Tubes

Aircraft, bamboo, bones, bridges, drinking straws,

feathers, insects, pipes, submarines and tunnels

17th June 2001 Back to Home Page

Arch Beam Box Girder Cable stayed bridges

Cantilever Pre-stressed bridges Truss Oscillation

Damselflies and dragonflies have many features of a typical insect. One of these is a cuticle, or exo-skeleton, made of chitin. Like many natural materials, chitin is very versatile. It can be thick and rigid, or thin and flexible. It even penetrates inside the insect in the form of breathing tubes, which are inward developments of the cuticle.

The abdomen of these insects is tubular, which is advantageous for breathing, because the animals have no lungs. Air simply diffuses, helped in some cases by movements of the abdomen. So an insect can never be enormously big, because every part needs to be near the surface. A tube is just the right shape for a creature that spends much time flying, and needs much oxygen.

The long tube, with a relatively large moment of inertia, may help to stabilize the insect against the motion of the four large wings.

The wings of dragonflies are based on tubes, arranged in a fine network. More "advanced" insects have fewer veins, often very few indeed, and they also tend to use only two wings instead of four, just as aircraft evolved from biplanes to monoplanes.

Even those with four wings have often evolved to the point where one pair acts only as a cover for the other par. In butterflies the two pairs are linked in flight by their shape. Some moths have hooks for the same purpose. In flies, the rear pair of wings have evolved into oscillating halteres, which function in balancing, much as a gyro does.

The word "advanced" is not entirely suitable, for flies, dragonflies and ourselves all made it to the present, after descent from a common pair of individual creatures, which must have lived several hundred millions years ago. So in a sense, we are all equally successful – so far. Which will be the last to survive – the descendents of ourselves, flies, or dragonflies?

The dragonflies were around long before the flies, which in turn preceded people by many millions of years.

Returning to the tubes in the wings, these have two main functions. When the insect has emerged from the previous skin, the wings are tightly crumpled, and are very small. If you watch an emerging dragonfly, you can see fluid moving around the body, through the transparent cuticle. The fluid expands the body to its final shape, and it is pumped into the veins of the wings In the first picture the Libellula depressa has completed the expansion. In the second picture it is almost ready for flight, but it it will not get its body colour for some days.

The abdomen of the male is mainly pale blue when mature. That of the female is yellowish brown.

The main wing spars of a large airliner are usually in box form, which is very convenient for carrying the fuel, as well as for stiffening the wing. Carrying the engines and the fuel on the wings may seem wrong when the aircraft is on the ground, but in the air, the weight is better distributed than if the fuel and engines were carried by the fuselage. The bending moment in the wings is much less of some of the weight is carried by them directly.

On the ground the wings have to take a reverse force at the roots.

People have sometimes tried to make aircraft with everything inside a thick wing, but such schemes have never been as successful as those based on division of function among specialist parts, such as fuselage, wings, tailplane, and fin.

All animals that are highly evolved for flying or gliding have separate organs for flying. Animals such as gliding frogs, squirrels and snakes have rather poor gliding angles, have not developed proper wings. But who knows how their distant descendants might look?

As in aircraft spars or box-sections, the other function of the veins in the wings of an is to stiffen the cantilever that each wing actually is. The stiffening is greater near the front, so that even a crude up and down flapping will tend to produce lift and forward motion, as the angle of incidence changes during each flapping cycle. In fact the motion of the wings is very complex, and the attachment to the thorax, and the musculature and structure of the thorax, all contribute to the ability to fly. Some damsels can fly towards a spider’s web, collect the spiders’ prey, and fly backwards with the booty.

Most dragonflies can catch prey on the wing. They are superb fliers. Some can fly from dawn to dusk, catching all the food they need, and looking for a mate as well. Others only perch, and dart out when prey is seen. With up to 20000 ommatidia, or individual sensing tubes, in each eye, the dragonfly can sense quite slow movements and very small objects.

Making reasonable assumptions, it is not difficult to calculate the smallest size and speed of what a dragonfly can see.

The spiny legs, used for seizing prey and taking it to the mouth, are tubular like the body. As with all insect muscles, the leg muscles are attached to the exoskeleton. Living in a box or a tube may have limitations, but it is a successful mode of life, judging by the enormous number of insect species. The most numerous insect order in terms of species is the coleoptera, or beetles, which are especially well protected by wing cases and cuticle.

Where the insects lose out is in size – they can never become the type of giants that are sometimes seen in films.

Most dragonflies look very pale for a long time after their last moult, and may take many days to acquire their full colours. But an amazing thing happens with some which have iridescent colours – in the space of only minutes you can watch an almost transparent body become a metallic green, for example. These colours are not caused by pigment, but by interference of light at tiny structures on the surface.

Click here and here to find out more about iridescent colours.

It’s hard to look at something like a dragonfly without thinking that there must be something special about a creature that can fly non-stop for sixteen hours, find its own fuel, and reproduce. Yet in a sense it is not a perfect flying machine. It could fly just a little better if it didn’t have to carry digestive organs, legs and reproductive organs. But then it would die out. So it is not quite perfect at any one job.

At the other extreme is a female termite, which is little more a huge bulk for making eggs. It really cannot do anything else, except control the workers. It relies on the workers it controls to sustain it. It is almost a perfect reproducing machine. What the species has done is effectively to export some of the work of its organs by using other individuals. Some plants and animals take this to the extreme of parasitism.

We can admire the flight skills of the dragonfly, but there must be thousands of other attributes which are equally well tuned. What is most perfectly tuned by evolution is not any one attribute, but the probability of reproduction. In a hugely dimensional probability space, each species lies near a local maximum of probability. Perhaps the peak is never obtained, because conditions change,

if only because all other species are changing, and so the probability map is changing. So each species is a little behind perfection all the time. It would be utterly impossible to compute a prediction of the evolutionary path of a system of life-forms. In effect, life on earth is a giant Monte-Carlo program, continually trying and rejecting or using possible paths.

This aircraft, like many others, has a fuselage based on a tube, and wings based on box sections. In fact, since people have to be inside, almost all forms of transport use an exoskeleton, like insects, with the living animal inside. One famous car was even nicknamed "Beetle". The bicycle and the motorcycle are exceptional in that the user is outside the vehicle.

They are more closely related to the horse, donkey and mule. Both types of bike use tubes in their construction. Trains and cars can go through mountains or under rivers and seas in tunnels, which are circular tubes, like submarines, to resist the pressure, in places where the rock is not strong enough.

However, in many applications, it is simpler for fabrication and assembly to use rectangular or trapezium sections, as in the towers of many suspension bridges.

The feathers of this peacock, like those of other birds, are based on tubes.

This makes it easy for an austringer to repair a bird’s feather, by "imping". This means putting a wooden splint inside the shaft and gluing it in place.

Many of the bones of a bird are hollow, with stiffening struts joining the walls. Again the tube provides lightness with strength.

The bamboo and many other plants use the same principle. The bamboo stems are stiffened at intervals by thicker parts. The tubes also carry vital fluids, a good example of dual function, like the wings of an aircraft, which often carry fuel. A bonus is the improved weight distribution, reducing the bending moment in flight. The bicycle frame is a familiar tubular structure.

Torsion and Bending – How do tubes work?

If you examine a cardboard tube, such as a mailing tube, you will find that it is resistant to torsion, bending, compression from the ends, and even compression from the sides.

Yet the same amount of cardboard in the form of a flat strip is quite flexible. What is going on?

If try to twist a tube, every part of it is subjected to an equal shear stress, and so it will try to twist. But the shear is in the direction in which the tube is large, and so the tube can resist.

Effectively we are trying to distort a large rectangle into a parallelogram.

If we try to twist a flat strip, a much smaller area has to take the stresses. Furthermore, much of the material is near the neutral axis, where it cannot have much effect on the result.

The diagrams below illustrate these points.

The next diagram shows two hoops joined by twenty rods. A relative rotation has been introduced. Does this tell us anything about stresses in a tube with applied torques?

The second diagram shows the helical strain (exaggerated) experienced by a twisted tube.

Let us look at bending. If we bend the flat rectangle that we used above, we are only resisted by the thickness of the material. As described in the page about beams, what matters is the force the cardboard can generate, multiplied by its distance from the neutral axis.

If we consider the tube, we see immediately that the average distance of the material from the neutral axis is much greater. So the tube can generate greater moments to resist the external bending moment. Therefore it is more stiff.

A similar argument applies to compression. If a thin rectangle is pushed at each end, it can easily buckle.

But if the same amount of material is rolled into a tube, then from whichever side we view it, there will be appreciable thickness.

Resistance to buckling will therefore be greater.

The diagram at left shows a cross-section through a tube. If the tube is bent in a horizontal plane, the green line is the neutral axis. Orange represents compression, and blue represents tension.

The intensity of the colour represents the effectiveness of the material against bending, being proportional to the stress.

The first graph shows the distribution of material as a function of distance from the neutral axis.

The second graph shows the total stress as a function of distance from the axis. We can see that much of the material is in effective positions.

If we imagine the tube unrolled into a vertical flat strip, we can see that all the material would be near the neutral axis, where is can develop little stress to resist the bending.

Analysing the bending of a tube is not easy because it can deform into an oval cross-section. On the other hand, if you make a curved tube with an oval cross-section, it can be made to change its curvature in response to the difference of pressure between the inside and the outside. This, with amplification of the tiny movement, is the basis of the Bourdon gauge.

The cross-section of this lamp standard is polygonal rather than circular. Why do you thick this was preferred to a circle? Note the method of joining the tapered tubes by inserting the narrower end of one into the wider end of the next.

The tubular chimneys in this tower are supported by a truss made of tubular members. Simple fabrication techniques make the use of tubes economic for many applications. For large tubes this is not necessarily the case. The tubes of the Firth of Forth railway bridge were assembled on site from curved plates.

The tubes of the first Saltash bridge were also assembled from curved plates. Large tubes will need flanges inside for stiffening. More on this below.

The arches of this bridge comprise welded steel plates, arranged so that no more than three meet at any point.

The bones and feathers of the swan, like the tank and the chimney, are based on tubes.

The jib of this crane comprises a series of square-section tubes with rounded corners. This construction is rigid and strong.

The tubes are not as light as trusses could, but are very much simpler to use in an extending crane.

The reeds on the left are holding up their heads to allow pollen to depart and arrive on the wind. Some of those on the right have collapsed. The sudden collapse is typical of cantilevers, including box girders under construction, and of tubes. Tubular beams and cantilevers need stiffening flanges at intervals, like a grass or bamboo. In order to bend the metal pipes that are used for plumbing, without buckling, you sometimes have to insert a strong and stiff helical steel spring.

During the 1960s and 1970s, the bubble chamber was a powerful tool in particle physics, for it made visible trails of bubbles where charged particles had passed. Now the trails are detected electronically, reducing operating times from about one second to microseconds, or even nanoseconds. This picture shows the escape vent for a very large bubble chamber, which was based on a large tank of liquid hydrogen, chosen because the nuclei of hydrogen contain only one particle, which simplified analysis of the results.

A tube uses material efficiently to make a strong, rigid member. Tubes are used extensively in the natural world as well as in manufactured articles. The veins of an insect’s wing are filled with fluid under pressure to unfurl the wings after emergence. Subsequently they provide a stiff network, designed to allow the wings to bend in just the ways that efficient flight demands. Many plant stems and bones are tubular.

The principle is to get the material as far from the axis as possible. This construction resists torsion very well, and resists bending because the tension and compression are far apart, providing a large moment. Other examples in bridge building are the Forth rail bridge and the Menai Straits tubular bridge.

It is one thing to say that a tube is the ideal shape for a compression member, or strut. It is another thing to implement the idea. Certainly a tube achieves the ideal of getting the material as far as possible from the neutral axis, but a large tube is not a simple or a cheap thing to make. In the Forth railway bridge and the Saltash railway bridge, the tubes were built up by riveting many curved plates together. This was labour intensive. Furthermore, if a tube is very large it will need internal circular flanges, and possibly longitudinal flanges, to stiffen it. It is not easy to connect these to a tube without introducing unwanted strains.

In a sense, there are local axes from which the material needs to be separated in order to increase the stiffness.

Click trusses and skew arches for more on tubes.

The nest of the polistes wasp has many hexagonal cells, which help to support each other. The hexagon is the shape that minimises the volume of material used. The angle of 120 degrees is found in soap bubbles and in cracks that have developed simultaneously.

The principle of moving material away from the neutral axis finds use in corrugated sheets, used in both construction and in packaging. These sheets are rather like many parts of tubes added side by side.

Here are some examples.

The corrugations only stiffen the material in one direction, but used with battens, purlins or stringers, the sheets make very good cladding. The Nissen hut was an early example, which used curvature to obtain rigidity. Some trucks and small cars have used the principle. In the fourth picture we see that moss has colonised the roof. This not the only life-form to have used the roof. When a narrow pipe was left on it, a leaf-cutting wasp made a nest in it. Because the garage is rather ugly, a climbing plant has been planted.

Some aircraft made by Junkers before 1920 had corrugated skins, with the grooves parallel to the line of flight. Unfortunately, the airflow over both fuselage and wings is not parallel to the axis, and so the drag was high. Furthermore, the skin could not take stress at right angles to the grooves. The ability to stress the skin of an aircraft is extremely valuable, as long as it does not tear, because these stresses enable the skin to contribute greatly to the strength and stiffness of the structure. This will be discussed later in this page.

Some large rockets have have corrugated skins. Because there is no lift, the airflow, neglecting turbulence, is in principle along the axis. Furthermore, for an object that will spend only a short time in the atmosphere, drag may not be the primary concern. The corrugations do nothing for rigidity at right angles to the axis, and so frames are needed, as in an airliner.

Corrugated cardboard usually comprises a sheet of corrugated paper with flat card one or two sides. Surprisingly strong containers can be made of this material. Another method of stiffening is to use a hexagonal array like a honeycomb, sandwiched between flat sheets. This can be thick and very rigid.

In insects, the skin is not only specially shaped in many places – it even helps to control the flapping of the wings. The idea of corrugations must not be confused with the idea of riblets, which are minute ridges that seem to reduce drag in aircraft.

A thin sheet that is flat has little rigidity. It is useful as a drum-skin, sound-board, wobble-board or thunder simulator.

Returning to the subject of tubes, we find that corrugated tubes or bellows are ideal when flexibility is required. many loudspeaker cones have a roll surround that is about one half of a torus. Many loudspeakers also have a central support behind the cone, in the form of a corrugated disc. An aneroid barometer uses the same principle.

In this picture, the left hand unit is a fairly cheap wide-range drive unit, using a surround with roughly circular cross-section. The right hand unit is a high quality bass and mid-range unit by KEF, using an asymmetrical section for the surround.

We have seen that corrugations can be used both to stiffen things and to make them more flexible.

Let’s look at tubes more carefully. We have already seen that corrugations can be used to stiffen tubes or to make them more flexible. So, as usual, things are more complicated than they might appear. The diagram below shows a set of concentric rings.

All the rings, both blue and green, have the same area, and so they represent cross-sections through tubes that all use the same amount of material, which is the same as for the rod in the middle. The large tubes will be more resistant to overall bending, but we can see the walls getting thinner. Particularly in the longitudinal direction, we are losing local rigidity, even though the tube is stiffer on a large scale. Putting it another way, as the ratio of the radius of curvature to thickness increases, small areas behave more like a flat plate. This is why corrugations or flanges are needed in large tubes.

So simple rules such as "The bigger the radius, the stiffer the tube." need to be qualified. With any rule, knowing the area of applicability is as important as knowing the rule itself.

The smallest known tubes are probably the "nanotubes", which are made of carbon with a curved graphitic structure. These were discovered as a result of the investigations of buckminsterfullerene, which has a molecule comprising sixty carbon atoms with icosahedral and dodecahedral symmetry, which are in fact the same thing.

Right, circles are good. In a symmetrical world, yes. But our world has less symmetry than empty space – gravity provides an axis everywhere. Only in the world of the very small, especially under water, do we see living structures in which up and down do not matter. Even then, they are rare, because the pressure gradient and light gradients provide an axis, as do the variations in dissolved gases and in food supplies. Some organisms, such as volvox, are based on spheres, but even the humble daphnia has a top and a bottom.

Let’s look at a couple of pictures. In the left one, a brick is supported by the packet of a bar of chocolate, which has a triangular cross-section. A cylinder would probably have done as well, because gravity acts along the tube, so symmetry is maintained. But look at the second picture. Are you sure that a paper cylinder would have supported the brick when the forces act across the tube? Try a little experiment. Take a long narrow piece of paper and fold it neatly on the long centre line. Give the paper an L-shaped cross section. Now rest it on supports at each end, firstly with the ridge upwards, and then with the ridge downwards, making a trough. Is there any difference in the behaviour? If not, make a new version, longer or narrower than the first. Bridges have been built with triangular cross-sections. Brunel built one. One footbridge on the A1 autoroute going north from Paris has a triangular cross-section. It has the ridge at the top, as in the photograph of the chocolate wrapper.

Connecting to Tubes

It is much easier to connect to a plate girder than to a tube. The same problem arises when a tube has to be connected to another member. Achieving this it satisfactorily is not easy. Look at a photograph of the foot of one of the great towers of the Forth rail bridge. The method of connecting the tubes of the towers and of the cantilevers is quite complicated, in order that the stresses could be transmitted satisfactorily from the cantilevers to the piers, and from the piers to the foundations below. In fact a structure can even be made weaker by adding "strengthening", if the additions introduce undue strains, and therefore stresses, that were not there before. This can happen if the resulting structure is over-determined and poorly constructed. See also the page on attachments.

These tubular masts are stabilised by struts which in turn are held by wires anchored at each side of the yacht. The reduction in the effective length of the tube means that it can be laterally narrower, this reducing weight. Where the struts join the masts, there must be some reinforcement to cope with the stress concentration. The masts, struts and wires in a yacht like this can be subjected to very high stresses in heavy weather, which can in extreme cases dismast the boat.

Cracks

Another problem with stress concentrations occurs in designing the fuselage of a pressurised aircraft, or the deck of a ship, when material has to be removed from the ideal tube. The fuselage of an aircraft has to be pierced by various holes for doors, windows, wheels, antennas, and so on. The openings have to be designed carefully, to prevent stress concentrations. The Comet 1 airliner suffered explosive decompression when fatigue, starting at a hole, resulted in catastrophic spreading of cracks. This phenomenon is now much better understood, and all designs would now include measures to reduce the probability of cracks being generated, and also measures to prevent their propagation over long distances in the structure.

Sharp cornered hatches in the deck of a ship can result in stress concentrations which can be the source of cracks, which can propagate if not stopped. As a result of considerations about construction, tubes are not employed very often in bridges. What is beautiful to the engineer, the aesthete, and the financier may differ quite strikingly. Fritz Leonhardt, in his great book "Bridges", explains the desirability of reaching a satisfactory resolution of these questions.

Nature does not experience the same constraints as people. Nature’s constraint is that each step in evolution be attainable from the previous one, and that it should be a slight improvement in some way. The improvement need not be one that can be recognised millions of years later, when the use of an organ may be completely different from a previous use. Improvements that require a temporary set-back, however small, in overall probability of reproduction, cannot happen. Evolution has no foresight.

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 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.

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 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.

Such a time-dependent failure had been envisaged in the novel "No highway", in 1948, by Nevil Shute, though he may have thought in terms of time rather than the integrated cyclic stress.

More About Cracks (including some of this material again)

See also Hampton bridge and Saltash bridge

Arch Beam Box Girder Cable Stayed Cantilever

Pre-Stressed Truss Oscillation

Photograph of Forth Bridge tube –

http://mulder.umist.ac.uk/civil/research/historic/joe/forth_bridge.htm

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