Energy When the stress on an object is changed, the change in stored energy is the product of the force times the distance through which it acts. In most substances, the force increases as the deflection increases, and so integral calculus is needed. The stress will probably vary with position also, and so the energy has to be integrated over space as well as time. If material can take a large force with little change in size or shape it will store less energy than one which changes more for the same force. Let's look at a few graphs. The lower graphs show deflection versus force for two materials. The one on the right deflects less for a given force. So the force has to do less work. The upper graphs show that the energy stored is less, for a given force, in the second case. So if both materials had the same maximum safe load, the one depicted on the left could safely absorb more energy. For a given force, the stiffer material stores less energy than the more flexible one. Archers use bows that can bend a very long way without breaking, to store a large quantity of energy. In fact, bows can be pre-stressed so that that even before they are pulled back, there is a force. The energy stored is then much greater. Note that the energy stored is always positive, whether the deflection is positive or negative. In other words, it doesn't matter which way you twist or bend something, or whether you push it or pull it, you will use energy, and the material will store some of it. If the material is perfectly elastic you will get it all back when you remove the force. Most materials absorb a little energy, which heats them. A tyre becomes quite warm at high speed, since a wave of deformation is rotating around the tyre, as seen in the frame of reference of the tyre. This means that during oscillations, kinetic and elastic energy vary at twice the frequency of the displacement. This is explained in the page about oscillations. Since the amount of energy that an object can absorb is related to the quantity of material, we could draw a graph of strain versus stress below, and then the upper graphs could represent energy per unit volume. That doesn't mean that we always choose the material that will absorb more energy. If you are bungee jumping, trampolining, or falling off a crag, you expect some energy to be absorbed. But you don't expect a bridge to sag very much when you drive over it. Now let's draw the graphs the other way round, again with the stiffer material on the right. Now we see that for a given deflection, the stiffer material absorbs more energy. Returning to archery, an archer can only pull a bow as far as he or she can reach, so the deflection is limited. By making the bow as stiff as possible, as long as the archer can still pull it all the way back, more energy can be stored. |

How many forces are needed? How many forces are needed to create stress? According to the diagrams below, we need at least two forces to compress, two to stretch, three to bend, and four to shear or twist. Obvious, isn't it. Obvious, but wrong. Look at the picture of a space craft below, floating in a region of negligible gravity. The only force on it comes from the pressure of the exhaust gases. Yet the ship is very slightly compressed by that one force. In fact, if the thrust of the rocket is exactly equal to the weight of the craft, the compression is the same in both cases. But isn't there a difference? When the space craft is resting on its tail, on earth, it experiences two forces, its weight, pulling it down, and the reaction from the earth, pushing it up, just as Newton's third law predicts. Where is the second force in the diagram? Is there a second force? What we learn from this is that when we read or hear a statement, we must be sure that we understand the range of conditions in which it applies. Is there a second force in the diagram? We could equally ask, is there a second force in the space craft waiting to be launched. Yes - gravity. But Einstein thought about these two cases, and realised that the people in the space craft cannot distinguish between the accelerated ship and the ship sitting in a gravitational field. (We are ignoring the vibration caused by the noise part of the rocket's thrust.) So Einstein decided that there was no second force in either case. He eliminated gravity as a force by deciding that "gravity" is actually a local curvature of space-time. He took a long time to work out how to calculate the effects of this idea, and in fact he only did it with the help of mathematical collaborators. We got off the subject there, but not as far as you might think. To see why, let us go back to the 19th century, when Maxwell discovered his electromagnetic equations. They looked quite complicated, though there was great symmetry among them. Later, they could be written down much more simply by using vector notation, which also made them easier to manipulate. In Einstein's case, the mathematical technique was tensor algebra, which is can also be used to calculate stresses. Notation may not seem to be a very profound subject, yet the use of suitable notation can make an enormous difference. Try multiplying or dividing with Roman numbers. The introduction of the Arabic system, together with zero, led to great progress. A little later in the 20th century than Einstein's theory, Dirac's equation, which led to the prediction of anti-matter, could also be written down in a long form and a short form. Dynamic Effects There is another aspect of forces and stresses which would not be apparent when building most structures. If we used a trampoline instead of a plinth and we let go of the bust when it just touched the trampoline, we would get a nasty surprise. The bust would accelerate downwards at first, then slow to a stop, and then do the same in reverse. It might oscillate up and down several times before coming to rest . So forces also produce acceleration. This is extremely important in structures because there are live loads - people walk across upper floors, trains and vehicles go across bridges, and the wind can blow from any direction. These forces can set the structure vibrating at its resonant frequencies - the ones at which it naturally oscillates. If the forces vary at one of the resonant frequencies, things can go wrong. Breaking step on a small, flimsy bridge is good practice. Even worse, more than one resonant frequency could coincide. Worse, still, a small but fairly steady force can produce oscillations because of the interaction of the structure with the air. Tacoma Narrows bridge is only the most famous example of several collapses. In the year 2000 the Millennium bridge in London was closed very soon after it was opened, because of oscillations. The effects of forces are never instantaneous. They always take time to propagate through the medium. As the speed of waves in structural materials is so high, we are normally not aware of this. What happens dynamically is the result of the relationship between inertia and elasticity, that is between kinetic energy and potential energy. Oscillation Compression and Tension Compression differs from other stresses in producing inherent instability. Something is stable if it tends to return after a deflection. But if you take a thin plastic ruler and push the ends together, you will find that when a certain force is applied, it will suddenly bend. So struts in compression have to be much thicker than tension members. Look, for example, at the piers and the cables of a suspension bridge. Look also at the four piers that support the tower and spire of Salisbury Cathedral - they are noticeably curved. A strut will generally buckle before it reaches the compression limit that would make it fail by crushing or shearing. A catastrophic buckling failure caused the collapse of the first attempt at a bridge over the St Lawrence River at Quebec. Several large steel box-girder bridges have collapsed during construction, because the stresses were greater than those expected in the complete structure. These stresses were obviously not well understood. Here is a dramatic example of the difference between compression and tension. Each of the four legs of the Severn suspension bridge carries about a quarter of the weight of the bridge. Each of the four main cable sections visible here has to carry about an eighth of the weight of the bridge, but as they are nearer to horizontal than vertical they actually experience more force than the towers. Nevertheless the legs of the towers are much wider than the cables, and are stiffened inside by flanges. Such a bridge has be designed in such a way that the resonant frequencies of the towers, cables and suspended structure are all different. It has also to resist resonance while being built. The partly finished towers are rather like a giant tuning fork. Torsion Torsion is not a good idea in a bridge. It is a twisting of the structure. In a long suspended span, the presence of many heavy vehicles travelling in one direction, with few on the other side, will cause the deck to twist. Torsion is particularly worrying in plate girders and box girders, if the calculations are done on the assumption that all flat surfaces remain flat under load. Deflection from the ideal can greatly reduce the stiffness of the structure. A long motorway over-pass may supported by single columns to reduce the visual and practical effects on roadways below. This design might be chosen for a highly skewed bridge. The deck must then be stiff enough to transmit asymmetrical loads to the abutments without undue torsion. The result of torsion is to set up shear strain throughout the material. A stiff component needs the material to be concentrated where the shear is greatest. A tube is the ideal shape. More on torsion Shear Shearing is a change of shape in which a rectangle tends to become a parallelogram. You need to apply forces on four different surfaces to get this effect. The most visible effects of shear are seen in faults like the San Andreas, where continental plates are moving relative to each other. This causes deformation at the edges, where sliding is inhibited by friction or bonding between the slabs. Eventually, the stress builds up until at some place it is too much, and the interface ruptures. This passes more stress to neighbouring parts, and the separation spreads catastrophically. The stored energy is released into kinetic energy of the ground. Like any other elastic solid, it oscillates until friction damps the motion. People have reported seeing visible waves running along the surface. How can you know if there is shear at a surface. Imagine a complete split opening up at that surface. If a sliding motion would occur, there must have been a shear stress. Bending Bending would seem to be a fairly simple business, but it isn't. If you bend a long rectangular rubber eraser it tends to bend in two different axes. This effect is very small for structural materials, but we do need to remember that bending a beam causes compression on one side and tension on the other. There is a neutral surface where the length remains unchanged. More on bending Fatigue Once we know the elastic moduli and the breaking stresses of a material we are not yet in full possession of of all the facts. If you repeatedly bend a paper clip in different ways it will soon break. In fact it may break under a stress that earlier it would have withstood. In other words, the material after many cycles of stress is not the material we started with. The crashes of the de Havilland Comet 1 were caused by the repeated expansion and contraction due to pressurising the cabin, combined with stress concentration at rectangular openings in the hull. Such a time-dependent failure had been envisaged in the novel "No highway", in 1948, by Nevil Shute, though he may have thought in terms of time rather than the integrated cyclic stress.You can test the effect of stress concentration by cutting a V shape in the edge of a piece of paper. That is where it will break when you pull the paper apart. The perforations in sheets of postage stamps and in rolls of tissue are designed to allow for easy separation when required, while preserving integrity during handling. More on fatigue These three pictures show a piece of foam plastic which has been strained by pushing objects against it. 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. The final configurations are those which minimise the total strain energy. |