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Fatigue Analysis - lite Andrei Lozzi 2014

1.1 What are the problems in estimating the effects of variable loads on the ‘fatigue’

strength of a component? Early in this piece we need to define our terms, how do we

characterise variable loads and what is meant by fatigue strength. But let us be warned that this

field of study has over time provide us with an evolving and changing perspective, a perspective

that can be look quite different even today between English and non-English engineering worlds.

There is almost no mechanical component that in practice is subjected to a truly steady load, and

consequently steady stresses, within that component. Hence static strengths, such as Su , Sy, Sus & Sys

etc (the ultimate tensile and yield strengths in tension and shear), of the material from which a part

is made from, are not immediately applicable in calculating the likelihood of failure under variable

loads, since these strengths are determined under steady loads. We somehow need to be able to

estimate the strength of the material at any particular location of a part, while the material at that

location is subjected to some variable stress.

Now, variable can mean infinity of patterns, so we begin by

taking possibly the simplest possible view. We resort to

separating the variable load into its Fourier components. We

thus arrive at the representation of a general variable load as

the sum of a steady load and a range of sinusoidal loads

acting together. Fortunately in real-world situations this can

be reduced to focusing on the steady load and just one

sinusoidal element added to it, that is, the one element that is

potentially the most damaging, as represented on Fig 1.

Hence we may restate our objective as this: we want to be able to estimating the degree of safety

against failure by a variable load, at any particular location in a component, where the stress

variation may be represented as on Fig 1. The locations that we analyse are usually clearly critical,

where the stress conditions are the highest in the component. We will leave out of these

considerations how we may deal if there is more than one significant oscillatory component acting

concurrently, or one acting after the other, and random loads that may not be easily separated into

components.

1.2 Tests that determine the strength 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 its fatigue strength - Sf.

We will see later that the steady and fatigue strengths are each diminished, if their stresses coexist.

We will first consider some of the means of determining the fatigue strength, which we will see can

have a multitude of expressions.

1.2.1 Moore rotating beam test. Below at left in Fig 2 is shown a pictorial view of a

Moore test piece and at right a side view of the same component. These pieces are usually carefully

manufactured. They are 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 any machine component. There is a low level of stress concentration at the

central waist of the part, but it is expected that such test pieces will give a reliable indication of the

highest fatigue strength available from that material. We also expect that all use of that material in

real components will exhibit lower fatigue strengths, because in general components shapes and

manufacture will be considerably less fault-free than these test pieces.

Fig 1

2

ω ω

-σa

σa

δ

F

T

Fixed end

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

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 us with just the fatigue strength Sf of the material in

tension, that somewhat meets the requirements of section 1.2 above.

1.2.2 Pulsating tests. It is more common in modern numerically controlled test machines, that tests be

carried out on specimens loaded as 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, as shown on Fig 4 b).

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).

Fig 4 a) 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 of the

test to the maximum stress. Whereas in bending tests: Fig 3 a) only one fibre, that at the outside

of the bend, will be subjected to the maximum stress. Pure tensile tests are more severe than

bending tests, but again the results from either can be used provided corrections is made.

1.2.3 The root cause of fatigue failures (a somewhat simplistic explanation).

Fig 2

F

3

F

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, for

the machine components that we need to analyse, is a tensile failure originating at some fault on a

surface, the other is a shear failure that may start below the surface.

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 from b) is repeated the bottom corner may open further and

deeper into the component d). For ferrous alloys if the level of the alternating stress is kept below

some prescribed limits, the growth of such imperfections can be reduced to such a low level that

will produce an illusion of ‘infinite fatigue life’. For other engineering materials such as

aluminium alloys, such growth cannot in practice be kept to essentially zero level. Bridges if

maintained well, can be expected to have ‘infinite’ life, while aircrafts on the other hand have to

be carefully monitored to ensure that they are operated within a finite 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 expensive repairs, this means that careful crack inspections are

carried out at regular times.

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 improper materials or worn

tools leaving sharp corners etc and

by the inclusion of unexpected

residual stresses. 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 typically leads to

production improvements

Fig 6

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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, manufactured 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 to 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.

Fig 7 Examples of ferrous alloys are shown here, the 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.

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Fig 9 Aluminium alloys. At right is

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

supports the concept that fatigue failures mainly

arise from the surface of a part and are tensile in

nature. 7075 Al alloy is often used to make light

weight and heavily stresses parts and vehicles

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.

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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:

Sn = aN b Eq 1

or log Sn = log a + b log N Eq 2

where:

e

m

S

S

NNb log

loglog

1

21

Eq 3

log a = log Sm – b log N1 Eq 3

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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:

A b

Ground 1.58 -0.085

Machined or cold worked 4.51 -0.265

Hot worked 57.7 -0.718

Forged 272 -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.

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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: 586

134log

uS

Eq 8

where r is the radius (in mm) of the notch or convex corner:

r

q

1

1 Eq 9

giving a stress concentration factor: 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.

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2.0 Predicting safe fatigue loading

The alternating and steady von Mises stresses for 2 dimensional conditions may be expressed as:

22'

3 aaa Eq 11

22'

3 mmm 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:

1

2''

u

m

e

a

SFS

S

Eq 14

One has also to ensure that the point ),( ''

ma is also below the yield line:

1''

y

m

y

a

SS

Eq 15

Or, vary the loads or dimensions such that the stresses meet the ASME ellipse condition:

1

2'

2'

y

m

e

a

SFS

SFS

Eq 16

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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 222

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eatTatM STKMKa and 222

43

umm STMb Eq 17a, b

we get:

21

2

22

6

42

322

aabab

bFSD

Eq 18

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.

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