ŷ������

It is confidence that causes accidents and worry which prevents them.

all of us are really using mathematics through every moment of our lives. When we play tennis or walk downstairs we are actually solving whole pages of differential equations, quickly, easily and without thinking about it,

What we find difficult about mathematics is the formal, symbolic presentation of the subject by pedagogues with a taste for dogma, sadism and incomprehensible squiggles.

it is perfectly normal for any and every structure to deflect in response to a load. Unless this deflection is too large for the purposes of the structure, it is not in any way a 'fault' but rather an essential characteristic without which no structure would be able to work.

pressure acts in all three directions within a fluid while the stress in a solid is often a directional or one-dimensional affair.

Numerically, the stress in any direction at a given point in a material is simply the force or load which happens to be acting in that direction at the point, divided by the area on which the force acts.

the stress in a material, like the pressure in a fluid, is a condition which exists at a point and it is not especially associated with any particular cross-sectional area, such as a square inch or a square centimetre or a square metre.

Just as stress tells us how hard - that is, with how much force - the atoms at any point in a solid are being pulled apart, so strain tells us how far they are being pulled apart - that is, by what proportion the bonds between the atoms are stretched.

Like stress, strain is not associated with any particular length or cross-section or shape of material. It is also a condition at a point.

Nature seems to be a pragmatic rather than a mathematical designer; and, after all, bad designs can always be eaten by good ones.

A deep, intuitive appreciation of the inherent cussedness of materials and structures is one of the most valuable accomplishments an engineer can have. No purely intellectual quality is really a substitute for this.

In our material world, every single happening or event of whatever kind involves a conversion of energy from one into another of its many forms. In a physical sense that is what 'happenings' or 'events' are about.

This quality of being able to store strain energy and deflect elastically under a load without breaking is called 'resilience', and it is a very valuable characteristic in a structure. Resilience may be defined as 'the amount of strain energy which can be stored in a structure without causing permanent damage to it'.

A simple but interesting example occurs in an ordinary spider's web. The web is subject to impact loads arising from flies blundering into it, and the energy of these blows must be absorbed by the resilience of the threads. It turns out that the long radial threads, which form the main load-carrying part of the structure, are three times as stiff as the shorter circumferential threads which have the duty of actually catching the flies.

it is quite possible to break a bow by 'shooting' it without an arrow. What happens is that the strain energy which was stored in the bow can no longer be disposed of safely as kinetic energy in the arrow, and so some of it is employed in producing cracks within the material of the bow itself. In other words the bow has used its own strain energy to destroy itself. The broken bow is, however, only a special case of all kinds of fracture.

In a 'brittle' solid the work done during fracture is virtually confined to that which is needed to break the chemical bonds at, or very near to, the new fracture surface. As we have seen, this energy is small and amounts only to about 1 J/m2.

In a tough material, although the strength and the energy of any individual bond remains the same, the fine structure of the material is disturbed to a very much greater depth during the breaking process. In fact it may be disturbed to a depth of well over a centimetre: that is, to a depth of about 50 million atoms below the visible fracture surface. Thus if only one in fifty of these atomic bonds is broken during the process of disturbance then the work of fracture - the energy needed to produce the new surface - will be increased a millionfold, which, as we have seen, is about what really does happen. In this way molecules living deep within the interior of the material are able to absorb energy and to play their part in resisting fracture.

on the whole, we can use high strength metals and high working stresses more safely in small structures than in large ones. The larger the structure the lower the stress which may have to be accepted in the interests of safety.

our bones do not become fully calcified until some considerable time after birth. Naturally, young children are mechanically vulnerable, but on the whole they tend to bounce rather than break, as one can see on any ski-slope.

However, all bones are relatively brittle compared with soft tissues, and their work of fracture seems to be less than that of wood. This brittleness limits the structural risks which a large animal can accept. As we have already pointed out in connection with ships and machinery, the length of the critical Griffith crack is an absolute, not a relative distance. That is to say, it is just the same for a mouse as it is for an elephant. Furthermore the strength and stiffness of bone are much the same in all animals. This being so, it rather looks as if the largest size of animal which can be regarded as moderately safe is somewhere round about the size of a man or a lion. A mouse or a cat or a reasonably fit man can jump off a table with impunity; it is distinctly doubtful if an elephant could. In fact, elephants have to be very careful; one seldom sees them gambolling or jumping over fences like lambs or dogs. Really large animals, like whales, stick pretty consistently to the sea. Horses seem to present an interesting case. Presumably the original small wild horses did not very often break their bones, but now that man has bred horses big enough to carry him without tiring, the wretched creatures always seem to be breaking their legs.

As far as our legs are concerned, muscle is not only bulky but heavy, and the object seems to be to arrange for the centre of gravity of our legs to be as high up in the body as possible. The reason for this is that, in normal walking, the leg operates as a pendulum swinging freely in its own natural period and therefore consuming as little energy as may be.

It is because we have to force our legs to oscillate faster than their natural frequency that running is so tiring.

It may be interesting to note that the various tissues of the larynx are among the few soft tissues in the body which conform approximately to Hooke's law; most of the other body tissues obey quite different and rather weird laws of their own when they are stretched

The larynx contains the 'vocal cords', which are strips or folds of tissue whose tensile stress can be varied by muscular tension so as to control the frequency with which they vibrate.

the higher frequencies of the voices of women and children are caused, not by higher tensions in their vocal cords, but simply by the fact that the larynx is smaller and the vocal cords therefore shorter. There is a surprising difference in this respect between grown-up men and women, the relevant larynx measurements being about 36 millimetres for men against about 26 for women. However, the larynxes of both boys and girls are of very similar size up to the age of puberty. The 'breaking' of boys' voices is due, not to any change of tension in the cords, but to a rather sudden enlargement of the larynx around the age of fourteen.

to contain a given volume of fluid at a given pressure will require a greater weight of material if we use a cylindrical vessel than if we use a spherical one.

Even at a mundane level we may reflect that the cost of usable space, per cubic metre, is about twenty times as high in a small ship as it is in an ordinary house; the cost of space in aircraft is a great deal higher still.

The strains to which present-day living membranes can be extended safely and repeatedly varies a good deal but may typically lie between 50 and 100 per cent. The safe strain under working conditions for ordinary engineering materials is generally less than 0-1 per cent, and so we might say that biological tissues need to work elastically at strains which are about a thousand times higher than those which ordinary technological solids can put up with.

the longitudinal stress in a cylindrical vessel, such as an artery wall, is just half the circumferential stress; this will always be the case, whatever the walls of the container are made of.

the egg-shell itself, with its rounded domed shape, is difficult to break from the outside but easy to break from inside.

The process of cooking meat therefore consists in converting most of the collagen fibres into gelatin (which is jelly or glue) by roasting or frying or boiling.

The shape of the stress-strain curve for most animal tissues - such as skin - is very much like that of a knitted fabric, which it is almost impossible to tear.

An arch collects the vertical loads and turns them into lateral ones.

The strength of any structure which is liable to fail because the material breaks cannot be predicted from models or by scaling up from previous experience.

If we are concerned with the plain, semi-circular masonry arch which was widely used in Roman and medieval times, then one of the facts of life is that the rise of the arch must be about half its span.

the skin of young people has a low initial Young's modulus and a low shear modulus and therefore conforms easily to the shape of the body.* In later life the skin becomes stiffer in shear, with obvious results.

the strength and stiffness in twisting of a tube or torsion box depends upon the square of the area of its cross-section.

Cast-iron beams are usually made thicker on the tension face than on the compression face because cast iron is weaker in tension.

Would it not be much more convenient, for instance, if men were made like octopuses or squids or elephants* trunks? One view of the question, which was put to me by Professor Simkiss, is that animals never really meant to have skeletons at all; what may have happened was that the earliest bones were simply safe dumping-grounds for unwanted metal atoms in the body. Once animals had produced solid mineral lumps inside their bodies, then they might as well make use of them as attachments for muscles.

In one sense a structure is a device which exists in order to delay some event which is energetically favoured. It is energetically advantageous, for instance, for a weight to fall to the ground, for strain energy to be released -and so on. Sooner or later the weight will fall to the ground and the strain energy will be released; but it is the business of a structure to delay such events for a season, for a lifetime or for thousands of years. All structures will be broken or destroyed in the end -just as all people will die in the end. It is the purpose of medicine and engineering to postpone these occurrences for a decent interval.

One of the most insidious causes of loss of strength in a structure is 'fatigue': that is to say, the cumulative effect of fluctuating loads.

In a world which has an unreasonable admiration for reason we are apt to forget that the human mind is rather like an iceberg. The rational part of our minds, of which we are conscious, is quite small, and, like the visible part of the iceberg, it is supported from underneath by the subconscious mind, which is much larger.