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Strains and Stresses
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Translation - Italiano
Introduction We all use the words "strain" and "stress", but as with many other words, they have specific meanings in engineering and science. So firstly we must define what we mean by strain and by stress. Some of the earlier airliners, and also aircraft such as the B-47 and B-52, had rather flexible wings, which could move up and down quite alarmingly. Yet both stress and strain were actually quite low. Consider a rope lifting a weighty object. No matter how thin or thick the rope, the force in the rope is equal to the weight of the object, if the movement is steady. But the stress and the strain are different in different ropes. Stress Stress is defined in terms of a force and an area across which it acts. Force is measured in Newtons (N) and area is measured in square metres (m^{2}). Stress is measured in Newtons per square metre, which can be written as N/m^{2} or Nm^{-2}. Note that although pressure is measured in the same units, which are then called Pascals (Pa), we do not measure stress in Pascals. Let's take an example. Suppose that the force in a cable of a suspension bridge is 10000000 Newtons, or 10 megaNewtons, that is, 10 MN, and the cross-sectional area is 0.25 square metres. The tensile stress is then 10000000/0.25 N/m^{2}, or 40 MN/m^{2}. To make this easier to visualize, one Newton is roughly the weight of 0.1 kilogram, so a kiloNewton is about the weight of 0.1 tonne, and a megaNewton corresponds to about 100 tonnes. More exactly, 1 kg weighs about 9.8 N. If we were to reduce the area of the cable to 0.1 square metre, the stress would increase to 100 MN/m^{2}. The stress is not related to the type of material: only to the force and the dimensions. Strain Suppose we have a cable of length one metre, and a cable of length one hundred metres, and we apply the same force to both. We would expect the change in length of the long cable to be greater than the change in the shorter one, simply because the longer one includes the length of the shorter one. If the shorter cable stretched by 1 mm, we might expect the longer one to stretch by 100 mm. In order to obtain a measure of the deflection which is independent of the size of an object, we define the strain as the deflection divide by the original dimension. If the stretch of an object were very great, as in an elastic band, we might worry about what length to divide by, but this problem seldom arises in structures. Tensile stress = change in length / total length. Strain is not measured in any units, because it is the ratio of two lengths. As with stress, its definition says nothing about the material. Types of Stress The simplest types of stress are tension and compression, which are illustrated below, red showing compression and blue showing tension. Bending is also familiar: we often need to bend something, and sometimes things bend when we don't want them to. Most bridges are stiff enough to prevent us noticing any bending. The next picture illustrates the compression and tension in a horizontal beam, ignoring the stresses at the supports. Shear is a stress which is perhaps less easy to imagine: it results from the opposition of forces that are not aligned. The diagram attempts to illustrate this. The next two pictures show a flat rock face. The central part, between 30 and 90 cm wide, shows effects from shear forces. This type of relative movement of rock sections is called a fault. For this deformation to have happened, the layers in the central region must have moved relative to each other, so that any rectangular area in the original rock has become a parallelogram. That is what is meant by shear. If you don't know what kind of stress is present, try imagining a cut through the object. If it pulls apart, there was tension: if your tool jams in the cut, there is compression: if the parts slide relative to each other, there was shear: if the parts rotate, there was bending, and if the parts twist relative to each other, there was torsion. As we have seen, bending implies tension and compression: similarly, torsion, which twists things, implies shear. In real structures, all forms of stress may be present at any point, and they may be acting in any direction, and calculation my be complicated. In a structure such as a pinned truss, all the stresses, except near the pins, act along the members, and the calculation, while complicated in practice, is simple in principle. Let us consider an imaginary cutting experiment with a beam. If we make a vertical cut near the end of a beam then it is clear that this will start to happen - But for a cut in the middle, the two halves will fall identically, with no offset. So there is shear stress near the ends of the beam, but none in the middle. There is also no shear at the surface, even at the ends. A calculation gives a picture like this - Purple and green represent opposite polarities of shear. From these diagrams we see that at the centre of a beam there is no shear and no tension or compression. We could put a hole there to save weight. In fact, some beams, such as castellated beams, have holes all along the major axis, and a truss is a means of making a lighter structure in which the top and bottom members take the compression and tension, leaving the sloping members to deal with the shear. An I-beam is another solution to the same problem. Let's look at a simple example. The diagram below represents a part of a long bar of metal which has two slots cut in it, such that the central section has one half the width of the bar. We now imagine heating the central part while keeping the remainder at the original temperature. The heated part will be forced into a compressive state, shown by red colour, because unrestrained it would expand. This is not the lowest possible energy state: if the outer parts are put into tension equal to half the original compression, while the compressive stress is halved, the total energy is halved. The reason is that the energy is proportional to the square of the stress. Halving the stress makes reduces the energy to a quarter, but we have doubled the energised volume. This is all very well, but in both diagrams there is no means of creating the stresses. What actually happens is more like the next diagram, in which coloured lines represent the directions of the stresses. Only one half of the picture is shown, together with an expansion of the earlier diagram. The stresses are not really in separate lines - the lines are merely a means of showing the direction and strength of the stresses. Where the lines are more closely spaced, the stress is greater, and where they are sparser, the stress is weaker. The third diagram is a part of the first one - bear it in mind when beams are discussed below. What we have done is to illustrate the load path, the route by which the forces are transmitted through an object. The diagram is not exact: the red and blue lines ought to cross at right angles, just as they reach the surfaces at right angles, but it does illustrate the way that the compression and tension are related. The regions where both are present are in fact in a state of shear, so we see that even simple structures can include regions where the stresses are not purely of one kind. This illustrates the fallacy of thinking in terms of equations or geometry. Engineering is not geometry. The curves illustrate another feature of materials and stresses - continuity and discontinuity in the properties of the material must be reflected in the continuity and discontinuity in the distribution of stresses. This shows that the diagram is not correct, because it contains discontinuities in curvature where the straight lines join the curves. In reality, none of the lines would be quite straight, except the central one, and there would be stress concentrations around the ends of the cuts. Later in this page we will see similar effects in beams, and we will see that the geometrical simplicity of a continuous box girder with piers is superficial. Having explained that compression and tension can be present along with shear, we must also mention the fact that "pure" strains are not common. If you compress or stretch a rod, the width will probably change, by an amount that is specified by Poisson's ratio. If you twist a rod, not only will its width probably change, but its length will probably change as well. The full description of stresses and strains is very complicated, though luckily most structures are strained so little that the more subtle effects are not important. On the other hand, the small amount of strain that large stresses produce has another side - if you force a poorly fitting member into a structure you can set up very large stresses. It can even happen that adding a "strengthening" member can make a structure more vulnerable. See indeterminacy. This effect has its uses - you can split a large piece of wood with a thin axe or wedge. These lines of stress may remind you of magnetic and electric lines of force. Indeed, in the 19th century, some people believed that magnetic and electric fields were carried by a medium, called the "aether". It was even capable of transmitting waves, at a speed we now know to be the speed of light. The aether had to be incredibly light, since it did not seem to impede motion, yet the colossal speed of light meant that it had to be unbelievably rigid, since speed is proportional to the square root of rigidity divided by density. Not many people still believe in it. This digression reminds us that compressive and tensile forces in solids are not transmitted infinitely fast, but at the speed of sound, which is much faster in solids than in air. Bending motions, as in the Tacoma Narrows bridge, travel much slower, but still at hundreds of kilometres per hour. We have not mentioned anything about the properties of materials. Using the ideas we have already mentioned, we can look at the properties of materials. For example, if we attach an elastic band to a steel wire, and pull the ends, we will see that the band will stretch much more than the steel. We divide the stress by the strain, and the result is called the modulus of elasticity. It is measured in Newtons per square metre. A larger value corresponds to a more rigid material. Moduli for solid materials are - Young's modulus for compression and stretching Shear modulus for shearing and twisting Compressibility or bulk modulus The moduli don't tell us everything about a material. We are interested in the ultimate strength as well as the elasticity. A material can be very rigid, but unable to stretch far before breaking. In other words, it cannot absorb much energy. Many natural materials can absorb large amounts of energy, and release much of it when the stress is released, as in an archery bow. Kevlar is an artificial example of such a material. The idea of a rigid material sounds very attractive for building structures, and so it is. But there is a dark side as well. If we have a very rigid deck, supported by hangers or piers, and the lengths of these are not adjusted correctly, the rigidity of the deck will prevent it from conforming to the faulty lengths, and this will lead to stress concentrations. It is even possible to weaken a structure by adding material in an unsuitable fashion. For example, stresses can remain after welding or rolling. A catastrophic accident destroyed a DC-10 airliner after an engine pylon was mounted using an incorrect procedure, which overstressed the parts when they were very slightly misaligned. Let's look at a simple example that shows the effects of errors in construction when the materials are very rigid. Here is a tied arch bridge over the M42 motorway. It was built on land to the left of the motorway, and rolled into position as a complete unit in the course of a single night. We see near the left side, arrows representing some of the forces on the deck - blue for the tension in the hangers and green for the effects of gravity, that is weight. Although each hanger takes the weight of one section of the deck, the great rigidity of the deck transfers loads between hangers, especially live loads. The hangers, too, are rigid in the sense that they do not extend very much when loaded. Suppose we operate the hangers with an elongation of 0.3 percent of their length. This means that if one hanger is too long by only 3 parts in a thousand, it will experience no tension at all, placing extra stress on its neighbours. And if it is too short by the same amount, it will experience double stress, which would be very serious. These conclusions would be modified by the response of the whole system to the error, but they are roughly right. Errors in the fabrication of the attachments, the deck, or the top chord of the arch, would produce similar effects. In practice, the hangers might be provided with means for adjusting their lengths. What causes what? In many materials, if they are not stressed too much, the strain is proportional to the stress. The values at which this is no longer true is called the elastic limit. Does the stress cause the strain or vice versa? If you hang a heavy weight on a cable, you are stressing the cable, and then it stretches, so it appears that the stress causes the strain. But suppose you fix a cable very firmly between two very rigid supports and you then cool the cable with ice. A free cable would shrink slightly, but a fixed cable cannot do this. What will happen is that it will build up a tensile stress, because effectively you are stretching it, because at the cooler temperature its natural length would be less. You could say that the strain has caused the stress. In fact, there is no value in thinking of cause and effect: strain and stress occur together - they are inseparable. If you pour very hot water in to a thick-walled glass vessel, the thermal expansion of the inner part may shatter the vessel. The result of the expansion is to place the inside into compression and the outside into tension. The thermal expansion coefficient for soda glass and lead glass is about 9 X 10^{-6}, while for Pyrex glass it is about 3 X 10^{-6}. Some artefacts made of glass and plastic have built in strains and stresses, which may be made visible by the use of polarized light, if the object is transparent. To see these effects you need to send polarized light through the object, and you need a polarizer for viewing. In this example, the toughened glass windscreen is strained and stressed throughout its volume. When specially treated glasses break, they may do so in ways which are unlike the normal shattering into dangerous shards. If the windscreen of a vehicle shatters like this, the driver can easily punch out a viewing hole without risk of being cut by sharp edges. This grindstone is made from a number of pieces of stone which are held together by a metal rim that has been placed around the stones when hot, and then allowed to cool. The result is a rim that is under tension, and a set of stones that are compressed together. The hole in the centre was originally square, to fit the square wooden drive shaft. Expansion and contraction are sources of concern in almost any structure, and complicated measures may be needed to cope. With a structure weighing thousands of tonnes, which has to carry heavy loads, designing attachments to the outside world can be extremely difficult. A Question If we subject an object to compression or tension, squares become rectangles. If we subject an object to shear stress, squares become parallelograms. This is shown in the next diagram. Now look at the next diagram, where the smaller squares have been drawn in different orientations from the larger ones. The smaller square under compression has become a parallelogram, and the smaller square under shear has become a rectangle. Does this mean that the types of stresses are not fully independent or distinguishable? We really ought to be thinking in terms of the stress tensor. |
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How much stress? How much strain? How can we measure stress and strain? Strain looks easy - because it is simply related to deflection, perhaps you can simply measure an object before and after the application of a force. Yes, but you will only know the overall strains, which is certainly of great importance. The strains that allow the wings of aircraft to bend in flight, and indeed on the ground, would be utterly unacceptable in the fuselage or in a bridge span. But that is only a part of the story. We also need to be sure that no part of a structure is strained beyond a safe limit. The means of measuring local strain is the subject of another page. On the subject of wings, they may be bent under tests by amounts that look very alarming, but in fact the strain remains quite small. The wing-tips may be a metre or more from their normal positions, but that is irrelevant. Strain is measured by the relative local deflections over very small distances, as a fraction of the distance being measured. For example, suppose a hundred foot wing is bent by a total of six feet, a fairly substantial amount. For simplicity of calculation, let us assume that it is one foot thick throughout, which is of course unrealistic: a real wing tapers. The distance from the neutral axis of the surfaces is half a foot. These surfaces lie on curves which are longer and shorter respectively than the neutral axis. By how much? The displacement of six feet in a hundred feet means that the angle at the tip is about 12/100 radians, or 0.12 radians. This is a fraction of a complete circle, namely about 0.12/2pi, or 0.06/pi. So the 100 feet is 0.06/pi of the circumference, which is therefore C =100pi/0.06. Now we can calculate the radius, because C = 2piR, or R = C/2pi, which is then 50/0.06, a large value. The radius of the two surfaces differ from this by half a foot, and so the strain is 0.5/(50/0.06), or 0.5 x 0.06/50, which is 0.03/50, or 0.06/100, ie 0.06%, a small value. Thus the appearance of objects does not immediately tell us about the strain, unless we are used to interpreting them. So much for strain. Stress is another matter. Measurement of dimensions tells us nothing. We need to measure forces. Alternatively, we need to know the moduli of elasticity. For example, to bend a strip of steel and a strip of card with identical dimensions, we need vastly different forces, and the stresses differ in proportion. |
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Strains, Stresses and Atoms Until well into the 20th century, nobody knew the nature of the bonds that held atoms together to make molecules and materials. Consequently, the design and production of materials was based on empirical results rather than theory. Not until the advent of quantum mechanics could people begin to understand the forces between atoms, and hence the stresses and strains in materials. It is natural that if we want to understand how atoms behave, we find that the methods that we use for large objects do not work. Imagine cutting a piece of iron in half, many times, photographing one half each time, until we get a photograph of one atom. Before starting a project, it's a good idea to know as much as possible about what it will entail. Suppose we start with one kilogram of iron. How many times must we cut in half to get down to one atom? A reference book tells us that the number of atoms in one kilogram of iron is 6 X 10^{23} / 56 = 1.1 X 10^{22}. So if we were to divide by 10, we would have to do it 22 times to get an atom. Dividing by two would require more like 73 operations. The problem is that the final result would be very disappointing - a photograph of an atom looks very fuzzy. Atoms are composed of minute nuclei, and electrons, which are so small that nobody has any idea of their size, or even whether they have a size. Quantum mechanics tells us that the idea of localizing such a small particle is meaningless. All we can do is to compute a picture of the probability of finding the particle at each place. For a multi-electron atom we can do the same thing. We get something rather like the next picture. Atomic nuclei give a similar picture, except that they have a rather more definite radius, especially the larger ones. And the atoms in a cubic crystal might be arranged like this. Although the atom appears to be a rather indeterminate sort of thing, it turns out that the distribution of the electrons can be calculated with great accuracy. It also turns out that the positions of the atoms in a crystal can be determined very accurately as well. One of the beautiful aspects of quantum mechanics is that although it forgoes the ability to calculate quantities that "common sense" suggests to be real, it gave, for the first time in history, the reason why atoms have the sizes they have, are spaced as they are, and interact as they do. Quantum mechanics gave not only that, but a complete means of understanding chemistry and the solid state of matter, which has led to a great variety of substances and components, on which a great variety of devices are based. Electronics, for example, has expanded so much since the invention of the first transistor, that we can now see that using valves (tubes) enabled people to create the early computers before a really suitable technology was available. Charles Babbage, in the 19th century, hoped to create computers using steam powered machines. They would have worked, but they would have been enormous, and would have consumed vast amounts of energy. With computers, people have been able to design things with much more efficient use of materials, and to design things that would have been too difficult to calculate at all without computers. Thus technology feeds on technology. All this and very much more came from a theory which was based on de Broglie's idea that electrons can behave like particles or waves, and had no immediate application. To return to the behaviour of structural materials, what you are doing when you stretch, compress, bend or twist an object is to change the relative positions of the atoms, displacing them from their normally stable positions. We can get a good idea of the behaviour by imagining the atoms as masses connected by springs that resist being either compressed or stretched. From this model we see how forces on an object are resisted: the object is deformed until it can generate exactly enough force to balance the imposed forces. We see also that an impulsive force will take time to propagate, and that objects can vibrate or oscillate, as the energy passes between the compressed or stretched springs and the moving atoms. We must remember that models (and theories) are just that, and are not reality. We must also remember that models, and most theories, have limitations, and using them outside their range of applicability leads to error, or even catastrophe. Extrapolation from the known to the unknown has led to many failures. This is one reason for making practical tests before taking such steps. Computer simulations are powerful means of investigation also, as long as the modelling is realistic. On other hand, some physicists who study fundamental interactions, or at least those that they believe to be fundamental, would like to create theories, or better still, one theory, that explains the existence of all known "elementary" entities and all the "forces" between them. These physicists have not yet found a "theory of everything (TOE)", nor have they found a "grand unified theory (GUT)". Let us leave them to their search. The next diagram tries to give some idea of the energy relationships for two neighbouring atoms. What we are doing here is to investigate the energy of an atom as we vary its distance from another atom. The distance up the graph represents energy, while the distance from left to right represents distance between the atoms. The red and blue curves represent the energy of an atom. The equilibrium position is the one of minimum energy, where the red and blue curves meet. We can see that if we try to push atoms together by compressing the material - red curve - the energy increases rapidly. Tension also increases the energy, but not as sharply, and eventually we can increase the distance between the atoms as much as we like with no input of energy - we have broken the object. What are the dots? They are intended to suggest the range of energies and positions in a real substance. All the atoms in a substance are in a continual state of movement. The quantity that we call temperature is in fact a measure of the kinetic energy of the atoms. Along with the variation in energy goes a variation in distance between the atoms, and if we compare the upper diagram with the lower one, we can see that higher temperature means higher average distance between atoms. So we understand thermal expansion. The anomalous behavior of water between 0 C and 4 C is not an exception to this rule: what is happening is that as we cool the water, the internal arrangement of the molecules is already changing towards the structure of ice, which has bigger molecular spacing than water. |
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Here are some diagrams which suggest what happens to atomic spacings under various types of strain. In practice, the strains in any region of an object is likely to be a mixture of two or more types of strain. For example, in the diagram for bending, we see that in any small region, the effect is either compressive or tensile. It is only when we connect the little regions together that we see the overall effect. Einstein faced a similar problem when he was working on the general relativity theory, in which space-time is "distorted" by the presence of objects with mass. The connection of all the small regions and the description of a large region of space time required the use of tensor algebra. The same algebra is used to calculate stresses, by means of the stress tensor. What is a tensor? It is a means of manipulating many equations at the same time, without having to write each one out again and again as it is altered by the steps in the calculation. The nature of the tensor has to be such that the relationships between all the equations is correct. It may be instructive to compare the results of various types of strain and stress. In tension and compression of a bar, the strain will be roughly uniform throughout. The same is true of shear, and therefore of torsion in a tube. But in bending a beam, the material around the neutral axis is stressed much less than that near the edges, and the same is true for torsion of a bar. Much ingenuity has gone into designing shapes that optimise the use of material, such as I-beams, box-girders and trusses. We must always remember that an object will tend to move to a position of lower energy, which may not be the one we designed for. A strut in compression may buckle rather than shrink, a box girder may buckle rather than bend, and a plate may buckle rather than take shear strain. Much theoretical and practical effort and material goes into combating the tendency to buckle. |
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Time There is a lot in this web-site about space and its three dimensions and some of the objects, such as bridges, that occupy space, but there is little about time. What has time to do with bridges, apart from the time it takes to obtain approval from agencies and local populations, the time it takes to design them, and the time it takes to build them? But time enters into bridge behaviour in a much more subtle way, one which can have serious, even catastrophic, results. Imagine a large truck arriving at a bridge at 80 kilometres per hour - about 20 metres per second. If the wheel-base is 4 metres, it will take a fifth of a second for the entire weight of the truck to move on to the bridge. How long does it take for the parts of the bridge to respond? Is the response instant? No, it isn't. The main effect of the truck is to deflect the bridge downwards, but another effect is the longitudinal thrust from the tyres. All the effects of forces on an object are transmitted at speeds that depend on the type of stress and the properties of the material. If we hit a nail with a hammer, we expect that the effects are transmitted instantly through the nail, but in fact, a compression wave travels down the nail at the speed of sound, which in iron is around 5000 m/s, varying with the type of iron. So the propagation time in the nail is minute. For a 10 cm nail, the time would be 0.01/5000 = 2 microseconds. Bending waves travel more slowly than sound waves, but may still go at several hundred km/hr, which can result in a time of several seconds for a long bridge. So if we hit one end of long a bridge with a hammer, the result could be heard significantly later. Waves in a structure are generally damped to some extent, that is they weaken as they travel, because energy is absorbed by the steel. What happens when a wave reaches the end of a bridge. The discontinuity at the end causes most of the energy to be reflected. Several reflections can occur before the energy is completely dissipated. This normally does not matter, but in some cases there have been serious results. If energy is supplied more or less continuously, waves are continually generated and propagated. Again, this will normally not matter. But consider waves with such a frequency that after being reflected twice, they are exactly in step with the next cycle. The old cycle will reinforce the new one, and so on. The amplitude will build up until the rate of energy input is equal to the rate of energy dissipation. Again this does not usually matter. You can hear this resonance effect by putting a bottle or a large cup or mug near to your ear, and listening to the sounds, which are based around a resonant frequency. Incoming energy excites the resonance, so the container acts as a frequency filter. Danger threatens when the input of energy occurs at the same frequency as the resonance. This is the reason why soldiers are told to break step on bridges, and it is the reason why the London Millennium Bridge had to be closed for modifications. The pedestrians of London are generally not known for walking in step, but an unforeseen peculiarity of the motion of the bridge was that walkers unconsciously adjusted their stride to match the oscillations of the bridge. The Tacoma Narrows bridge was downed by a steady wind, but that steady wind induced vortices at a regular frequency. These, together with the extreme slenderness of the bridge, helped to cause its destruction. See also the page about oscillations. That is where time comes into bridge building - the time it takes for something to happen. Time also appears in the phenomenon of creep; slow changes in strain as a result of continuous application of force. Time also appears in the phenomenon of fatigue, in which stresses that would be easily held if constant can prove destructive if applied cyclically. |
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