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Invisible Cantilevers

The integral bridge has deck and end piers cast as one rigid structures. Thermal expansion is catered for by the piles below, which can move slightly against the soil. To this extent the piles act as vertical cantilevers.

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? Many plant leaves have this same concave-up shape. 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.

Some leaves use a similar principle, aided by the extra thickness provided by the thickness of the stem. Here are some sections through an iris leaf, which assumes a graceful curve from root to tip, rather than collapsing like the rule. e 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.

The giraffe needs a long neck to reach its food. It needs big teeth to chew the food. It needs big jaws to hold and manipulate those teeth. So it has a big head on a long neck, though the head may not be as big in proportion to its body as for some other ungulates. The neck is a good example of a natural cantilever. Like every other animal and plant, the giraffe is a mass of compromises between differing requirements, resulting in the design most likely to reproduce itself.

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.

Here is a much older example, much larger, but with a far smaller storage capacity. The discs themselves are cantilevers that are spread around an axis. Larger wheels must be much thicker to preserve rigidity. The London Eye, like the traditional bicycle wheel, uses the third dimension to provide stiffness via the tensioned spokes. The tension is resisted by the rim, which is in compression, like an arch which makes a complete circle. The bicycle is an apparently simple machine, but it includes a number of basic engineering concepts.

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.

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

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 and its barbs illustrate 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 to its right.

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

This bee-fly, poised for lift-off, 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, its 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.

More antennae.

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 trees in the first picture, on the west coast of Jersey, are exposed to strong winds off the Atlantic Ocean. They could have grown vertically, but their slope probably reduces drag and turbulence and the consequent stress. In many high mountain regions the shrubs lie almost prostrate on the ground. The reduced air speed probably reduces water loss from the plant. The trees in the second picture are in the Cotswolds, forming a line at right angles to the prevailing wind. Two trees have been blown down, and one has been truncated. These are marked by crosses. Note that not all the trunks are not straight: at the bottom, some are nearer to the vertical, and they curve over more or less smoothly.

The picture at left 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 strong gust of wind, or perhaps a series of gusts, has uprooted this tree. The cantilever was strong enough, but bending, tension and shear have eventually broken the roots on the upwind side.

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.

Many of the leaves of this iris have collapsed. Over the whole plant, most of the leaves that collapsed, did so at about the same position. Whatever the cause, it seems that the behaviour of the leaves was fairly consistent, except of course that some did not collapse at all. This an example of the not uncommon situation in which small variations in history or structure can lead to large differences in behaviour.

Here is a tree that needs extra support from a prop, which shortens the cantilever arm and reduces the turning moment at the base.

The shape of this antler has evolved for lightness and strength. The large antlers of the red deer, a much bigger species, have to look impressive, to intimidate as many rivals as possible without the need for a fight. They have to be strong enough for an actual fight. They have to be as light as possible because energy is needed to carry them. Would you think that deer which live in woods tend to have smaller antlers than ones which live in the open?

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.

Here is a bees’ nest in Sarawak which is cantilevered out from a rock surface. It appears to contain only one chamber.

Here are two wasps’ nests, the first being that of Polistes sp, a solitary wasp, in Europe. The second was in Sarawak. Both use a form of paper. The nest of Polistes comprises hexagonal cells, like the nests of the European honeybee, apis mellifera. The hexagonal array is one of only three regular two-dimensional tessellations, the others being square and triangular. The hexagonal array uses the least material of the three per unit area. Semi-regular tessellations exist, and of course an infinite number of irregular space-filling patterns. All use more material per unit are than the hexagonal array.

The myriad scales on a butterfly’s wings are like tiny leaves bracketed out on thin stalks.

Many species of fungus form fruiting bodies in the form of brackets.

But let’s allow the horse chestnut tree to have the last word on natural cantilevers. No photograph can do justice to this magnificent tree when bedecked with flowers.