Attachments and joints in bridges and other structures

For the want of a nail, the horse-shoe was lost . . . .

A part of the fuselage of a Boeing 737 blew off, after creeping failure of skin joints. The plane landed safely. One person, a stewardess, was lost. In 1974 the cargo door of a DC-10 blew off because a closure indicator showed that a bolt was home when it wasn't. Everyone died. In 1979 an engine came off a DC-10 because of a faulty maintenance procedure on the connection between engine pylon and wing. Everyone died. In 1985 the rear dome of a Boeing 747 failed, destroying the fin and the control systems, because of a faulty repair to a joint in the rear pressure dome. Only four people survived. Deadly and expensive failures can result from simple faults in joining parts together.

On May 10th, 2002, a railway carriage was derailed at Potter's Bar station, causing deaths and injuries. Within a few days it was reported that some nuts were missing from a set of points, or switch. At this time, the reason was unknown.

On a simpler level, if you make a bunt in a glider, and you forgot to strap yourself in, your next action will be to pull the rip-cord, assuming that you have a parachute. It has happened. The pilot dropped something on the floor. To reach it he undid his buckle. In bending forward, he pushed the stick. The plane went down suddenly. The pilot didn't - Newton's first law kept him going almost horizontally.

Note, in the picture at left, how the joints are wider than the bars. This picture shows a joint in the chains of the Clifton bridge, near Bristol. Look at the "knees" (actually the ankles) of a flamingo, or even your own ankles. The difficulty of making joints, especially moving ones, was a factor in the long period which elapsed between the invention of variable sweep wings by Barnes-Wallis, and its use in actual aircraft, such as the Tornado, F111, F-18 and B-1. However, numerous aircraft have had folding wings for use on aircraft carriers.

The picture of the Clifton bars hints at some of the complications that can arise in design. Compare the concave curves with the concave corners: the curves are needed to prevent stress concentrations which could take the material beyond the safe limit. The forces within any object are distributed in a way that minimizes the total strain energy. The convex parts need not be rounded off because the forces do not penetrate that far.

It might be imagined that minimization of energy could be achieved by having all the forces taking the shortest possible route, but this is not the case. In that situation, most of the structure would be unstressed, while a small part would be highly stressed. Since the energy is proportional to the square of the stress, it turns out that a lower energy state is achieved with some spreading of the forces. So the stable state is a compromise between the shortest path and a uniform distribution.

A second consideration of attachment design, a result of the first, is that in or near joints, all simple ideas about struts, ties and beams do not apply. In those idealized structures, we can calculate the distributions of stress quite simply, often using analytical methods but around a joint, a numerical method, such as finite element analysis, may well be needed.

Consider, for example, a deck stiffened arch. We may think of the bare arch as containing purely compressive forces, aligned exactly along the arch. But if we add the vertical spandrel pillars and the deck above, the weight of the deck adds vertical force to the arch at each support. Somehow, the effects of these forces must be transformed into forces along the arch, so that in the middle of each segment, the forces are roughly aligned with the arch. In fact, even in the bare arch, the fact that the force in the arch varies along the length shows that force is being injected. It comes, of course, from the weight of each successive part.

No matter how strong the parts of a structure, they have to be joined to other parts, in a way that does not reduce their effectiveness. This can present problems that are far from trivial. Some airline passengers probably wonder what stops the wings falling off. In fact, of course, the wing structure goes right across through, over, or under, the fuselage, forming a continuous beam, to which the fuselage is attached. The attachment must transmit the forces in a way that does not produce too much stress concentration. In fact, the two wings may be bolted to the central beam. The bolts have to withstand enormous forces caused by the bending moment.

If wings are to move, as in some carrier-based aircraft, which can fold or over-sweep the wings for storage, and as in aircraft types with variable sweep, the weight penalty of the joints and mechanisms may be severe. It may be partially offset by a reduction in the requirement for high-lift devices for take-off and landing. Imagine the structural needs of the wings of a bat, bird or beetle, both of which fold the wings when not in use. The wings of an earwig fold in a very complicated way, into a very small space, but few people have actually seen an earwig land or take off.

Fixed wing aircraft often have devices to increase lift at low speeds, like the leading edge flaps and trailing edge flaps of this 747 which has just taken off. The weight and complications of these devices and their attachments is such that on smaller aircraft they are usually simpler than this, and very small aircraft they may be absent altogether. This aircraft is in the process of cleaning up, with the wheels almost folded away. The bogies and their fittings weigh several tons each, and like the flaps, they are only needed for short periods during each flight. The differences between aircraft and birds can be attributed to the great difference in Reynolds' number, the difference in materials, the difference between growth and manufacture, and the difference between evolution and design. Even smaller Reynolds' numbers are found in the flight of tiny insects such as aphids and midges.

To appreciate the difficulties faced by the designers of movable connections, we need only look at the human ankle, the legs of a horse or a flamingo, the universal joint in the transmission shaft of a truck, or the hinges of the blades on a helicopter. And think about how many problems, especially in old age, and in people who take part in games and sports, are caused by wear and tear or disease in joints. We may not think of feet as attachments, but in a sense they are, and the glue is gravity. We instinctively realise why they are so big, but to see this explained more formally, click here. You will also see how accurately the flamingo must position its body when asleep.

The jib of this crane is assembled by bolting together a series of short sections. These are revealed by the widening of the top member, especially where the sloping bar holds the whole thing up.

This example is very much older - in fact it is derelict and rusty. Perhaps that is the reason that the joints are much bigger, because increasing knowledge often allows more efficient use of material. This knowledge not always about what is needed: sometimes it can be about how to achieve it. Very often the ideal shape is more expensive to make than one which is rather less efficient.

Here is another example from a crane. Each of the two lower members is simply joined using four bolts and a fishplate, because the butting struts take the compression, and the fish plates take the shear, leaving the bolts only to hold everything together. But in the top member, the attachment has to take the tension from one member to the next, and the attachment is relatively large.

One of the problems with cable-stayed bridges is to achieve a neat appearance for the attachments of the cables. On the Jackfield bridge, not far from the famous Ironbridge in Coalbrookdale, the cable attachments were made quite obvious. The picture also shows the pin at the end of the deck, which is in compression, and the base of one of the two main legs.

This example shows joints in the three legs of a tower. Larger tubes can be joined in much neater ways. The fuselage of a 747 airliner is made in sections, but this is not apparent either from the inside or from the outside. The booster rockets of the NASA space shuttle are made in sections. Because the tubes can oscillate in various modes which change the shape of the cross-section, it is difficult to make these joints gas-tight, especially with the high temperature and pressure within the rocket engines.

Here is another

example of a

flange joint in a

tubular mast.

See also this bridge and the Darcy Group

Tie bars at NEC Arena in Birmingham, and tie bars of the Clifton Suspension bridge, showing the expanded joints. Before techniques were learned for forming this type of connector on a bar, they had to be welded on. The weld had to be of high quality, as it took the tension from the bar to the connector.new cars

This picture shows an example of riveted construction, once widely used in iron and steel structures. The parts of the Tour Eiffel and the Forth rail bridge were riveted. Many rivets of the Tour Eiffel have needed replacement, because they have been deformed by shear forces, and some of the old ones are on sale in the tower. Steel rivets may not suffer the same fate, because after the heads are hammered down while the rivets are hot, the contraction produced by cooling squeezes the riveted plates together, and they may be held in position by friction.

Below are some pictures taken on a very dark morning after a severe storm. Most of the tree-branches broke at or near the point of attachment. Why? Why do you think that in so many cases, the branch brought away a long strip of the trunk, instead of just snapping off?

And here is the same phenomenon on a much larger scale, showing a common type of fracture.

The pictures above show an example on an even bigger scale - a branch with a diameter of around 40 cm has broken from a tree. The first three pictures hint at the enormous forces that were unleashed. The fourth suggests the sequence of events. At the bottom of the fracture zone is the pale area of the final break. Above that are dark regions where the crack had spread, over a long period of time, starting at the top of the joint, where the tension was greatest. Ingress of water, and the effects of frost, would have forced the crack to widen. As the remaining cross section of the joint was reduced, the tensile stress would have increased, and the growth of the crack would have accelerated. Eventually, a critical point would have been reached where a gust of wind, or the weight of raindrops, would have pushed the structure over the tiny remaining potential barrier. Almost all the failure is in tension or shear. The crack producing moment would also have been increased by the outward sag of the branch as the are of fixture diminished. In addition, a cyclic increase would occur each year with the extra weight and length of the new growth, and the weight of the new leaves. The final break in fact occurred in early summer.

The pictures below show some of the sawn up wreckage on the ground. The right hand picture clearly shows surfaces of different age, the most recent being on the right.

Here are more pictures where branches have been torn off. The way things fail may give important clues to the causes. The crash investigators who look at the wreckage of an aircraft are very experienced in the art of looking and interpreting. The entire sequence of a disintegration can sometimes be deduced from little things like scratches. If two pieces of a component have aligned scratches, the object that scratched them must have done it while the parts were still joined together.

It may seem odd to show so many pictures of broken things, but in fact the visualisation of forces is very difficult except when things begin to fail. Strain gauges can be attached to surfaces, and if the material is linear, we can work out something about the forces. The shapes of electrical and magnetic fields can be made visible using fine insulating or ferromagnetic powders, but the nearest we can get with mechanical forces is to send polarized light through a transparent model. Click here to see a page which includes an example.

Modern calculational techniques using computers and graphics programs can now produce images which can be drawn in colour as well as in apparent 3-D. These simulations are of course only as good as the assumptions on which they are built.

This picture shows a view from above of a gate post made from a tree trunk. Much of the middle has rotted away, but five conical parts remain, pointing inwards and downwards. These are the remains of the anchorages of five branches that formed at that height on the tree, as can be seen from the "knots" on the outside. The next pictures show slices through trees near the attachment of branches. The almost elliptical shapes are caused by the angle of the branches from the vertical. The pictures are not very clear because the lighting was diffuse.

This tree has split right down the trunk, which is seen to have been hollow. The break probably started high up at a branch point. The tree may have been struck by lightning.

Nature seldom joins things together in a crude manner. Look at the way that a tree grows its branches - if you cut through the wood, you see the lines of force well inside the main branch, showing where the subsidiary branch grew out. The first two pictures below show how palm leaves grow.

The other pictures show pieces of wood cut from a place where two branches grew out, together with a computer simulation. If you look at an old fallen tree you can often see clearly the flow of the stresses to which its growth was a response. Given the enormous time-scale of evolution, we can assume that natural structures represent good compromises between all the requirements for survival and reproduction.

The next pictures show roots of beech trees on limestone. These roots do not grow deep: they spread. Here, they have been partially revealed by soil erosion.

A plant or an animal is never strong enough to resist all possible forces, for it would be at a disadvantage in more usual circumstances, having used unnecessary resources of energy. The evolved structure is strong enough that on average the species continues. As many as a third of adult gibbons have broken bones because of misjudgments or breaking branches. Were they stronger, and therefore heavier, their speed would be reduced, and they would probably be less successful in feeding, reproducing and escaping. On the other hand, some male animals, such as elephant seals, have gone to the other, massive, extreme, and have evolved suitable mating behaviour along with the increase in size.

Given that most animals have to reproduce by sexual means, and that many, from spiders to tigers, are fierce predators, ingenious means have evolved to achieve coupling without suicide. Nevertheless, in some species such as mantids, male promiscuity is not usually possible. Plants, being mainly immobile, have evolved an enormous range of mechanisms to get their gametes together, often using animals to transport pollen. In fact, many flowers have probably evolve together with the corresponding insects and birds.

The same is true of many mammals and their fleas, which have also evolved together, so that the fleas cannot live apart from their mammal. Lichens are symbiotic pairs of fungi and algae. We ourselves contain mitochondria, which may represent ancient species which have become almost a part of ourselves. These species bonds may not be physical, but they are strong nevertheless.

Returning to more tangible ideas, the difficulty of transmitting huge forces can be appreciated if we think of an elephant, the Eiffel tower, a Saturn rocket, or a Boeing 747. In each case, huge forces are at some places transmitted through quite small areas.

Stress concentrations at bolts, rivets, rivet holes, welds, and other joints, are major sources of concern. Deformed rivets from the Eiffel tower are on display in the shops in the tower, showing dramatically the effects of continuous long term stress. Look at pictures of the wreckage of the first Tay bridge to see the effects of poor connections between the iron piers and the masonry below. Connections do indeed begin with the foundations, and end only at the top. Examples - Tay bridge 1 and Tay bridge 2.

Watch a weight-lifter and look at the great care taken to ensure the best possible connection of the hands to the bar. Anyone who has carried heavy shopping bags will know a bit about this. The connection formed by a handshake is in fact a symbol of goodwill between people, and the term is even used in electronics and communication to denote the correct exchange of information. We sometimes speak of marrying parts together. Other forms of touching, such as holding hands, kissing and hugging, are also used by people.

Some common phrases referring to joining or breaking are - "buttoned up", "coming apart at the seams", "divide and rule" "get a grip on yourself", "hang in there", "hold on", "let these persons be joined together in holy matrimony", "losing his grip", "nervous breakdown", "a screw loose", "we must stick together", "a stitch in time saves nine", "stitched up", "tied to his mother's apron strings", "unbuttoned", "unhinged", "united we stand, divided we fall".

Here are some common requirements -

Within materials -

Cohesion - cast iron, spider web, steel, etc

Within parts -

Reinforced concrete, pre/post-stressed concrete, fibreglass, tufnol, and other composites - avoidance of separation, delamination and cracking (see web-page about cracks, in this website).

Between parts -

Joints - Compression, Tension, Sliding fit, Rotating joints

Here are some commonly used ways of joining and holding things -

Air-bearing, Anchor, Arc welding, Ball and socket, Ball race, Belt, Bluetack, Bobby pin, Bolt, Brazing, Buckle, Bulldog clip, Bush, Button, Cement, Chain, Chuck, Circlip, Cleat, Clevis pin, Clip, Clip-board, Clothes peg, Collet, Contact adhesive, Contact welding, Cotter-pin, Crimp, Crocodile clip, Cyano-acrylate adhesive, Door-bolt, Double sided tape, Dovetail, Dowel, Drawing pin, Drift, Duct tape, Electromagnet, Electrostatics, Epoxy, Expansion bolt, Explosive bolt, Eye-bolt, Flange, Friction, Friction welding, Gasket, Gecko feet, Glue, Grub-screw, Gummed paper, Hasp, Hat-pin, Hinge, Hook, Hook and eye, Impact adhesive, Jubilee clip, Jumar, Karabiner, Kirby grip, Knot, Lace, Latch, Lock, Lock-washer, Magnet, Magnetic chuck, Match casting, Mooring rope, Mortar, Mortice and tenon, Nail, Nut, Olive, O-ring, Paper-clip, Passepartout, Peg, Picture hook, Pin, Piton, Plain bearing, Plug and socket, Popper, Pop rivet, Post-it, Rawlplug, Rawlbolt, Redux bonding, Rivet, Rope, Rubber band, Safety belt, Safety pin, Screw, Sealing wax, Self-tapping screw, Sellotape, Set screw, Shrink fit, Shrink wrap, Solder, Split pin, Spot welding, Stamp hinge, Staple, Stitching, Strap, String, Sucker, Superglue, Surface tension, Tenon, Tendon, Tent peg, Thermite welding, Tie-clip, Tie-pin, Toggle, U-bolt, Universal joint, Velcro, Vice, Wedge, Weight, Welding (arc, contact, friction, spot, etc), Wire, Woodscrew, Wringing, Yorkshire, Zip.

Within these groups, think how many types of plugs and sockets, and how many types of screws, bolts, nuts and washers there are.

Some of these types of fastenings may require water tightness, permeability, gas tightness, electrical insulation, electrical conduction, heat insulation, heat conduction, stiffness, flexibility, corrosion resistance, vibration resistance, pressure resistance, damp resistance, temperature resistance, sliding, rotation, inspection, reliability in inaccessible places, and so on. And all parts of a joint must be compatible with each other, without unwanted effects, which may be binding, slipping, chemical action, electrolytic action differential expansion due to temperature or damp, and other problems. A joint may need to be reliable for many years, yet easily demountable for inspection, modification, or repair.

Parts of the famous iron bridge in Coalbrookdale are connected in ways which would not be used today, and were probably hardly ever used again. The designer may have decided that innovation in materials was a sufficient leap of faith in itself, without inventing new jointing methods. Certainly the history of projects that have tried to innovate in every possible way has not always been happy. On the other hand, trying to invoke a new technology without supporting it with new design and construction techniques may nullify some of the potential gains. The answer is thorough research and development throughout.

Here are some examples of attachments in bridges -

The appearance of the cable fixtures can be a big problem in cable stayed bridges. In the Jackfield bridge in Coalbrookdale, shown above, they are features in the design. Note also the tapered foot of the main strut, and the pin at the end of the deck.

Click here to see attachments in a modern arch bridge.

Attachments that we don't see are the roots of a tree, and the foundations of structures, which have to withstand the steady weight of the dead loads, the forces caused by soil movement and traffic vibration from nearby roads, and intermittent live loads from wind or traffic.

Joints

Here is a diagram which highlights the problem of joining two narrow members. The gusset plate points the way to the idea of triangulation, which is the basis of trusses. Without the plate, the large ratio of L and W means that an applied force produces a much larger force at the joint.

If joints are required only to provide connection, and not rigidity, they can be hinged or pinned, as here. This is one of a row of wooden struts which support a canopy on a building near the docks in Bristol. These struts remind us of the masts of Brunel's ship, the Great Britain, which is being restored nearby. The ship provides a wealth of technical and social insights. Masts can be hinged like this, and held in place by rigging, or stepped into the hull, in which case they can be self supporting, though support from rigging is desirable. On board the Great Britain you can see hinged masts. The second picture also shows the @Bristol exhibition, where you can find an enormous range of interactive technical exhibits.

If a joint is not required to produce rotational constraint in any plane, it can be a ball and socket. Such joints can be used in bridges when there is a pair of supports on each pier. This picture shows a support for a cantilever bridge. This example is a hinge.

Here is a turbofan engine as used on many large commercial aircraft. The engine is fixed to the wing by a strong pylon, hidden inside a fairing. This pylon has to support the weight of the engine and the forward and backward thrusts it produces during flight and landing. It also has to cope with the variable stresses of turbulence and touchdown. And it carries a multitude of services to and from the engine.

This position for the engine is excellent for maintenance, and keeps it clear of the airflow past the wing. The engine can also act as a mass balance during wing oscillation. Once the stresses are known, the design seems easy. Make it strong enough, make two accurate holes at the top, and bolt it firmly to a flange on the wing.

So why, in 1979, did the left engine and pylon of a DC-10 become detached from the wing during take-off, causing an accident that killed all on board? The engine rotated up around the leading edge, and landed on the runway behind the aircraft. The plane had lost one third of its thrust, but should have been flyable. But the torn off mass had severed hydraulic lines in the wing. The leading edge slats closed, reducing the lift catastrophically. Only a massive increase in speed could have saved the plane, but the pilots did not know, could not have known, what had happened. Aileron input would have increased the drag on that wing, and did not work. The plane may have started into a spin, and it hit the ground before that spin became established.

The design was good, in principle, but the airline had used an engine demounting and remounting procedure that saved a lot of time. Unfortunately, the procedure produced a high probability of misalignment during re-assembly, which with the tight tolerances of holes and bolts, led to cracks being induced in the pylon. During subsequent flight stresses the cracks would have spread, until in one aircraft there was a disaster. Cracks were then found in the pylons of some other DC-10s.

The 747 was a massive increase in airliner size, and although it looked somewhat like a scaled up 707, at least one feature caused immense difficulty during the early months of production. None of the engines would give the required maximum thrust on an aircraft, after working perfectly on test. The fault was the attachment of the pylon to the engine, which was, quite naturally, near the top of the engine. This had always worked before, but the powerful thrust of these massive engines, applied against an asymmetrical support, produced a slight bending of the engine. Slight, but enough to change the minute clearances between fans and casing, enough to decrease the thrust unacceptably, enough to ground the rapidly increasing queue of undelivered aircraft. So the mountings had to be moved nearer to the axis of the engine. The problem disappeared.

Scaling up is always fraught with possibilities. The next big step in airliner size is the Airbus A380. No doubt the designers have expended vast amounts of thought, testing, and computer time on every part, big or small, and the expectation is that the project will proceed very smoothly.

Foundations

Every bridge has to rest on the ground in at least one place. The supports have to be placed so as not to move unacceptably, which means that the stresses must be reduced to values that can be supported by the ground. On hard, strong rock, the supports can be narrow, but in weaker ground, a wider foundation may be necessary. In animals it is called a foot. The feet of moorhens, lily-trotters and camels are specialised for weak surfaces.

These three pictures show a piece of foam plastic which has been strained by pushing objects against it, to represent a pillar resting on the ground. The strains are revealed by the square graticule that was drawn with a fibre pen. From the distortions we can deduce the following facts -

The strains are concentrated near the point of application.

The strains are spread over a large area.

There is tension, as revealed by the curved upper edge, which is longer than the original straight edge.

There is shear, as revealed by the angles which are no longer ninety degrees.

And of course there is compression.

The final configurations are those which minimise the total strain energy.

The picture below shows a paper-clip floating on the surface of water. The weight is held by tension, not compression. The surface of the water is dimpled like the surface of the foam shown above.

This picture tries to give a rough idea of the way that pressure diffuses through the ground under a heavy weight. The ground has to able to withstand the stresses at all points without giving way, either quickly or by creep. In any volume where this is not the case, the ground must be replaced by a structural material.

Click here for a website dealing with earth structures and related matters. It also includes numerous links to websites about engineering and science.

Similar considerations apply at the top of a structure. Here the load on a roof beam is spread by means of a piece of wood. Old Chinese and Japanese buildings often have complicated arrangements of brackets which spread the loads at the tops of the pillars.

Attachments are not always the strongest and most rigid available Roland Huntford, in his magnificent book "Nansen" - ISBN 0-349-11492-7, explains how Nansen lashed together the parts of his sledges, so that the resultant flexibility would allow for the absorption of energy transmitted when the sledges were dragged over rough ice. In a page about indeterminacy you can read about the effects of badly designed joints. The ability to absorb energy, or even to store it and release it, is characteristic of many natural materials.

The currently smallest known means of joining is represented by gluons, which are considered to transmit the interaction which holds quarks together in particles such as protons, which have a dimension of about 10^-15 m. In spite of the immeasurably small "size" of the quarks, the force is reckoned in tonnes weight.

The largest known means of joining is represented by gravitons, if indeed they exist, mediating the gravitational force, which is believed to reach across the entire universe.

The strongest known means of joining are probably the forces between quarks, mediated by gluons, and the forces holding everything inside a black hole. The calculation of the results of gluon forces is complicated immensely by the fact that gluons themselves are held together by gluons, leading to the possibility of "glueballs" - particles containing gluons without quarks. Calculation with gravity is also complicated by the fact that the field itself has energy. Some discussion of these points can be found in the pages on physics. Let's se how strong the gluon force really is. If you try to separate two quarks, at a certain point they pull apart, and a quark-antiquark pair is created. The lightest particle that can be made in this way is a pion, of mass about 140 Mev/c². The distance at which the split occurs is probably of order 1.5 10^-15 m. A very crude idea of the force between the quarks can be obtained by dividing these two results.

Force in Newtons = 140 Mev/c² / 1.5 10^-15

= 140 X 10⁶ X 1.6 10^-19 / 1.5 10^-15 = about 15000 Newtons.

This is equivalent to the weight of about 1500 kg, or 1.5 tonnes.

This illustrates the care we need in using words like strength, rigidity, energy, force, and so on. Imagine a sheet of aluminium, a sheet of glass, and a sheet of steel, all a metre long, 10 cm wide, and a few mm think. We could bend the aluminium easily into a U shape, and it wouldn't break. The glass is about as rigid, but it breaks when it has bent very little. The steel would also be very rigid, but we could not break it. The glass breaks because it can absorb very little energy. Its breaking stress is low. The aluminium could eventually be broken by repeated bending and straightening, by metal fatigue. The aluminium is ductile, while the glass is brittle.

Returning to the quarks, a peculiarity of the forces between them is that three quarks can form a stable system, as can one quark and one anti-quark, but other combinations have so far not been found to exist. Another peculiarity, making calculation extremely difficult is that gluons, the particles that mediate the forces between quarks, have similar forces acting among themselves. A proton or a neutron, which is nominally a three-quark system, behaves as if it were a seething mass of particles.

The forces between protons and neutrons, which enable atomic nuclei to exist, are a kind of left-over of the quark forces, and are not really understood in any deep sense. Outside all nuclei, electrons are held in a very well understood way by electrostatic attraction. Perhaps, by means of quantum electrodynamics, these are the most well understood forces.

Inter-molecular forces, like inter nucleon forces, are a kind of left-over. As we look at bigger and bigger molecules, we find not only forces between individual atoms of molecules, making chemical bonds, but also shape dependent effects, as in the enzymes that catalyse the processes on which life depends. Each enzyme has a specific shape which can connect properly only with a specific type of molecule.

From quarks to enzymes, we see a remarkable variety of ways in which entities can be attached to each other. Most of the time we don't need to know about this: we simply buy the materials whose properties are suitable for the job we are doing, whether it be building a shed or constructing a bridge. But the creation of these materials has needed a knowledge of basic physics along the way. Without the weird theory of quantum mechanics, vast numbers of the materials we use would not have been created.

We return now to the macrocsopic world.

Rough values for the elasticities of the materials mentioned earlier, in GMm^-2 are -

Aluminium 70

Glass 75

Steel 200

Rough values for the tensile strengths in MNm^-2 are -

Aluminium 90 to 150

Glass 30 to 90

Steel 400 to 500

The weakest known force is gravity, at the energies with which we are familiar. But some physicists believe that at high enough energy, or temperature, the strong, electromagnetic and weak forces reach comparable strengths, though at lower energies they span a range of about 10¹³ in magnitude. Perhaps gravitation, another factor of about 10²⁶ weaker, will one day be joined in this unification.

Weak though gravity is, it is enough to have prevented anyone from earth, so far, from travelling much further than the moon, though unmanned probes have escaped from the solar system. The energy for for distant travel was obtained partly by the clever use of gravity, using the sling shot effect, in which the space craft makes a hyperbolic orbit near a planet. Although, relative to the planet, it does not gain anything, it gains relative to the solar system, because the planet is moving. This is not unlike the motion of a ball bouncing off a tennis racket, a baseball bat, or a cricket bat. The bounce is almost symmetrical in the rest frame of the implement, but the implement is moving relative to the earth, and so imparts energy to the ball.

Crampons, ice-axe and rope are sometimes useful for coping with the effects of gravity . . . . They are the equivalents of a harness in an aircraft.