nm0827: now today we're going to talk about fatigue and that doesn't mean that 
i'm going to pack the lecture in half way through because i'm worn out although 
with you lot the ability to feel knackered after a lecture is very easy i will 
say that er you've all got i hope this piece of paper these notes come on which 
i've given out take this please we're not going to get on to this yet but we 
will in a moment now fatigue is a strange phenomenon and it's to do with 
loading which varies it varies so in other words it's oscillating well here's 
some good words oscillating or fluctuating loads so a classic example of this 
would be a car going over a bumpy road what's er does the er suspension 
experience what about an out of balance piece of machinery some pieces of 
machinery of course are deliberately out of balance er if you want to crush 
things you very often will have an out of balance thing to give a 
sort of hammer effect and things like jackhammers that you use for er making 
holes in the road these also are er they're not out of balance but they're c-, 
controlled by air yep we're on air so be on your best behaviour so [sigh] 
rotating machinery is a very important example of this and we'll come on to an 
example of that at the end as a sort of case study which is what you've got 
your er notes on now the simplest way to understand fatigue and indeed to 
understand the history how how it came to be a topic is to consider what 
happens in a spoked wheel so you have some sort of wheel it could be a horse 
and cart wheel it could be a bicycle wheel it could be a gun wheel i don't know 
anything you wa-, name it think of what it looks like now obviously if you have 
a wheel er let's say er right you'll get shot let's say that's the front wheel 
of your bike or something or other your weight on the frame pushes down on 
there so 
there is a vertical loading or a er an inclined loading on the axle which is 
here okay so you've got that now the question then is what prevents the fork of 
the bike going into the ground well obviously because the wheel is pushing back 
on the end of the fork so that if we take that spoke as an example this spoke 
is pushing back but because the hub if you like does come down a bit because 
you've got a tyre it's squidgy the spoke above is coming down so if you think 
about it the upper spoke is in tension and the lower spoke is in compression at 
that p-, particular instant and so if the vehicle is stationary that is a 
stress pattern which will be there er as long as you leave it in that condition 
if you consider other spokes at angles well they have some load in them er w-, 
i'm not going to bother to work out how you er calculate what f-, force is in 
the spokes but the general idea is that below the middle there is compression 
and above the middle there is tension now the 
next question is what happens when this thing starts to move so that this now 
goes forward so that that point there er if we move the wheel along we rotate 
it through er a hundred-and- eighty degrees what happens well what we find of 
course is that that blobbed point there is now down here and so here are the 
spokes et cetera et cetera and so what it means is that that spoke which was 
above the axis is now below and what was below a hundred-and-eighty degrees 
later is now above but the stress state what the wheel has to do to hold the 
thing up is exactly the same as before so this bottom thing is in compression 
and the top one is in tension still but this one was up there so what was in 
tension is now in compression and if you roll it another a hundred-and-eighty 
degrees it goes back to where it was and so the tension has become compression 
has become tension has become pompre-, er progression [whistle] tension has 
become compression has become tension so if you were 
to plot this out as the force experienced by the spoke force in spoke against 
er rotation revolutions whatever you want to call it of the wheel what you'd 
have is a load up there if we consider one spoke tension it drops down to 
nothing when it's half way because the horizontal spokes don't have anything in 
them because they're not resisting any horizontal motion if the force is truly 
vertical and then going on a bit more it goes below and goes to an equal value 
but in compression and then it does this and so on and so forth and this you 
can show with a bit of trigonometry i'm not going to do it today but in fact 
it's a sine curve so the process of rolling or rotating a wheel or an axle or 
anything like that produces alternating tension and compression so that the 
spoke of the wheel is experiencing this phenomenon of a varying tension 
compression alternating fluctuating oscillating whatever the those words you 
wish to use and the question is 
well so what why should we be worried well we have to be worried because it 
turns out that this oscillating loading can break things can fail things at 
much lower stresses than you would work out on the basis of a simple single 
monotonic pull or push that's why it's important who first thought about this 
well you might think that people with horses and carts go up to the Museum of 
English Rural Life up the top of campus have a look in there you might have 
thought that farmers might have been interested in this and perhaps they were 
but if you think of the state of roads before eighteen-thirty forty or fifty or 
whatever they were sort of rutted tracks even the so-called turnpikes and 
therefore if anything broke it was probably because there was a big pothole in 
the road in the way that we all have problems with cars and bikes if we go over 
a what appears to be a puddle but in fact it's a like a bucket-shaped hole full 
of water and you damage your front 
suspension and so on so and what you've got is the problem of er rotating 
machinery rotating wheels in the old days there were fractures but they never 
thought about it what made this an important subject was the growth of railways 
because for the first time you had a smooth track and you had wheels going 
along that smooth track and of course the engineers of the day although they 
were not er given all the materials that we have today we've done a bit of that 
a bit of the history wrought iron malleable iron et cetera et cetera Bessemer 
steel not coming along until the eighteen-fifties this in fact held up the 
growth of railways because all of the manual methods of making ductile er steel 
before that han-, before that time er nevertheless you had a reasonably smooth 
thing system with a reasonably smooth wheels and all the rest of it and what 
they discovered was that these axles were breaking the axles were breaking not 
so much the wheels they might have had spokes but 
they might have been solid wooden blocked wheels it doesn't matter why were the 
axles breaking well let's look at this picture but looking at it from the end 
this way and let it not be a bicycle wheel any more let it be er two wheels on 
an axle like a railway axle so this look at it sideways and you've got this 
sort of device and there is the axle and here is the wheel er and there is the 
the rail do you know by the way some clever character said there was a method 
of preventing trains coming off rails and what he said was that when you look 
end on like this what you actually need is a wheel which looks like this and 
the you know the rail is in there and so you know the wheel wouldn't come off 
that a good idea well it isn't a good idea but you know the original 
explanation for why it wasn't a good idea because they said that's no good 
you'd have to go to the end of the track to get the vehicles on [laughter] 
think of it right now if you have this system and if you have outside 
bearings so it's like a goods wagon with springs and so on there's the axle box 
there here's the spring the force is pushing down there so what does that do 
well obviously if you think about it it must bend the axle i'm exaggerating it 
but that's what happens the axle gets bent what happens when you bend a beam 
you've had namex's lectures last year when you bend a beam that bit goes into 
tension the underneath goes into compression that is a hogging beam remember 
the word hogging from pigs sagging and hogging that's hogging tension 
compression what happens a hundred-and-eighty degrees later well this bottom 
bit has gone on the top the top's gone on the bottom it's exactly the same as 
this it's the top and bottom of the axle now not the upper and lower spokes and 
so the stress state in that axle is exactly the same as the stress state in the 
spoke and in fact in those wheels if they were spoked wheels the spokes were 
being fatigued and the axles were also 
being fatigued anyway these axles began to break for no apparent reason and so 
the people in charge of the railways s-, thought well either we're not making 
the wheels very well or there is a problem that we don't understand and it was 
a problem that they didn't understand and the problem was this thing called 
fatigue and the man who started all this work off was an Austrian whose name 
was Wöhler that name keeps coming up W-O-umlaut-H-L-E-R Wöhler who was not 
sure what he was might be some sort of chief engineer of the Austrian railways 
one of the Austrian railway companies anyway he started to investigate this 
business and he very cleverly came up with a sort of test which in many ways is 
still used as a fatigue test even today and the 
best way for you to think about it is to think of going downstairs into the 
workshop and going to one of namex namex's lathes and saying okay here is a 
chuck on the lathe can't spell chuck and what you do you have a normal round 
tensile test piece which you are used to so here is this round tensile test 
piece but instead of putting it in one of namex's machines and pulling it until 
it breaks or whatever you're doing what you do is you stuff one end into the 
chuck so you lock it in the chuck and if you turned the lathe on the thing 
would rotate but what Wöhler then did was to say right at this end i'm going 
to put some sort of collar with roller bearings or ball bearings it doesn't 
matter so here's looking from the end elevation there is the specimen here is a 
sort of collar going round it with ball bearings and things like that and from 
that you hang a weight you put a mass on it and so 
obviously the weight on the end of this thing what is it it's a cantilever and 
it's like the things that you solved in part one so you can work out the 
deflection you can work out the bending stress and you can do everything else 
and what happens well the cantilever the top is tension the bottom's 
compression hundred-and-eighty degrees later the compression has become tension 
and the tension has become p-, compression et cetera et cetera and old Wöhler 
very clever he started to do tests what sort of tests did he do he put 
different weights on these test pieces so he changed this W weight and he 
rotated the lathe and he found out how many rotations it took before the 
specimen broke all right and he then plotted the load against the number of 
cycles N cycles to failure and failure here is breakage i say it's breakage 
because of course you could define failure and in part three i'll teach you 
plasticity and failure of course could be permanently being bent obviously if 
an axle permanently gets bent it's not much good as an axle er so you always 
design within the elastic stress range we've already been through this in this 
course so he's got load he's got number of cycles to failure now if you have no 
cycles it's really just bending bending bending bending bending until it breaks 
and that's like a static quasi-static test and you'd have a a value there say 
if you then increase the weight and start the machine up and wait for something 
to happen you'll find that the lifetime or the number of cycles to failure is 
lower in other words if you want the thing to last a hundred cycles a thousand 
cycles it will only take a certain load and then it'll break he puts more 
weights on and he finds that he can go out there more cycles lower loads so 
these loads are dropping all the time and he finds that sort of behaviour and 
that is a classic sort of so-called fatigue curve and in the literature you'll 
find these things called 
S- N curves now we these days use sigma for stress er and h-, originally of 
course he used load but you can convert it to stress verily easy easily using 
bending theory so in the books you'll find that called very often an S-N curve 
and it means a fatigue curve now because these numbers of cycles in fact are 
very high they go into the millions you don't normally plot it on Cartesian 
coordinates you can have load or stress here in Cartesian coordinates but here 
you have the log you do it on a semi-log plot of log numbers of cycles and what 
you find more or less is that the data follow some sort of falling line like 
that and maybe that's ten-to-the-six so that's a million cycles ten-five ten-
four ten-three thousand cycles hundred cycles ten and et cetera one remember 
there's no-, not a zero on a log scale what's the log of zero come on what what 
is it 
sm0828: 
nm0827: log of zero no the log of one is zero what's the log of zero 
sm0829: 
nm0827: it may be in Dublin dear boy 
but it's minus-infinity in the rest of the world [laugh] [laughter] the log of 
zero is bloody big it's minus-infinity so the zero on a scale is over there 
somewhere so never put a zero on a log scale now old Wöhler f-, was playing 
around with steel and he discovered that after about ten-to-the-six after about 
a million cycles the thing levelled out that was pretty interesting really and 
so that was called an endurance limit and remember he was dealing with steels 
or wrought irons or some ferrous object anyway and that was for simple so-
called reversed bending and reversed bending is like Wöhler and his cantilever 
where you have that so it goes from plus- T to minus-C those are equal 
magnitudes and you go through zero in the middle now if you have an endurance 
limit that's great if you think about it because what it means is according to 
this diagram that if you have that sort of loading system providing you keep 
your stresses below this endurance limit then you have an infinite life so you 
can have a life forever providing you recognize that there is an endurance 
limit or what is sometimes called a fatigue limit i'll explain there is a 
difference and i'll explain that in a minute but fatigue limit or endurance 
limit and that happens for steels now if you don't test it just in bending 
which is what we've been doing here you might for example want to test 
something in axial fatigue so you would have your specimen in the usual way but 
you would oscillate it in the axial direction this would be like trying to 
estimate what's going on with the spokes 'cause they're going tension 
compression tension compre-, but in unidirectional loading not in bending is 
there a difference can you make a machine which does this well of course you 
can do it on the types of testing machines down in namex's lab these are screw-
driven machines and if you think about it you just drive 
the screw one way drive the screw the other way and you can do this you might 
be saying to yourself well er it's all very well talking about rotation speed 
of oscillation frequency of oscillation but does how fast you do it matter is 
there a difference between sort of oscillating very slowly and oscillating you 
know like a i don't know a er bird flaps its wings or something else something 
very very fast and the answer is yes there is and it's particularly important 
in things like plastics where because you are oscillating the load you're 
obviously doing work force point of application that work becomes heat in a 
metal the heat tends to be dissipated but in plastics it tends to be localized 
where the thing is happening that then alters as we saw the other week the 
properties the strength varies with the temperature and the speed at which 
you're doing it and so on so the speed does matter but whilst in the lab with 
the namex type machine the oscillation rate is pretty 
limited pretty limited you can have very fast v-, axial oscillations and the 
firm that invented the method of doing this is a firm called Amsler who are in 
Switzerland and what they did was really very clever because they attached what 
amounts to a whacking great big tuning fork to the bottom of the specimen i 
mean a big one really big the biggest Amsler machines the tuning fork would be 
about the size of where i am here and you can get that resonating and of course 
it puts the er device into oscillation and you by altering the amplitude of the 
tuning fork you alter the load you can set this on the machine and then you can 
generate er these same sort of data well what do you get what do you get what 
do you get well it's like this but this plot if i say this is bending pure 
bending if you have the axial thing this says the same sort of thing but goes 
to a lower value before it levels out so this would be axial fatigue and you 
might also say er well what about other methods of 
loading what about if we twist things you've heard of torsional suspensions on 
motor cars what happens if you think of having a tube or a rod which itself is 
not being twisted monotonically but is fluctuating in the twist what happens 
there well er the same sort of thing but this time it goes down h-, even lower 
down to there not drawn that very well so that this endurance limit although 
it's very good the values that you have depend on the mode of deformation so 
bending to axial er to torsion to twist now er before the days of what's called 
fracture mechanics which is the subject i shall teach you next term where we 
are talking about the progression of cracks through bodies before that time 
people were limited in a sense as to what they could use for data and they 
tended rightly or wrongly to reference everything to the simple tension test so 
in the simple tension test which you know about load against extension or 
stress against strain er engineering stress and 
strain there is your yield there is the ultimate tensile strength we discussed 
this in the corresponding lectures in part one so what these chaps did from 
about eighteen-fifty up until well up until the second war and even now in fact 
for a for an empirical sort of corre-, correlation they said what's that value 
bend or axial or torsion as a proportion of something that i can easily measure 
and the thing that they could easily measure was the ultimate tensile strength 
maximum load over the starting area all right so when you look at these types 
of diagrams you will find labelled on to the diagrams what these levels are so 
if i just rub that out put them in again you'll find that the bending-only one 
is about and it's very rough about point-five of the U-T-S so in other words if 
you had a piece of steel with a U-T-S of about three-hundred megapascals er 
that value for pure bending pure reverse bending Wöhler bending would be about 
er three-hundred-over-two which is a hundred-and-fifty if you have the axial 
fatigue that value is about point-four-three of U-T-S i don't know that i 
believe the point-three i suppose i believe the point-four but it's that sort 
of thing and if you go down to torsion the lowest one there it's about point-
three of the U-T-S so again a three-hundred megapascal very weak low carbon 
steel in torsion would not take more than three-hundred-over-three er about a 
hundred megapa-, pascals reversed twist so you need to 
think about that if you're designing a torsional suspension for a motor car now 
then this is all very well but is it true for all materials and unfortunately 
the answer it is not true for all materials what i've drawn there with these 
endurance limits are true only of irons and steels as for practical purposes 
titanium and one or two other funny things perhaps er er have these sort of 
features but the important thing is and in particularly in terms of normal 
engineering is that aluminium alloys do not have there's no fatigue limit so if 
you were to plot these types of things for aluminium alloys you would have this 
just going down and down and down and down and down and so if ten-to-the-six a 
million cycles is the endurance limit for er steels then it doesn't level out 
for most aluminium alloys you check in the books which particular metals obey 
the endurance limit idea and which don't the important ones 
as i say are the ferrous and aluminium alloys but what does that mean what does 
that mean if you want to design an aeroplane which is made mostly out of 
aluminium alloys 
sm0830: has to be scrapped after ten years 
nm0827: [laughter] ten years no well good idea well the implication of this is 
that of course you cannot design for infinite life that is very true and what 
it does say is that you have to be very very careful about inspections you have 
to look for cracks it's cracks that we're dealing round to and i'll be showing 
you some pictures of fatigues in a minute you have to look for er cracks now of 
course if you have a piece of steel and you are in an application where you 
don't want it to last more than a million cycles then you don't have to limit 
yourself to working below here you can work up here and a typical example of 
that would be a gun barrel think of the old naval guns you know twelve inch 
shells whatever i'm reading a book at the moment by Edmund Blunden called 
Undertones of War 
about all the ghastly things in the trenches in the First World War and he goes 
on about these whizz-bangs that go over and you know these are ten inch shells 
which just plop in the mud and they're duds they don't go off if they do go off 
he wouldn't have been around to write the book anyway these big gun barrels 
you're not going to fire that barrel a million times naval guns if you read 
Jane's Ships and things like that naval guns those big guns were not even after 
things like Jutland were not fired that many times so you might argue that you 
could be in a thousand cycles and so you could design at the higher stress 
level if you design at the higher stress level of course you want less material 
so it's this weight business that comes into this and this is important in 
relation to aluminium because of you know space vehicles aerospace light weight 
materials very high specific strengths toughnesses and so on and so forth so 
you don't have to go 
below there but if you want the thing to last for a long time and you don't 
want your customers coming back and saying oi i you know bought something from 
you and it's broken you don't want to have the reputation of being the Arthur 
Daley of er engineering design or whatever it is then you have to think about 
these things and of course there are standards which say you must design below 
these stresses if it satisfies this particular standard but you don't have to 
and in some circumstances you needn't but the problem is that although this 
part of the diagram is called the finite life region and of course this is the 
infinite life region you have a problem with the 
aluminium alloys 'cause you don't know where you are because it's all finite 
life there is not an infinite life with those things you've still got a design 
you still have to tell people how good or how bad aluminium alloys are so what 
happens is you still quote this value here in the same way that you quote this 
value here even though it doesn't plateau out and you don't call it the 
endurance limit you call it the fatigue limit which was the other word you see 
er those of us like me who tend to be a bit sloppy we use the things in-, 
interchangeably but strictly endurance limit is the ferrous thing and fatigue 
limit is a chosen value at a given number of cycles