially made from
fault-free material, have no residual stresses and are finished to the highest practical surface finish
that is applied to machine components. There is a low level of stress concentration at the central
waist of the part, but it is expected that they will give a reliable indication of the highest fatigue
strength available from the material from which the specimens are made. We also expect that all
use of the material in real components will exhibit a lower fatigue strengths because they will not
be made so carefully.
Fig 2 a)
b)
1
The Moore test machine is somewhat like a lathe, the two opposite ends of a test piece are grasped by two
initially coaxial chucks. During a test a bending moment is forced on the piece, by setting up an angular
misalignment of one chuck with respect to the other, by an angle δ, as shown on Fig 3 a) below. The piece
is then rotated transmitting no torque but subjecting the material of the test piece to alternating bending.
On the outside the bent test piece is subjected to tension on the inside to compression, with the highset
stresses occurring at the waist or middle of the specimen, where some stress concentration will take place
σa
δ
-σa
ω
ω
Fig 3 a)
b)
Just as one expects for a section that is being bent: the normal stress varies from a max tension σa on the
outside of the bend to a max compression -σa on the inside; and is zero in between. A point on the surface
of the specimen will experience a normal stress that will vary sinusoidally as the test piece is rotated, as
shown on Fig 3 b). This test obviously provides just the fatigue strength Sf we may have expected from
section 1.2 above.
1.2.2 Pulsating tests.
More commonly in modern numerically controlled test machines, tests can
be carried out on specimens similar to that shown on Fig 4 a). The tensile load F can be arranged to
pulsate between zero and the maximum sinusoidally, so that only one frequency is present. Alternatively
the specimen may be subjected to torque T either in one direction about the long axis of the piece
(pulsating), or the torque T may alternate equally in both directions (alternating).
F
Fixed end
Fig 4 a)



1.2.3

T
b)
The results from pulsating tests is represented on Fig 4 b), showing that there is a mean stress
present, that is equal in magnitude to the alternating stress.
Both test results represented by Fig 3 b) and Fig 4 b) can be used, provided we make due
adjustment for the presence or otherwise of the mean stress.
Fatigue tensile tests as shown in Fig 4 a), ie with force F, subjects the whole of the surface test of
the piece to the maximum stress. Whereas in bending tests: Fig 3 a) only one fiber, at the outside
of the bend, will be subjected to the maximum stress. Tensile tests are more severe than bending
tests but the results from either can be used provided some correction is made.
The root cause of fatigue failures (a simplistic explanation).
Tension and or shear stresses. There appears two means of initiating and bringing about a
fatigue failure, which can act independently or sometimes in unison. The more relevant one to us
is a tensile failure originating at some fault on a surface, the other is a shear failure that may begin
on the surface but may also start below it.
2

Surface faults. Fig 5 a) shows an idealized concave surface irregularity that finishes at its bottom
with a sharp corner. When subjected to a tensile force, the stress concentrating effect of the sharp
corner causes the material in its vicinity to yield or fail and open the corner up until the
surrounding material can support the additional stress - b). If the tensile force is release or
reversed the corner will close up but will not weld shut - c).
Fig 5 a)
b)
c)
d)

Infinite fatigue life. If the cycle is repeated the bottom corner may open further and deeper into
the component - d) . For some engineering materials such as aluminium alloys surface
imperfections are such that they will grow continuously in the presence of variable stresses. For
ferrous alloys if the level of the alternating stress is kept below prescribed limits, the growth can
be reduced to such an imperceptible level that will produce an illusion of ‘infinite fatigue life’.

Growth rate. The growth rate of such imperfections, that leads them to become cracks, is best
described as exponential functions of cycles. While they are very small they grow very slowly
and may not be detectable at all for billions of cycles. As soon as they reach visible sizes their
growth may become explosive and bring failure in a few cycles. For many critical parts, the
failure of which would lead to large costs, this means careful crack inspections during rebuilds.
1.3
Component fatigue testing.
Beginning in paragraph 1.1 we started a process that
separated steady from alternating loads and speculated that each had a strength that can be associated
with it. We then looked at methods of determining the fatigue strength Sf of a material by using near
perfectly manufactured test pieces. Later we will consider how this fatigue strength Sf is reduced in
real parts by poorer surface finish, sharp corners and a multitude of other strength reducing factors.
There is another way of determining the fatigue strength in an actual part other than beginning with
test pieces and so on. We expose samples of a manufactured apart to representative loads that they
will experience in service. That is, we can test the part itself !
Testing a whole part like the spring
at left or a suspension subassembly
at right is more expensive than just a
test piece but will reveal the fatigue
strength of the weakest point. The
strength may be surprising low
because of unexpected residual
stresses, improper materials or worn
tools leaving sharp corners etc. It is
often hard to determine why real
parts are weaker than expected, but
it is usually very worthwhile to do
so. Testing the whole part should
lead to production improvements
because they check the actual endproduct not the theoretically
expected product.
F
Fig 6 a)
b)
3
1.4 Experimental results. We will now look at a range of experimental results that will be used in a
simple mthematical model for the stress analysis of machine componenets, made and finished by
different means and subjected to wide range of stresses. The original references are mentioned in
these notes so that you will be able to inform yourself further.
Provided the bending moment imparted to a Moore test piece is sufficiently large, tensile fatigue
failure originating on the surface, near the waist of the part, will occur at some number of cycles. As
you may have reasonably expected the number of cycles to failure is inversely proportional to the
stress level and as mentioned already the strength at failure is a random variable of the number of
cycles. This situation allows us in only being able to predict failure using an average fatigue strength
and a standard deviation of its distribution. An unexpectedly early failure or at other than at the waist,
would indicate inadequate preparation or in-homogeneities in the material.

Ferrus alloys. Fig 7 below is a normalised distribution of fatigue strength Sf and the
expected cycle life N for wrought steels of strengths. Sut < 1400 N/mm2 . These steels display a
regular drop in fatigue strength from 0.9 Sut at 103 cycles, to 0.5 Sut at 106 cycles. There it
levels out to a strength referred to as the ‘endurance limit’, or the strength for ‘infinite life’ of
the steel - Se. Data from R C Juvinall, “Stress Strain and Strength”, McGraw Hill 1967.
Fig 8 at right shows the
characteristics of higher strength
steel, here for 4130. Data from
NACA Tech Note 3866 1966.
The specimen here is reported to
be of Sut ~ 860 N/mm2. Note that
1 kpsi ~ 7 N/mm2
Predicting failure in the low cycle
range, ie <103 cycles, involves
stress levels that will typically
exceed the yield condition. The
analysis that we will consider here
presumes elastic behaviour only
and therefore cannot be extended
into this ‘low cycle’ fatigue range.
4
Fig 8
 Aluminium alloys. Fig 9 at right
displays the fatigue behaviour of
aluminium alloys: wrought, moulded
and sand cast, of a tensile strength
Sut ≤ 590 N/mm2. The strength
disadvantage of the random grain
structure resulting from smooth cooling
and the lack of work hardening is clear
for components manufactured by
casting. Hence aluminium castings have
to be stressed with due care. From R C
Juvinall. Note that Al alloys do not
exhibit a clear levelling off or endurance
limit of their fatigue strength. In part
explaining why aircrafts have a limited
service life unlike trains, trucks etc
Fig 10 at left is a set of SN curves for 7075 Al
alloy, showing the effect of mean stress. As one
may have predicted, the presence of a mean
tensile stress decreases the expected fatigue life,
but a mean compressive stress increases it. This
supporting the concept of fatigue failure mainly
arising from the surface of a part are tensile in
nature. 7075 Al alloy is often used to make light
weight but heavily stresses vehicle parts and
aircraft components. This figure can be used to
estimate fatigue strength from 104 to a greater
number of cycles. For Aluminium alloys the
value for Se is often given as Sf at 108 cycles.
Fig 11 The data at right shows a safe
estimate of the fatigue limit for Al
alloys, plotted against their ultimate
tensile strength. The plot is of Se versus
Su indicating a limit of Se ≤ 130 N/mm2
for all aluminium alloys. Although the
information provided here is a little
confusing, one can see from Figs 9 & 10
that Se for 7075 to reach 160 N/mm2 for
some grades.
5
1.5 Shot peening and plating
Fig 12 The graphs at right show that the
beneficial effect of compressive stresses can be
take advantage of by inducing compressive
residual stresses on the surface of components.
This may be done by such processes as shot
peening or work hardening the surface by
hammering or rolling with hardened rollers.
Note that on the figure at right, electrochemical
plating has a detrimental effect on fatigue
strength. This may be in part because plating can
leave behind microscopic irregularities that may
lead to surface defects. If plating were to be
done slowly followed by stress relieving and
surface grinding the damage need not be at all as
shown here, but the expense is considerable,
hence it is done to special parts or to repair for
excessive wear.
1.6
Distortion energy stress
Fig 13 at right shows a typical set of results that support
the view that for steels and Al alloys used in machine
parts, the distortion energy stresses can be used to
predict quite accurately the onset of fatigue failure, as it
can for static failures. The data is not new and the range
of tests available to the public seem to be limited but
there is a general adoption of the use of von Mises
stresses, for mean and alternating stresses, in the
prediction of fatigue failures. This make the analysis of
components subjected to combined shear and tensile
stresses more straightforward. Equations 11 & 12 below
show the relations between von Mises tresses and 2D
shear and tensile stresses.
1.7
Fatigue strength for intermediate cycles: Sf at 103 cycles and Se.
Fig 14 at left shows a simplified SN diagram that may be used to
calculate fatigue strengths Sf between N1 and N2 cycles. Letting
the fatigue strength at N1 cycles be represented by Sm, then the
straight line from N1 to N2 cycles is:
or
Sn = aN b
Eq 1
log Sn = log a + b log N
Eq 2
where: b 
S 
1
log  m 
log N1  log N 2
 Se 
log a = log Sm – b log N1
Eq 3
Eq 3
6
If N1 = 103 & N2 = 106 this equation simplifies considerably, but care should be taken to not simplify
the calculation unnecessarily. Should N2 occur at 108 the fatigue strength of the material would be
understated and the potential of the material would be underutilised.
1.8 Combined alternating and steady stresses
Fig 15 at right shows the diminished
strengths Su and Se in the presence of mean
and alternating stresses σm and σa. The
straight line has been used to provide simple
means of estimating safe strengths Su and Se
in the presence of combinations of σm and
σa. On this diagram this line may be referred
to as the normalised Goodman line. The
actual data falls on a curved path above that
line and it may be more precisely
represented by a parabola or a quarter
ellipse. Aluminium alloys have very similar
properties. Data from P G Forrest, “Fatigue
of Metals’ Pergamon Press.
Fig 16 left shows similar data
to that of Fig 15 except that here
the mean stresses σm are
extended into the compression
range. It is clear that mean
compressive stresses reduce the
likelihood of tensile fatigue
failures to an extraordinary
degree. The use of straight lines
that would simplify calculations
was almost a necessity before
the development and adoption
of PCs.
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1.9
Fatigue strength modifying (ie reducing) factors.
The fatigue tests pieces should be so relatively well prepared that almost no real machine part will
exhibit as high a fatigue strength as they. Their results represent the upper limit of the strength that can
be expected of that material when used in any machine component. The fatigue strength provided by
those tests has nearly always to be reduced to provide us with a means of estimating the strength and
the life of a part. Shigley (7th ed p 328) ascribes to A Marin the simple equation shown below, where
there is a linear effect of these factors upon the initial tests results, but no effect upon each other. That
is, for example size is assumed to have no effect on stress concentration. Here we follow the
nomenclature used by Norton 2nd ed p348, to calculate for the reduced fatigue strength. We will here
on use the primed Se’ to represent the strength obtained by test pieces and Se as that calculated for the
strength at a particular location in a part.
Se = Cload Csize Csurface Ctemp Creliability (1/Kf) Se’
Eq 4
Sf & Sf’ may replace Se & Se’ for cycles between N1 and N2 .

Cload Type of load. This factor allows for differences by which the fatigue strength is
determined. A stress history of the part as represented by Fig 3 b) is different from that in Fig
4 b), as mentioned in 1.2.1 tensile fatigue tets are more severe than bending tests.
Alternating loads ( as in bending tests) Cload = 1
Pulsating loads (as in tensile tests)
Cload = 0.7
Eq 5

Csize Surface area. With increasing size, or more specifically surface area, comes the
increased likelihood of including significant surface defects in the component.
For 8 to 250 mm dia use
Csize = 1.189 d-0.097
Eq 6
The allowance for non circular parts such as I or C channels has to be related to the size of
the surface area subjected to the maximum stress.

Csurface Surface finish. This factor should both reflect the surface roughness and the residual
stresses on and below the surface caused by the machining operations. Fig 17 below provides
an easy but subjective grasp of the detrimental effect of commercial finishes. The analytic
expression Eq 7 on the other provides the means of precise and rapid calculations.
Csurface = A Sutb
Eq 7
For the machining operations:
Ground
Machined or cold worked
Hot worked
Forged
A
1.58
4.51
57.7
272
b
-0.085
-0.265
-0.718
-0.9956
Fig 17. The effect of plating and work
hardening the surface has been
introduced in Fig 12 above. There are no
generally accepted parameters to allow
for these processes, one may have to rely
on experiments.
8

Ctemp Temperature effect. This is complex effect to deal with in part because it is very
dependent on the metal alloy being used. For machines using normal lubricating oils the
temperatures should be always below 120º C which does not alter the properties of steels, but
may effect some Al alloys.

Creliability Reliability effect. As mentioned above the
occurrence of fatigue failures exhibit the
characteristics of random variables, we can apply
statistical modelling to predict the likelihood of
failure. On Fig 18 at right, for the typical 8%
standard deviation of the distribution of failures for
steels, strength reducing factors are provided to assure
ourselves of better than 50% reliability.
Note again that we cannot nominate a fatigue stress level at
which no failures will occur. Note also that what may be an
otherwise arbitrarily chosen FS ~1.6 means approximately 1
failure in 106 instances of that design, within the number of
cycles that the calculations are done. This Creliability factor gives
rationality to the otherwise traditional ad hoc values of FS.

Kf Stress concentration. this stress concentration factor is made up by two parts, the first is
the surface stress concentration ratio Kt brought about by the change in shape and proportions
of the part. This ratio may be read from any one of a large number of graphs provided in most
text on design and engineering materials. Note that torsion, bending and tension each
generate different surface stress concentrations ratios on the same component shape. Many of
the graphs showing theses ratios confusingly use the same symbol Kt.

q Notch sensitivity. The second part is the sensitivity of the material to the notch or the step
in the profile of the part. Generally speaking the harder and more brittle the steel the more
notch sensitive it is and as a consequence Kt is applied on a sliding scale reducing in
magnitude from hard to soft materials. There are a number of ways to calculate this effect,
most require the use of tables, but to automate calculations an algebraic expression is required.
N E Dowlings ‘Mechanical Behaviour of Materials’ p 428, provides us with the following
method to be used for steels Su < 1520 N/mm2:
q - is the notch sensitivity of the material
β – is the Neuber’s constant, decreases with steel’s hardness
using the empirical equation:
where r is the radius (in mm) of the notch or convex corner:
giving a stress concentration factor:
log   
q
S u  134
586
1
Eq 8
1 
r
Eq 9
Kf = 1 + q (Kt -1)
Eq 10
The gist of this formulation is that q is normally less than 1. For harder steels β decreases in
magnitude, the denominator of q tends to 1 and the stress concentration fatigue factor Kf tends to the
same value as the geometric ratio Kt. Conversely for softer steels only a fraction of Kt applies.
The reason why Norton and others leave these factors out when calculating the reduced
fatigue strength, is that it is more simply introduced in the equations that calculate the critical
dimension of a part. For example in Eqs 11 & 12 the concentration factors can be entered as a stress
increasing factors, ie we use: KfT ·Ta in place of just the alternating torque Ta, and KfM ·Ma in place of
just alternating moment Ma.
9
2.0
Predicting safe fatigue loading
The alternating and steady von Mises stresses for 2 dimensional conditions may be expressed as:
 a '   a2  3 a2
Eq 11
 m '   m2  3 m2
Eq 12
where the  m may be the sum of the stresses from several sources like a tensile load and a bending
moment.
Fig 19. The experimental data shown on Fig 15 is here represented by a range of straight
and curved lines. The Soderberg (USA 1930), Goodman (UK 1899) and yield lines have been used
in the past to indicate safe combined mean and alternating stresses. But, as seen from Fig 15 these
simple lines underutilise the available fatigue strength. We are told that the Gerber parabola
(Germany 1874) and ASME ellipse both follow closely along the mean of experimental results.
Hence conditions that meet both the Gerber and Yield lines or that meet the ASME ellipse appear to
be acceptable, with the first being slightly more conservative.
We may set up the solver to vary the loads and dimensions that can be varied in the design, such that
the resulting von Mises stresses, as determined by Eq 11 &12, meet the Gerber parabola condition:
2
 a'
 ' 
  FS m   1
Se 
Su 
One has also to ensure that the point ( a' ,  m' ) is also below the yield line:
 a'
Sy

 m'
Sy
1
Eq 14
Eq 15
Or, vary the loads or dimensions such that the stresses meet the ASME ellipse condition:
  a'
 FS
Se

   m'
   FS

Sy
 
2
2

 1


Eq 16
10
2.1 The traditional method of solving for the required dimension.
A traditional method
for solving for an appropriate dimension such as a solid shaft diameter, takes the form of entering the
moments and torques into Eqs 11 & 12, then enter those into eq 14 and rearrange to give the shaft
diameter. This gives us an expression which is in D6, simplifies to the equations 17 & 18 below:

Let a  K tM  M a2  3 K tT  Ta2
4
S
2
e

and b  M m2  3 Tm2
232 FS   b 2
4
S
2
u
Eq 17a, b
2
we get:
D 
6

2b  a  4ab  a 2

1
Eq 18
2
2.2 An easier method – let the PC do the work
As above we evaluate eqs 11 & 12 to arrive
at the mean and alternating von Mises stresses. Thereafter there is no need to substitute, reduce or
simplify any equation, because we can set the solver to vary what we can (such as shaft diameter,
step size or bearing spacing), so that the values of Eqs 11 & 12 meet a chosen safe fatigue condition,
such as Eq 14. That is the left hand side of eq 14 is required to equal the numerical value of 1/FS,
leting the numerical solver do the work.
For some design problems a negative or some other impossible dimension may give numerically
optimal answers. For those situations one has to provide constraints to ensure realistic solutions. In
some rare designs there may be more than one numerically optimal solution. For those, one can only
vary the start conditions to search for any multiple solutions because the solver in Excel only finds
the nearest one to the starting values of the variables provided by you.
2.3 Shear fatigue failures.
Shear fatigue conditions can be dealt by converting all stress
to the von Mises form. Some shear fatigue failures originate below the surface in such components
as the tracks of rolling element bearings and below the surface of gear teeth. Both these condition are
understandable from Hertz analysis. Rolling bearings and gears are usually in the hands of specialists
and do not warrant reviewing here.
2.4 Assurance against fatigue failure.
Since we cannot guarantee against fatigue failure we
have to use some other strategy if its outcome is unacceptable, very undesirable or just very
expensive. We can provide capable redundant load paths, so that should one fail the other can take
the load. It may be necessary that the failure of one load path should be very clear to the user of the
machine. The alternate load path may just provide a ‘limp home’ sort of function.
An example of the first countermeasure is where a component may break but be trapped in place to
provide a rudimentary and short lived function, like the pivots of the wishbones of a car. An example
of the second is the windows structure in modern airliners. The world’s first jet liner (the de
Havilland Comet) suffered fatigue failures originating from the corners of its windows. The window
apertures were cut directly into the fuselage skin, a crack staring at the window could propagate
explosively around the fuselage. Airliners since then have the window panel separate from the
fuselage skin and the panels themselves are made in parts so that a window section could fail but the
crack being limited to the boundaries of the panel.
Having a redundant load path, is a much more effective means of protection against fatigue failure,
than the use of a large factor of safety as ~ 2.
11
12
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FSAE design criteria.
Andrei Lozzi 12
1
I suggest that reasonable working and maximum loads be assessed for each part of the
car. That a reasonable number of maximum load cycles be allowed for each component or for
the whole car, whichever is appropriate. That a reliability factor be used that reflects the
undesirability of a failure for each individual component and that finally a small arbitrary
factor of safety be used above all that, eg 1.1 or 1.2.
2
The number of bends, accelerations and braking in the competition circuits can be
estimated conservatively from the on-board data logger. The car can realistically be expected
to used heavily for up to 2 years, maybe 3 years. Generous safety margins should not be
added at each estimation. All activities of the car/s should be recorded in a log book to
provide a more realistic guide for the future.
3
In my experience failures have occurred when some significant load was completely
overlooked. In any case, an arbitrary factor of safety cannot be expected to save the day,
unless it is much larger than everything else that has been considered, like an FS of ~20.
4
I feel that we can provide more security by providing alternate load paths or features
that are countermeasures to failures that may occur to critical components, to prevent injury
or serious damage. For example additional struts are sometimes added to wishbones to
prevent their entry into the driver’s compartment in the event of a rod-end failure.
5
The size of the reliability factor should reflect the consequences of failure of any
particular part. A suspension wishbone should perhaps be give a 1 in 99 chance of failure
over the life of the car, while a transmission component may be given only a 1 in 90.
Reliability factors are based upon the statistical chances of failure. The likely hood of some
failure occurring somewhere in the whole car, is the sum of the probability of all individual
failures (a sobering thought).
6
All preloaded commercial fasteners should be stressed as pre lecture notes of 2004,
these notes area based on the latest editions of Shigley and Norton. If the preload used is in
excess of 75% then the fastener should be scrapped when removed.
7
Shoulder bolts should not be used to take tensile loads, their intended function is to act
as a pivot or shaft, to take large shear loads or as a locating device like a dowel pin. They
have abnormally high tensile stress concentration features and small threaded stem. All tensile
fasteners should preferably have fine threads. Coarse threads lose about 25% of their crosssection and are much more likely to come undone from vibrations than are fine threads. Use
nyloc self-locking nuts, lock-wiring or split-pins
8
Finally when a car reaches the end of its designed life, or if it is to be disposed earlier,
then the frame and all parts have to be rendered unserviceable to any other possible user. The
cutting up and disposal of all major parts have to be recorded by photographs, entered in the
log book of the car and copies sent to me..
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