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 bones of human legs comprise an outer layer of compact bone, and an inner structure of cancellous or spongy bone, which consists of many small struts.  The shapes of bones and their are exceedingly subtle, reflecting the variety of stresses and attachments that they have to bear.  To design an artificial structure to such a degree of refinement would be incredibly costly, and would achieve nothing.  Whereas a minute small degree of superiority can be significant under natural selection, where the type of engineering costs are completely different from ours, human engineering refinement is halted by cost and time at a much simpler level.

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.

Here are some sections through an iris leaf, which assumes a graceful curve from root to tip, rather than collapsing like the rule. We see also the cellular construction, with the stem divided into numerous tubes, though this is obscured by the liquid remaining in some cells.  The walls of these tubes, like the rest of the leaf, are cellular as well.  In the lowest two sections the leaf has a double thickness.  In fact it is split on one side into two halves.  Between these halves another leaf emerges.  You can see the edge of this leaf in the lowest section.

Near the root, all the leaves are nested together, making a thick structure which is much more resistant to bending than the individual leaves.  But if there is no connection between the leaves, and they act individually, the stiffness is much reduced.  Friction between the leaves, acting over a large area, bonds them together to form a stiff structure.  This is aided by the prestressing forces in each leaf, which enable it to act like a pair of jaws, clamping the next leaf very firmly.  The picture shows two of the leaves, at left and right, which have been pulled off, allowing them to close up.  They were originally around the smaller part which is in the centre of of the picture: that part includes the two newest leaves.

These two pictures illustrate the way that the leaves are held up by just enough stiffness to spread them out.  A few leaves have developed hinges; in these cases the stresses have changed throughout the leaves, making them all much more straight.  

Here we see how the leaves spread and separate.  Each leaf, once the enclosed leaves have separated, closes its slot and becomes a single strong cantilever.  Some plants use a very different strategy: once the stems have separated, the two side curve over and join to form a very strong tube, enabling broad and heavy leaves to be supported.

This picture reminds us that no structure, man made or otherwise, remains untouched by outside influences, such as wind, rain and sun; and biting, cutting, stitching, boring rolling, sucking and tunnelling by insects can all contribute to changes in leaves.  But if the leaf is to be a habitat, it must not be so badly affected that it collapses.  Here, a spider has created a silken home for its eggs.

The reeds on the left are holding up their heads to allow the pollen to depart and arrive.  Some of those on the right have collapsed.  The sudden collapse is typical of cantilevers and of tubes.  Tubular beams and cantilevers need stiffening flanges at intervals, like a grass or bamboo.

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.  

As a general rule, to make something as rigid as possible, try to arrange that the movement you are trying to resist causes bending, torsion, stretching or compression in as much of the structure as possible, and by as big an amount as possible.

The diagrams below illustrate these points.

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

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.

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.

These pictures show some tubes that have been bent to failure, showing the typical widening, followed by buckling on the compression side.  This is one reason for incorporating longitudinal and circumferential flanges.  A plumber often introduces a helical steel spring into a pipe that is to be bent.  The spring is removed after the bend has been made.  

Compare this result with the joints in the legs of the spider crab.  The difference is that bending the tube made a hinge, whereas the hinge in the leg allows bending.

When we bend a tube, we see that there is a point of no return: the graph of stress against strain is discontinuous at this point.  This is a property of many mathematical and physical systems.  Catastrophe theory provides a framework for describing and calculating.  In thermodynamics, phase changes are the equivalent of buckling, though these changes can often be reversed.  Reversibility in thermodynamics also has a very strict meaning, in which entropy does not increase.  One statement of the second law of thermodynamics is that states which are infinitesimally different may not always be connected by reversible processes.

An irreversible change of a quite different kind is described in minute detail in the book "De l’amour" by "Stendhal", where the author describes in minute  detail the irreversible buckling of a person’s will under the spell of another.  People sometimes say that someone has "flipped", when they make a transition into a completely different mood, or even an apparently different personality.  The manic-depressive condition is an extreme example of a system with two quasi-stable states.  The flip-flop in electronics is a two-state device which is much used in digital systems.  Digital systems are much like analogue systems, though they spend much of their time in stable states.  But during the transitions, they behave like any analogue system.  In very fast systems, the waveforms don’t look like the neat, square signals beloved of elementary text-book writers: sometimes they are almost like sinusoids.   

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.

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.  Do you think that the sections could ever be separated? In all probability this would never be necessary in this application.

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.  These tubes have both longitudinal flanges and circumferential ones.  More on this below.

And the Forth railway bridge includes some truly massive tubes.  The diagonal tubes in the towers have a rounded rectangular cross-section.

This footbridge in Edinburgh is on a much smaller scale.

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

Sometimes it is more economical, or simpler to build, using rectangular sections.

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 be, but are very much simpler to use in an extending crane.

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.

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.

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.

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 a bridge at Windsor with an arch of triangular cross-section.  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.

This picture shows a tube of square cross-section.  Is this better or worse than a triangular tube?  What about a pentagon, a hexagon, or other polygons.  The more sides we use, the closer we get to a circle.  One criterion is the overall stiffness of the tube against bending and torsion.  The effect of the material is related to the distance from the centre.  For a given length of perimeter, the circle has the greatest average radius and the triangle the least, with the other polygons intermediate.  Furthermore, the greater the number of sides, the smaller the flat areas between the changes of direction that help with rigidity.  On the other hand, a square section has greater changes of angle at the corners than an octagon, for example, and its corners are probably more resistant to buckling.  We also need to consider the ease of design and construction, and the effect of that on cost.

Tubes are so ordinary and so ubiquitous that we take them for granted.  They are available in quantity in many sizes and many materials.  But imagine a world where they weren’t.

Two pictures showing tubes at work.  The tubular veins of the butterflies’ wings are used firstly to expand the wings, using fluid pumped through them.  The fluid is then withdrawn.  The veins, along with the creases, then act as stiffeners for the four cantilevered wings.  The abdomen, legs, and proboscis are tubular as well.  The proboscis, evolved from paired mandibular parts, is rather like a mobile concrete pump in reverse.

This is not an insect – it is an arachnid – but it also illustrates the point.  It lives in houses, in places where it is not easily seen.  In such places its fragility is not challenged by wind.

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, and this bridge.

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.

Sometimes a structure is subjected to forces for which it was not designed.  This lamp post was hit by a vehicle, and bent through a large angle near the bottom.  By giving way like this, the post reduced the deceleration of the vehicle as compared with a massive object like a tree, and probably reduced the severity of injuries to the occupants.  What about the long smooth curve?  Perhaps it was generated at the same time as the kink, because the inertia of the tall post prevented it all from moving at the same time.  Is this plausible?

Let’s look at insects again.  This butterfly has a tubular proboscis to suck nectar.  Actually it is made of two tubes side by side, evolved from mouth parts.  When joined, the two tubes create a third one down the middle, which is the useful one.  The cross sections means that the proboscis is very much more flexible in one plane than the others, and it can be rolled up tightly when not in use.  When extended, it has a sharp bend about halfway along, as the photograph shows.

The legs are tubular too, and the wings are based on tubes that mostly radiate from the roots.  These, and the slight corrugations, make the wings fairly rigid in one direction, but quite flexible in another. At one point in each flapping cycle, the wings are closed above the body.  They then peel apart, starting from the the leading edge of the fore wings, and ending at the trailing edge of the hind wings.  The round projections at the front of the rear wings force these wings to follow the motion of the fore wings, so that the behaviour is as if only two wings are present.  Moths achieve the coupling with a spine.  The quick opening of the wings creates a lowering of pressure, and a vortex, producing lift and propulsive force.  The antennae, too are based on tubes.

An excellent example of a tube in nature is provided by the strangler fig.  It climbs up a tree, gaining exposure for its leaves and flowers without having to build a strong an rigid trunk.  Eventually, it may completely surround the host, which may die.  But by this time, the fig has a strong and rigid hollow tubular trunk, made with minimal effort by using the host as falsework.  The fig is not a true parasite.  Parasites take the process of dependence much further, taking sustenance and energy from the host.  What proportion of species in the world is composed of parasites?

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 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 them "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 is a product of its history.  The Comets that crashed were not the Comets that had 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 had been manufactured.

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 total elapsed time rather than the integrated cyclic stress.

Comet web-site

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

Cellular towers of Akashi Kaikyo bridge

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