Pressurised Aircraft

Let’s calculate the total force due to atmospheric pressure on the fuselage of a large airliner, such as a Boeing 747.  Let’s use a cabin length 57 metres and a cabin diameter 6 metres.

The area of the cylindrical surface is roughly 3.14 X 6 X 57 m², which is about 1100 m².  If the cabin is pressurised to the equivalent of 2500 m, and the aircraft flies at 10000 m, the pressure difference across the skin is about 0.075 – 0.025 MPa = 0.05 MPa.  Multiplying this by the area gives the total force, about 50 MN, equivalent to 5000 tonnes.  

To find the energy stored, we can multiply by half the radius to get a very rough value, giving 75 MJ.  That’s a lot of energy.  If an explosive decompression occurs, the result is quite unpredictable.  A Boeing 737 survived the loss of a huge piece from its fuselage, forward of the wing, and was landed successfully.  But in other cases, quite small holes, caused by explosions or structural faults, have had catastrophic consequences.  The famous early example is the loss, in 1954 of Comet 1s G-ALYP and G-ALYY, caused by explosive decompression.  

Metal Fatigue

A test at RAE Farnborough, using Comet 1 G-ALYU, duplicated the cracking of the fuselage after many cycles of compression and decompression.  By using water instead of air the energy released was held to a low value, because of the minute change in volume of water when subjected to a change in pressure.  A decompressing airliner has to get rid of about two-thirds of the air in the cabin to equalize the pressure.

Since that time, examples of airliners are subjected to load simulations on the ground at a rate that keeps "aging" much faster than the ones that are actually flying.

The Comets are now a part of history.  But history isn’t bunk, and all designers now know about "metal fatigue".  When a piece of metal is stressed to a level below its elastic limit, and then let go, it will return exactly to its original size and shape.  But in some cases, the appearance is deceptive.  The structure of the material has changed, extremely slightly, but changed nevertheless.  After another application of stress it has changed again.  The effects accumulate.  The material contains its own history.  The Comets that crashed were not the Comets that left the factory.  In effect, at certain highly stressed places, such as at the corners of holes, they were made of weaker material than those that left the factory.

After the Comet came the Caravelle, an elegant airliner that actually used the same fuselage nose section.  Its windows had no straight lines, being shaped like bulging triangles with rounded corners.  A particle physicist might be reminded of a Dalitz plot.  The later Comets had oval windows, with the longer axis aligned horizontally.  Most airliners have windows shaped like the holes in this 35 mm film.  The Comet had four engines buried in the wing roots, but the Caravelle had its two engines attached to the rear of the fuselage.  If you have ever sat between the Avon engines of a Comet 4 you will understand this well.  The rear mounting of engines became fairly common in the smaller airliners, but for very good reasons, discussed in another page, pylons on the wings are favoured for larger aircraft.

This picture shows a Vickers Varsity, which had rectangular windows, like the Douglas DC-3.  These types were not flown at great altitudes, not having pressurized cabins, and so the associated stresses were absent.

The transition from the box-girder main span and the side spans of the road bridge at Chepstow is made using a half-oval shape under the bridge.  A more familiar type of transition is represented by the stress-reliever that is fitted to many plugs where the cable enters.  Whenever a structure includes a sudden change in dimensions, there will be a stress concentration, which can be a source of failure, especially when fatigue is possible.  Fillets and gussets are among the features that are used to reduce these concentrations.  The picture includes a side span of the railway bridge that runs close by.

Newspapers occasionally report the finding of cracks in aircraft.  In many cases these can be tolerated because the structure is designed to prevent the cracks passing certain boundaries.  Nevertheless the cracks are watched during periodic inspections.

If a square hole can generate stress concentration, what about adding a negative square hole, that is, doubling the thickness of a square area by welding to gluing a square patch to the surface?

See "Black Box" by Nicholas Faith (Boxtree) ISBN 0 7522 1084 X, and "Air Disasters" by Stanley Stewart (Ian Allen and Promotional Reprint Company) ISBN 1 85648 182 4.

This picture shows a plastic hose nozzle that was left outside through the winter.  The water inside it froze, and the expansion has cracked the moulding at the weakest point.  A pressurized cylinder will always tend to crack along a line parallel to the axis, because the tensile stresses are greatest around the circumference.  Can you see why this is the case?  Expansion of liquids and solids is hard to resist because their bulk modulus of elasticity is so high.  A gas pumped into a confined space such as a rubber tyre has a much smaller modulus.  Of course, failure occurs at the same pressure, whatever the fluid, but the point is that once a cavity is full of liquid, a small change in volume creates a big change in pressure.

Cracks in Concrete

Cracks in reinforced concrete bridges can sometimes be tolerated unless they allow the entry of so much moisture that the steel rods are endangered by corrosion.  They are caused by tensile stress, which stretches the reinforcing bars.

When cracks appeared in a concrete bridge designed by Robert Maillart, he didn’t respond by strengthening the design for subsequent bridges.  He realised that the concrete wasn’t actually doing much just there, and he removed a large section, resulting in an elegant design that evolved further during his lifetime and has been influential ever since.  See Robert Maillart  The discovery of a new phenomenon in engineering or nature has often resulted from an observation which could easily have produced a different response. 

Microscopic Cracks

A small glass fibre used for optical communication can suffer from the creation of cracks if it is bent too sharply.  Each crack can scatter some light out of the fibre, or at least in the wrong direction.  The loss in dB per km will increase. 

But very thin fibres of glass and other substances can be relatively very strong compared with larger samples, simply because the probability of a crack existing is small.

Here are pictures of cracks in a piece of glass.  Glass is a rigid material, but it is not useful for large structures, because it cannot absorb much energy before breaking.  In other words, although it does not bend as much as some other materials, it breaks under a smaller deflection.  Glass is not normally a crystalline solid: in fact it behaves in some ways like an extremely viscous liquid.  When heated, glass does not melt at a certain temperature, like a metal, it slowly softens.  In some very old buildings, glass windows can be found that are thicker at the bottom, where they have slowly crept over the centuries.

Here is one piece of of a large pane of toughened glass that has been broken into small pieces, each of which is itself composed of many little cells.

Glass is a sort of super-cooled liquid, with a viscosity so high at normal temperatures, that its flow is measured in terms of hundreds of years.  In fact, during such a period, many specimens of glass will begin to crystallise.  During the shorter times with which we are familiar, glass behaves much like a solid substance, though when it is heated, it does not melt at a specific temperature.  The picture shows a piece of glass that has been hit by a small projectile.  Like the liquid drop, it displays non-random behaviour – the spacing of the cracks is not far from uniform.

Stress Concentration

Why do holes have such a dramatic effect on cracks?  Look at the diagram below, showing a piece of metal with two small notches in it.

If we apply a tension to the object, we might expect that in the two planes including the notches, the stress would be slightly increased because of the slightly reduced cross-sections, with a stress flow somewhat as in the diagram below.  We would be wrong – very wrong indeed.

Remembering that the stress is greater where the lines are closer, we see that a large part of the bar has greater energy than it would have without the notch.  What actually happens is that any stressed object tends to the configuration of minimum energy.  In this example such a configuration of stresses is much like the one without a notch, but with a perturbation around the notch.  The area of higher energy is quite small.  It is called a stress concentration.

Let’s just take a break from cracks to see how things work to minimize energy.  The notch example is rather complicated, so we will do something very simple.  We will imagine a set of blocks connected by springs, which can represent atoms in a 1-dimensional solid.  The normal position is shown at the top of the diagram below.

In the second we move one block sideways, creating tension in one spring and compression in another spring.  This condition is unstable, because the two blocks next to the middle one experience unbalanced forces.  Blocks with a black arrow are fixed in position.  The stable condition is shown in the third row, where the blocks are equally spaced, except for the imposed discontinuity at the middle block.

We can compare the energy in the false middle case and in the actual case.  In the middle case, we have displaced the block by a distance D, and because energy in a spring is proportional to square the change in length, we can write E = 2 X A X D², where A is a constant.  But in the stable case, the change in distance in each spring is (D/4), and we can write E = 8 X A X (D/4)², = 0.5 X D², which is a quarter of the unstable value.  The stable condition is of course always the condition of minimum energy.  The same applies to 2-dimensional and 3-dimensional structures, in which every part of the structure ends up in a state with no net force on it, but these can be difficult to calculate unless the geometry is very simple.

In any system, there is no discontinuity in any of the physical variables except where there is a discontinuity in the type of material, or where there is an applied force.  The extreme type of continuity is the surface of an object, outside which there are no forces.  This means that the effects of an applied force can be detected throughout the system, unless something is done to prevent the spread.

Back to talking about notches.  Around a sharp notch the effect is rather like the diagram below left.

In the picture at right, a piece of paper has been provided with a semicircular notch and a V shaped notch, and pulled.  It failed at the point of the V, where the stress was greatest.  Note the similarity to the cracks at the top of this page, and to a flash of lightning, though the details differ.  All these examples are a combination of randomness with a preferred direction of propagation.

If a narrower crack propagates more easily than a wider one, what about a substance with no cracks?  It could be thought of as having an infinitely narrow crack, which ought to propagate very fast indeed.  Look at the wording of the question – it is wrongly stated.  In what way?  Nevertheless, if we can pull hard enough to separate one plane of atoms from another, we have made a very sharp radius at the tip of the crack.

The diagram below shows a simulation of stress lines around a hole in a large slab, only a part of which is shown.

Compare this diagram with the photographs at left.  The same diagram can represent an electric or magnetic dipole in a uniform field, and the laminar flow of a liquid past a circular obstacle.  Pictures and films of Jupiter show similar effects around the great red spot, albeit with subsidiary loops.  But we must not imagine that forces "flow".  

The arches shown at left, based on the ogee shape, suggest that their designers may have thought of guiding the "flow" of the vertical forces into the arch.  If so, they were wrong: forces cannot be deviated without an input of transverse forces.  Simply curving the material does not work.  A normal arch only works because the transverse force is supplied by the weight of the voussoirs  More pictures are shown below.

The cracking of wood does not always have to wait until the tree has been felled and cut up, as this example shows.  The tree has grown on an unstable slope, and movement of the ground has caused the tree to lean outwards from the slope.  This has moved centre of gravity of the large branch much further from the tree, creating a greater moment of the weight.  This has proved too much for the strength of the tree, and a very long crack has resulted in the trunk.  The tree is doomed.  Two trees in the background have already succumbed.

The smaller the radius of a hole, or the tip of a crack, the higher the stress.  So we can stop a crack from propagating by cleanly drilling a relatively large hole at its tip.  That’s not a very good solution for a boat or an aeroplane, but it’s better than having a crack go right round.  This contrasts with the use of holes to enable postage stamps to be pulled cleanly from a sheet.  Once the crack starts to lengthen, the remaining material is narrowed, and the stress increases.  Eventually the rate of movement becomes catastrophic.  The example shows a piece of paper in which cracks have been persuaded to go towards the holes.

Here is a real example.  A small telescope was accidentally dropped on a hard surface.  Two cracks have propagated to a small screw-hole.  As the telescope was black, the picture was made negative: even so, the cracks do not show up well.

The upper part of this picture shows one of four curved slots in the edge of a circular saw blade.  Each slot ends with a larger hole.  What is the purpose of the slots?  The lower part of the picture shows the centre of the same blade.  There are two complicated slots, or at least grooves.  What are they for?

Here are pictures of pieces of plastic fencing.  This type of fence usually has rounded holes.  Sharp corners would make the sheets much easier to rip.

So – we have seen that cracks can start at a hole and finish at a hole, rather like the case of the man in the fable, who blew hot and cold with the same breath.  There was of course, good physics behind his actions – cooling his soup, he was blowing away some energetic molecules that had escaped from the soup, creating disequilibrium, and enabling more to escape, and he was using the expansion of his breath from a narrow orifice to cool his breath.  Warming his hands, he had his mouth wide open, and was blowing air on his hands that was warmer than the ambient air.  The pictures at left shows postage stamps which had perforations all round the edges, to make sure it separated cleanly from its neighbours, though not so easily that a sheet of stamps would come apart through handling.  Sometimes the holes are made elliptical to make sure.  In fact, the tearing is not very tidy on a microscopic scale (why), but this has no practical effect.

This example, a bar of chocolate, also exhibits the property of being breakable in preferred places, but not so easily that it would break in transit.  Chocolate would be a poor structural material, not only because it is weak, but because it softens and melts at quite low temperatures.  After being left in a refrigerator for a long period it is quite hard to break, while in hot weather it is quite soft.  It mimics at low temperatures the behaviour of alloys in the turbines of gas turbine engines, except that those alloys are designed not to reach the soft stage during normal use.

Railway lines can crack.  If the crack becomes a break, a train can be derailed.  If the crack is detected while it is short, the surface of the rail can be ground off.  The rail may be "weaker" by a couple of mm, but in a sense it is "stronger", because the crack has gone.  Grinding off not quite enough is no good, because it leaves the deadly tip of the crack, ready to propagate again.  The combination of a crack and the intermittent loads from the wheels is a recipe for fatigue.

In 2001, after a derailment in England, a section of rail about 30 metres long was found to have broken into about 300 pieces.  That’s about one break per ten centimetres.  When you look at a rail, a long piece of tough steel, this beggars belief.  But think about it.  The carriages, and more especially the locomotive, are very heavy, and they rest on hard steel wheels.  Mathematically, a circular wheel meets a straight rail at a point, giving infinite pressure.  But of course what actually happens is that wheel and rail deform, elastically we hope, until the each is able to deliver the required pressure.  Nevertheless, the contact area is very small.

As the wheel rotates, the stressed area moves around the wheel, every part of the wheel experiencing an oscillatory stress, and an oscillatory strain.  The rail undergoes a similar torture.  This is a recipe for fatigue.  In older times, up to the mid-twentieth century or so, you could stand in a station and watch a man walking along a train, tapping all the wheels with a tool.  A cracked wheel, like a cracked bell, does not ring.  The defect damps the sound, and a short sound makes a wide bandwidth, producing a dull sound instead of a clear ring.  

This is a fundamental theorem in physics, and you can even find it in those unbelievably small objects, "elementary" particles.  Lives as short as 10^-23 seconds can be measured using not a stop-watch, but a weighing machine, because the weight is related to energy, and energy is related to frequency, and, as we saw with the bell, frequency bandwidth is related to decay time.  Many species of particles live so short a time that they cannot be detected, except by collecting the debris from their disintegrations and calculating the mass.  We’re way off the subject here, but that is what this web-site is really about:  everything is related to everything else.  You can go out and find your own examples – a better occupation than reading this.

The rubber tyres of a road vehicle or an aircraft also undergo cyclic stresses: you can see many a shredded tyre on or by the road, any day of the year.

If you stand in a station and watch a train slowly moving past, you can see what happens as the wheels cross the gaps in the rails, unless the track is a continuously welded one.  You see the track deform, producing a little step at the junction.  If the fish-plate is loose, the step can be quite big.  It’s not often that you can see the deflection of a structure, because it is usually too rigid.  Sometimes you can see the wings of aircraft bending, as they pass through air moving with different velocities.  The wings of a high performance glider may almost touch the ground when stationary, but as it gathers speed they start to straighten out and then they bend upwards.  During a launch by winch they bend even more than they do during normal flight: there is a launching speed limit for each type of glider.  You can see the same effect in a B52 at an air show.  The wings of the B47 were even more flexible.

Going back to the railway tracks, perhaps we can see why they are not just bolted down on to a "rigid" concrete road.  Usually the rails are bolted or clipped on to beams called sleepers, which rest on stone ballast.  Why do you think they are built like this?  Effectively each rail is a beam bridge on many supports, and the longest bridges in any country that has railways are the railways themselves.

What do you think were the causes of the cracks illustrated below left?  And what governed the direction of propagation?  Why didn’t they propagate vertically by zig-zagging among the bricks?  What about the one shown below left, where a wall butts on to a building?  In contrast to some of the other cases, this crack has gone right through several bricks.  The fifth example, where a tree has pushed a wall near a corner, shows very clearly the inability of masonry to take tension.

Although holes can be used to stop the propagation of cracks, they are commonly used to help the separation of postage stamps, kitchen tissues, and many kinds of forms which have a return slip.

These two pictures show large slabs around two polygonal fountains near a civic building.  In the first example, all but one of the twelve slabs is cracked right across.  In the second example, all the slabs are deliberately divided into two sections, none of which has cracked.  Which fountain do you think was built first?

These three pictures show parts of an old Cotswold barn.  The wooden beams and two of the capstones have cracked as a result of bending stress.

Thermal Expansion

Thermal expansion of long objects requires measures to counteract the possibility of damage.  Many railways have gaps between the rails to allow for expansion.  Many bridges have gaps between sections, with rollers to allow for changes in length.  Long pipelines have bends or meanders to allow for changes in length without cracking.  Large, fast aircraft such as Concorde expand significantly at cruising speed.  Since they employ so many different materials, great care must be taken in the design, especially of the longer parts.