Cantilever

If you have any questions please write to dlocke@brantacan.co.uk.

11th November 2001 Back to Home Page back to Bridges Severn Cantilevers

Arch Beam Box Girder Cable Stayed Pre-Stressed Suspension Truss

Antennae, antlers, arms, balconies, bats, brackets, branches, bristles, chimneys, cilia, cranes, ears, eaves, eyelashes, fences, fingers, fins, fishing rods, flag poles, golf clubs, hairs, horns, leaves, legs, masts, necks, petals, pillars, plants, poles, posts, pylons, rackets, shelves, spines, stamens, stems, tails, teeth, toes, towers, trees, trunks, tusks, twigs, walls, whiskers, wings.

Cantilever is a clumsy word, reflecting a basic problem for designers - how to achieve a satisfactory appearance from a bridge which includes two types of structure, cantilever and suspended span. The requirements of these are completely different. Perhaps only on the grand scale of the Forth bridge can the builders get away with it.

On a smaller scale there must be compromises.

Just as as a great medieval cathedral can include several different styles, something that would look silly in a small house, so the Forth bridge is big enough that the eye can move from the huge cantilevers to the suspended spans, which are sizable bridges in their own right, and not be too offended. It is in the smaller cantilever bridges, such as the ones over motorways, where subterfuge is needed to get a good shape.

The Forth bridge is unique - its fame lasts more than fifteen minutes, if only because of the painting. But of course it would hardly famous for that if it weren't a magnificent achievement, which has never been surpassed in cantilever spans, except for the one span of the bridge in Quebec. To carry a railway locomotive and train up to 260 metres from the nearest support was a daring project, completely unprecedented.

From the side the structure of the bridge is easy to read.

The great tubes take the compression, held up by the narrow top members, with bracing and strutting to keep the forces from buckling them. The spread legs of the towers suggest great stability, though in fact the outer cantilevers have counterweights at the ends to keep the balance. The outer towers could in fact have been hinged at the bottom, but that would have been out of keeping with the middle one, which has to be totally self supporting.

The arch-like lower tubes of the Forth bridge remind us that the various types of bridge are not so different as they might seem. Take an open arch bridge, put a tension member in the top deck, and you can cut it in the middle and make two cantilevers. Morph the parts and you can end up with something more like a normal cantilever shape. Morph the arch a bit more and you have a self-anchoring suspension bridge.

This is illustrated in the right hand column.

The picture shows how two halves of two arches could be held together by a cable, forming a pair of cantilevers, with exactly the same stresses as in the arch condition. But of course this is not an efficient way of making a cantilever, and it does not allow of building without a lot of falsework, which is one advantage of a proper cantilever. The white line represents a cable which holds the half-arches together. Arches such as the Eads bridge and Sydney harbour bridge were built without temporary supports underneath by the use of cables, enabling the channel to remain open.

One great advantage of a large cantilever bridge is that it can be built outwards from the piers without falsework. Then the suspended span can be lifted into place. Another is that it is inherently rigid - heavy railway trains are no threat. Railways have always been a problem for suspension bridges, one which the cable-stayed type has now overcome, as in the bridge which joins Denmark to Sweden near Copenhagen and Malmo.

The problem with appearance is that the cantilevers need to taper from the supports to the ends, whereas the suspended span needs to be narrow at the ends and thicker in the middle. In practice, for a motorway bridge, the suspended span is often a simple beam, often shaped to continue the line of the cantilevers. There are some interesting asymmetric cantilever bridges on the M1, which take advantage of the sloping terrain. Each span comprises a cantilever supporting the next beam, which itself projects beyond its pier to form the next cantilever.

Construction

Cantilever bridges can often be built out from the supports without blocking the channel below. Then the suspended span can be lifted into place . Because there is no rigid connection through the bridge, small vertical movements of the foundations are not as dangerous as they might be with other bridge types.

Note that the connections at the two towers can all be hinges, but if more than two cantilevers are to be strung together, only the outer two towers can be hinged. The central pier and cantilever of the Forth railway bridge can stand alone. Another solution for multiple spans is a continuous beam, which may look at first sight like a row of shallow arches or cantilevers if it is haunched.

Concrete Cantilever Bridges

Several examples of this bridge are found in Gloucestershire and Wiltshire. More about these later.

There are some slightly similar bridges across the M11 motorway, but those are much longer, and they are beams and not cantilevers.

The third picture distorts the shape of the image unpleasantly, but it does show the joint where the suspended span is hung. That span does not need to be deeper at the ends than the middle, but the bridge would look rather odd if the suspended span were deeper in the middle, which it ought to be, in response to the bending moments. This is discussed in the page about beams. Many road bridges are treated in the same way. The suspended span is made of seven separate beams for ease of transport and assembly.

Looking at the Forth railway bridge reveals the shape of a suspended span when engineering considerations are allowed to rule, as they have to in large structures. On such a scale, it would be difficult to do otherwise.

Here is a slender cantilever footbridge made possible by the use of steel stressing wires inside the suspended concrete span. It is a typical type used over a dual carriageway. As the road is in a cutting, the designer did not have a problem connecting it to the footpaths on either side.

Here are two solutions to the connection problem, long straight ramps and helical ramps, though neither bridge uses cantilevers.

If you recognize this as a footbridge over the M6 motorway, you may be puzzled by the lack of traffic. The picture was taken near mid-day, as the angle of the shadow shows. It was taken during a normal working day. In 1968. In this example the designer opted for steps on each side for access to the bridge. Sometimes ramps are used in a town, because the bridge will be used by people with push-chairs and wheel chairs. Designing these ramps is no easy matter - they need to be quite long if they are not to be too steep. Making a tidy job of this and integrating the ramps into the design has been attempted using long straight ramps, folded ramps, and helical ramps.

The picture below shows the first Thelwall bridge, over the river Mersey and the Manchester Ship Canal, east of Warrington. Recently it was refurbished, and a new bridge was built alongside to cope with the huge increase in traffic. This bridge has many welded plate girder beams, and a riveted cantilever span of about 335 feet over the canal. The river span is about 180 feet. The total length is about 400 feet, with about 36 spans. The first bridge was completed in 1963, and the second in 1998.

Firth of Forth Railway Bridge

This is a picture of a part of the Firth of Forth railway bridge, completed in 1890. The picture shows one steel cantilever arm and one suspended span. The difference between the tubular compression members and the latticed tension members is clear. Only the Quebec bridge has a longer cantilever span. Why were no longer cantilever bridge spans built?

Jacques Cartier Bridge

Like the Forth Bridges, the Jacques Cartier Bridge in Montreal had to have enough clearance for the tall masts of big ocean going ships. The effects of such a construction can reach far inland, especially for railway bridges, because the gradient of the approaches have to be kept within strict limits. This adds to the cost of the structure. Here the Empress of Britain is about to pass under the bridge.

Cantilever Foot-bridges

Here is a cantilever bridge of a type which is found in several places in Gloucestershire and Wiltshire. This one crosses the Barnwood bypass, east of Gloucester. There is another one about a kilometre to the east. The narrow roads leading to these exhibit cracks typical of slumping on embankments. The local ground is largely clay. Some pictures at right show this. The last picture shows cracking due to drying out of the soil in a nearby field.

Here is a slender cantilever footbridge made possible by the use of steel wires inside the concrete. See also pre-stressed bridges.

The diagram below shows how slumping can occur on a hillside or an embankment. The ground breaks along a roughly cylindrical surface.

These lines were made using pseudo random numbers to simulate random deviations from a line. There is a clear resemblance to the cracks, which roughly follow the stress lines, but are deviated by random variations in the ground.

The diagrams above represent a simple beam bridge, which has been affected by subsidence (exaggerated). One response is for the beam to remain so straight that it is only supported in two places, leading to a bigger effective span. Another is for it to bend. A third would be to break, if either of the first two conditions were unsustainable by the structure.

In practice the designers might include jacks at the base of the piers, to allow for adjustment.

What happens as a result of the movement is that the beam suffers stresses which were not in the design. In fact the problem exists from the start.

The four support points can never be perfectly aligned, but the alignment is of course made so small that the beam can adjust its shape without absorbing too much energy.

The penalty for a through beam is the over-determination. The benefit is the spreading and controlling of loads and stresses.

The diagram above represents the response of a simple cantilever bridge to subsidence. In this case the joints allow stress-free movement, so nothing is distorted.

Jacking might still be provided. In a very slender foot-bridge, the slightest error could be noticeable, and so some adjustment may be needed.

After a bridge has been completed, jacks may be concreted over, or they may be left as usable adjusters. The Eiffel tower is a good example of the jacking requirement. The stresses, and therefore the strains, at the base, changed markedly during construction. Jacking enabled the builders to compensate. This subject is developed further in Indeterminacy.

This is what happens if the load produces a moment that is greater than the cantilever can supply: fine in a see-saw, catastrophic in a bridge. The answer is to tie down the ends. They can in fact be pulled down with sufficient pre-stress that there will always be compression between the bridge and the ground. The outer ends of the Forth rail bridge are pulled down by counterweights, a trick that cannot be used elsewhere in a cantilever bridge. That is why the central tower of the Forth is so wide - even with the heaviest train in the centre of a span, the central cantilever is stable.

Why are the towers of big cantilevers so high? Consider the weight of the bridge and it load, pulling down at some point a long way from the tower. This creates a moment which will pull the bridge down unless it is resisted. An opposite moment is created by the tower pulling the top chord and pushing the bottom chord. Since any moment is the product of distance times force, increasing the height of the tower decreases the forces. But increasing height will eventually add more weight than is gained by the reduction in force. In practice, the ratio of length to height does not vary all that much. Look at some pictures of cantilevers and try to work out the ratio in each case.

Note that once the load has moved on to the suspended span, its moment at the tower does not increase even if its distance increases. Why is that? In that case, why don't bridges have much shorter cantilevers and much longer suspended spans?

Cantilevers can be used during construction of structures which, when finished, do not include them. This enables the channel below to be free for navigation. The penalty is the introduction of temporary forces that are not present in the final structure. This was dramatically demonstrated by collapses of several box girder bridges during construction in the 1960s.

The two halves of the Sydney harbour bridge were held back by cables until the time came for them to be connected. Cantilever construction has been used in trusses, box girders, truss arches and cable-stayed bridges. The diagram below shows an example of cantilever construction at two different stages.

Railway Stations

Railway stations often have cantilevered roofs. Here we see an early 20th century style on the right, and a late 20th century style on the left. Pillars are certainly undesirable near the edge of a platform, because anyone opening the door of a moving train so as to get off more quickly would risk a collision between door and pillar, or worse still, a nasty injury.

Balconies

Many modern buildings have cantilevered balconies, or even whole sides of the buildings. Frank Lloyd Wright built a house projecting over a river near a waterfall; this building is now famous. The balconies in the pictures at left project less beyond the supports than might be apparent. In multi-storey dwellings, the balcony is especially important, to provide an outdoor area for the inhabitants and their plants and animals.

In older times, as now, the balcony has had both personal and cultural significance. It is the perfect place for rulers to be seen, and sometimes heard, by their subjects, while remaining above and aloof - the antithesis of "the walkabout" "working a room", and "pressing the flesh". For other people, the balcony is a grand place to view the surroundings and take the air, inferior only to the roof, which is enjoyed only by those who occupy the top floor. Romeo and Juliet, and West Side Story, gave the balcony special significance.

Roofs and Eaves

Roofs are often supported by trusses, and where they overhang they become cantilevers. The Golden Pavilion at Kinkakuji is not very old, being the replacement for the original, which was burned to the ground in 1950. A fictional account of this event was written in 1956 by Yukio Mishima.

In principle, well overhanging eaves can counter-balance some of the weight of the roof, at the expense of greater weight on the walls or pillars. The roofs of many Chinese and Japanese temples are supported on series of cantilevered brackets which spread the load from the narrow pillars.

Here is a modern equivalent in England, with a water garden and huge eaves on the building. This is a part of a business park.

This picture shows a tall tower carrying antennas. When the wind blows, it acts as a vertical cantilever. Such towers are sometimes built in concrete. Another solution is a much narrower tower with guys.

Wings

Aircraft wings, fins and tailplanes are cantilevers. So are the front and rear of the fuselage. To obtain a very clean wing, some designers have placed the engines at the rear of the aircraft, as in this picture. But there are disadvantages. The air supports the wings throughout their length, though with a mean position much nearer to the fuselage than the tips. The load, apart from the weight of the wings, is of course carried at the roots, making for large bending moments. By putting much of the fuel in the wings, and hanging the engines from them, the bending moment is significantly reduced, though on the ground the situation is reversed. To obtain lift, a clean upper surface of the wing is more important than the lower surface.

Most large modern turbojet aircraft, right up to the huge Antonov 225, have the engines under the wings, which makes them easily accessible for servicing, but even now, numerous smaller aircraft have rear mounted engines. Some aircraft with rear mounted engines are Trident, Caravelle, 727, VC10, HS125, 111, CL-600, Citation, DC-9, MD-80 etc, MD-91 etc, Fokker 28, Fokker 100 Gulfstream III, Il-62, Jetstar, TU-134, Tu-154, Yak-40, Yak-42. Most of these designs are either old or fairly small.

From time to time, people try out the idea of an aircraft which is all wing, a "flying wing". Examples are a very old Northrop aircraft and the recently introduced B-2. If you try to ignore the experience of generations, you may produce a stroke of genius, or you may find out why everyone does it the same way. Putting everything in the wing would certainly help with bending moment, but where do you put the stabilizing tailplane and fin? Well, you can use a high-speed swept-back wing, and put fins at the tips, but you won't get much moment. And you can use washout at the tips instead of a tailplane. Active control of stability by computer can help.

But for an airliner, passive stability is very desirable in case things go wrong. Aircraft have occasionally been flown using only the engines for control, after the hydraulic system has been wrecked. In 1985, as a result of a faulty repair, a Japanese 747 lost all hydraulic control, and the tail fin, when the rear pressure dome failed. The pilots managed to fly it for about half an hour, using only the engines, but it then hit a mountain. Only four people survived. In another case, caused by the failure of an engine fan, causing a DC-10 to lose all hydraulics, a combination of air traffic control and skilled flying, again using control by the remaining engines, got the plane to an airport near Sioux City, and 184 of 296 people survived. Only with an intrinsically stable aircraft could this feat have been possible.

There is another disadvantage of a flying wing. Putting a pressurized cabin in a wing presents great structural problems, since it ought to be cylindrical for strength with lightness. Windows would not be possible, and emergency exits would be difficult to provide.

A turbo-jet aircraft contains many cantilevers - not only the wings and tail, surfaces, but the hundreds of smaller aerofoils that are found in the engines. Some are used in the compressors, to compress and accelerate incoming air, others are used in the the turbines at the back, where they extract some of the outgoing energy and feed it to the compressors to keep them turning. In a bypass engine there are also large fans that add greatly to the thrust. In addition to these blades, there are rings of stator blades interleaved with the rotating ones, each set of stators directing the flow on to the next rotor set.

The rotating blades are subjected to enormous forces that accelerate them towards the axis, and should something break, a blade flies of on a tangent. An uncontained blade can wreak havoc, and the unbalanced disc will probably cause trouble if the engine is not shut down. In addition to the forces, the turbine blades have to withstand very high temperatures, and are made of special alloys, and of course, like every cantilever, each blade is capable of oscillating.

Can you think of any natural flier, reptile, bird or mammal, that has not separated the functions of lift, stability, control and payload? Well, gliding snakes have not yet evolved very far down this path, but then, their glides are very steep. Nature, of course, cannot produce "strokes of genius", jumping from one design to a completely different one, because evolution can only move in the direction of immediate greater survivability in the hugely multi-dimensional space of variables. Crossing to another "valley" is impossible. Another solution, however much better it is, cannot be reached if it would involve even the smallest temporary decrease in fitness for reproducibility.

Designers, however, can "go back to the drawing board" and start again, though attempts to be different have not always been successful. Sometimes the reason is commercial or practical: sometimes it is technical. In the 19th century there were two main railway gauges in Britain - 7 feet, and 4.71 feet. The broad gauge had to be abandoned, because it was introduced when the "standard" gauge already covered thousands of miles of track. It was technically good, but commercially bad. Had it been used from the start, things might have been different. Actually, the broad gauge did have the disadvantage that it could not contain curves as acute as those obtainable with the smaller gauge.

"Received wisdom" is not always a poor guide: often it is the result of long experience. But occasionally a completely new idea, such as the Dyson vacuum cleaner, really does take off and make inroads into a market. This usually requires immense effort and perseverance, technically because the existing products are made using years of past experience, while the new one requires all the research to be done quickly, and commercially because of the existing marketing and sales structures, and the conservatism of many customers.

One of the two inventors of the turbo-jet engine, Sir Frank Whittle, experienced immense difficulty in convincing officials that his idea was worth pursuing, in spite of the fact that the promised gain in speed would have been decisive in both military and civil flight. With hindsight, to have started research into a turbojet airliner instead of building the Princess flying boat and the Brabazon might have been a better choice.

The wings of this airliner are large cantilevers, supported by the undercarriage on the ground, and at the roots when in the air. The underside of the wing is subtly curved. All aircraft wings are subject to the requirement of being able to support the aircraft over a fairly wide range of speed, in order to land at a safe speed. The wings are much larger than they need to be at high speed.

The first cruise missile, the V-1, had small wings, as do modern examples, because they can be launched at high speed. As the lift of a wing increases as the square of the speed, the problem of low speed flight would appear to be great. Luckily, by increasing the angle of incidence, and by the use of flaps and slots, the lift can be increased, at the expense of increased drag. In fact, at the slowest possible flying speed, the power needed can be much higher than at the most economical cruising speed. Glider pilots know well that there is an efficient range of speed, because they have to get power from outside. There are two optimum gliding speeds, one that gives maximum time in the air, and one that gives maximum range. Why are they different, and which is the higher?

Helical Stairs

The helical stair has been a popular way of getting steps into a small space since medieval times. If you climb the tower of an old cathedral or church you will almost always find yourself inside a narrow cylinder of stone, with stone beams bridging the space between a central pillar and the wall. These beams keep the pillar straight, in spite of its height.

In this modern example the metal treads are cantilevers, bracketed from a central tube, each step having a vertical flange for rigidity.

A Sloping Cantilever

The platform of this fire and rescue vehicle is based on cantilevers which are box girders. The lightweight ladder is a truss which gains much of its rigidity from the main girder. The cantilevers are held in place by hydraulic jacks which form struts.

A Small Cantilever

Many old towns in England include houses with overhanging upper floors, increasing the floor area without using more land. This one seems to be in need of restoration.

A Thin Cantilever

Here is a steel rule, which is slightly curved transversely. When the concave surface is facing up, the rule can sustain about a 76 cm cantilever without collapsing. With the concave side down, it collapses at a much lower span. Can you see why? The collapse near the support is typical, because that is where the bending moment is greatest. The first Quebec bridge did the same thing, and so did several early box girder bridges which were constructed as cantilevers, with the intention of joining the ends to make beams.

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.

A Very Small Cantilever

Here is a small cantilever, made to very fine tolerances, in the hard disc drive of a portable PC. Don't look at yours - it will immediately become useless because of dust.

Gravity Dams

In a plan view a gravity dam looks like a beam holding back the water. But for any but a very narrow dam this would not work.

For more details about gravity dams and retaining walls, see Gravity Dams.

The vertical cross-section suggests that a gravity dam is more like a cantilever. It is held to the valley floor by gravity.

It is very important that all water be excluded from underneath a gravity dam. A boat can in principle float in a cavity that exceeds its own dimensions by only a minute distance.

The pressure of water on the surface of an object can cancel some of its weight. In a dam this could result in uplift and overturning.

It is very important for any object that the line of thrust should meet the ground well within the base area. Otherwise there is a danger near one edge that the pressure on the ground might be reduced to nothing. In the case of a dam, cracks would let water in. A dam should satisfy this no tension condition throughout the filling of the reservoir.

See Top Ten dam sites.

Natural Cantilevers

Natural cantilevers are common. Many are plants. Usually rooted in the ground, they have to withstand the wind and the force due to gravity. Sometimes the forces are too great, and a noble beech is ripped out of the shallow limestone soil, or a great oak falls.

Each branch and twig is also a cantilever, holding out leaves to catch the light.

Each leaf is a cantilever. Some leaves are subdivided into smaller cantilevers, and some are subdivided yet again. Look at the detailing in the palm leaf below and in the leaf below right.

Sometimes you can see how the wood has grown to follow the stresses it has to live with.

Every mammal's neck is a cantilever, as is that of a striking cobra. The neck and tail of an animal such as a diplodocus are striking examples. The giraffe is a striking example today.

Every bird's wing, feather, and part of a feather, is a cantilever. The feather at the bottom of this page illustrates the second, third and fourth levels of cantilevering: the wing itself is the first.

One of these pictures of wood reminds us of the electric field of two unequal charges of the same polarity, shown in the simulation at right.

Then there are insect wings, antennae of arthropods, fins of fishes, and many other animal appendages, right down to microscopic cilia.

---a

This bee-fly, poised for lift-off in the first picture, is quite harmless. An insect equivalent of a hummingbird, it hovers in front of small flowers and drinks nectar. The wings and proboscis are little cantilevers. Note the tubular legs, and the tubular veins supporting the wings. The next two pictures show the wings in use, the back legs trailing, and the other four legs retracted, as the insect approaches a flower. The other two pictures show the bee-fly beginning to investigate the flower, it's wings still vibrating, and then, in the last picture, deeply engrossed in feeding.

This Libellula depressa demonstrates beautifully the engineering of the wings. Two cantilevers on each side, supported and controlled by tubular veins in a truss-like arrangement. The venation helps to control the flexure of the wings during flight. The veins at the leading edge near the root are relatively large and stiff, and non-coplanar, to transmit forces to the nodus, from which main veins radiate. The head, dominated by large eyes, is held on a narrow neck, which is sensitive to angular movements of the tubular body, helping the insect to coordinate its movements.

This insect is a crane-fly, or daddy long-legs. Note the halteres, which have evolved from the hind wings. Note also the very simple venation compared with that of the dragonfly.

The stems of this plant are bent under the weight of the berries, as they might under dew-drops, rain-drops, snow, or ice. The stems need only be strong enough to withstand reasonably common loads. Rigidity is not essential. Many natural structures are much less rigid than most man-made ones. We don't expect the lamp-posts in the background to bend, and if you do see such an event, in a high wind, usually when a post carries a heavy cluster of lamps, you will realise how unusual it is to see artificial structures flexing visibly.

In fact a characteristic of many natural materials, such as tendon, skin, chitin and spider-silk, is the amount of energy they can absorb in stretching, twisting or bending. When an energy absorbing artificial substance such as Kevlar is created, this is noteworthy.

The chitinous shell of the thorax of the magnificent Privet Hawk moth plays a major part in control of the wings. The elasticity of the material, allied with the power of the muscles inside, enables the wings to oscillate many times per second. The second picture shows the antennae of a different species.

When you see the May blossom or the resulting berries you don't think about cantilevers. Quite right too. The efficiency of a tree can be gauged by the small volume it occupies when cut up in to small straight pieces, compared with the huge volume it uses in getting all the leaves into positions where they can receive plenty of light.

The picture above shows a set of slices through a celery stem. The cross section at the bottom (right) is the result of the close packing as the stems emerge from the meristem. Moving upwards (to the left), we see the transition to a cross section which provides the required stiffness in both transverse dimensions.

A horse chestnut tree in flower is a magnificent natural spectacle, illustrating the effort that an organism will make in order to reproduce. To hold up this splendid array of flowers, the tree needs a massive and extensive root system, a mighty trunk, many strong branches, and thousands of little twigs.

There is a way of getting flowers up high without all that effort. Be a parasite, like these mistletoe plants. No heavy cantilevers, just a lot of tough and flexible stems, exactly what is needed to withstand the wind in the treetops. If parasites are so successful, why aren't there more of them? Well - what is the ratio of parasitic to non-parasitic species?

The umbelliferous plants are not the favourites of many people, and are seldom used by gardeners. They are not even coloured in most cases. Yet they are very interesting indeed. In the first picture we see four of the stems that have branched out from a bigger stem, and each of these fans out into many smaller ones. Each smaller one branches again into many tiny stems, each bearing one flower. The end product is a wide and highly visible mass of white or whitish flowers, often forming an almost continuous flat sheet.

A single flower of the same area might be much heavier. Furthermore, its pollen and nectar would probably be in the centre, requiring visiting insects to aim at the right place. Indeed, many flowers have guide lines, sometimes visible only in the ultra-violet, to help the insects. On an umbelliferous plant, an insect can find something almost immediately, a great advantage, given its exposure to predators while feeding. If an insect is interrupted shortly after landing, the plant still has a good chance of being pollinated.

The construction of deciduous trees exposes a great number of leaves to light. If you have ever cut down a small tree or a shrub, and have cut it into individual straight pieces, you will have seen how small a volume these require, compared with the original size of the tree. Many trees are more or less rigid, but some, such as the weeping willow, use a different strategy. Having reached the outside world, the branches then drop very long hanging stems, from which the leaves form a curtain to intercept the light. A beautiful picture, Willow and egret, by Suzuki Kiitsu, in the Shinenkan Collection, contrasts these passive curtains with the active flight of a heron, and the contorted but sturdy tree trunk.

Some animals too, such as termites, can make tall constructions, many times larger than themselves. Ants, bees and wasps too, can make quite complex structures, based on simple rules.

Nano-cantilevers

"Nano-technology" is a name for a number rapidly growing fields of miniature engineering. Tiny cantilevers, often built into silicon integrated circuits, are being used in a number of applications. These include weighing at the picogram level, detecting specific molecules using the deflection of cantilevers loaded with specific molecular traps, precision placement using piezo-electric effects, acceleration and vibration sensing, fluid flow sensing, and so on. After all, our ears use a vast array of cantilevers in the form of tiny hairs, to detect sounds of different frequencies.

If you got this far, try a superb game about bridge building - http://firingsquad.gamers.com/games/pontifex/default.asp .