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What is the correct curve for a suspension bridge? Many people know that the curves called "parabola" and "catenary" have some connection with suspension bridges, but rather than answer the given question, let's look at things another way. Many people know that the curve of a projectile is a parabola, which is almost right. There are two reasons why the curve is not a parabola. One is that air resistance changes the shape of the curve. The other is that it was never a parabola to start with, even on the airless moon of earth. The curve in a vacuum is an ellipse, with one focus at the centre of the planet, but the part above the ground is so close to the shape of a parabola that the difference is academic for golf balls. In the same way, the curves of the cables of a bridge are close to being parabolas and catenaries, but are subtly different. To gain some insight into the reason, let's ask other questions. What is the correct curve for the back of a ladybird beetle or coccinelle? What is the correct curve for the sinuous eel? What is the correct curve for any natural object? In all these cases, the object is not designed to fit a curve: whatever curve we find is the result of the object being fitted for what it does. And that is the way for a suspension bridge - out of the complex calculations done by engineers, the shapes of all the parts emerge. The shapes may not have names: they are just curves for that bridge. On the other hand, between each hanger and the next of a suspension cable, the curve really is a catenary, subject to the approximation that the cable has negligible stiffness with the respect to the forces acting upon it. What determines the stresses in a structure? One of the difficulties of answering this question is that we never actually see anything that has no stresses in it. We will return to that point later. If we take a beam and lower it on to its supports, or we remove the centring from an arch, the structures will deflect because they are being exposed to different forces from the original ones. How far will they deflect, and what stresses will build up in them? What happens is that every part of the structure will move relative to its neighbours until the entire structure is in equilibrium, meaning that there is no net force on any part. When any object changes shape, which is referred to as being strained, it will experience stress, which in most structural materials is proportional to the strain.
For example, if we lower a beam on to two supports, it will bend downwards, probably not enough to be noticed by a casual onlooker, but bend it will. The amount of bending creates exactly enough bending moment to balance the weight that is pulling downwards. Even a simple statement like this conceals a problem. The bent beam contains elastic energy, which came from the loss of potential energy as it sagged between the supports. The concealed problem is that if we calculate the loss of potential energy and the gain in elastic energy, we find that they are not equal. The loss is twice the gain. Yet we are taught that energy is conserved. If we were to drop the beam suddenly on to the supports, we would find out where the missing energy goes. The beam would fall into its deflected position, but on reaching that position it will have a velocity, and therefore kinetic energy. It will go on sagging past the final position until it comes to rest. But now it has deflected too much, and it will start to straighten out. Once more it will reach the final position, this time moving upwards. In fact it will oscillate, and the amplitude of the oscillation will slowly die away as energy is absorbed in internal friction. The beam will become slightly warmer as a result. When the beam has come to rest, we find the following. If E = elastic energy, P = potential energy and T = total energy, E = - T = - 0.5 P Exactly one half of the original energy is lost as heat. This is a general property of systems. The diagram below hints at what happens, and you can download a program by clicking here. The program simulates the behaviour of a beam that is lowered on to two supports and then released. The variations of potential energy, kinetic energy, elastic energy and total energy are plotted against time.
But why does lowering the beam very slowly make a difference? It makes a difference because the gently lowered beam is experiencing the forces from the crane as well as those from its supports, and so it is in a completely different situation. Earlier we mentioned objects with no stress. These can exist at great distances from any gravitating star or planet. Objects in orbit seem to be weightless if you are riding in them, but in fact every gram of the object experiences the same force of gravity, and so the object is unstressed. This is one reason why the manufacture of specialist materials may one day be done in space. To be very exact, our orbiting object is not without stress, because some parts of it are slightly further from the planet than others. But unless the object is very large, the effects are minute. If it is very long, it will experience forces which will eventually make it point at the centre of the planet. A bridge looks pretty placid and static from a distance, but if you stand on it, you may feel the movements as the traffic passes. In time of flood you may feel vibration from the water flow. Many kinds of forces act on bridges. A bridge is often at its most vulnerable when it is incomplete. During construction, some parts may not be connected in the same way as in the complete structure, and they may exert greater forces than in the final design. Parts may be free to swing in the wind. The list below is mainly about forces on a complete structure, but some will apply to incomplete structures also. Steady forces on a bridge The weight of each part The forces from the supports The weight of stationary loads Drag from a steady wind Upward or downward lift from a steady wind Forces from smooth, steady water flow Forces caused by dimensional changes resulting due to temperature Transient forces on a bridge The weight of moving loads Longitudinal forces from accelerating or braking loads Transverse forces from moving loads on curved bridge Forces as loads enter and leave Forces from gusting winds Forces from turbulent water Transient periodic forces set up by vehicles entering or leaving vehicles changing speed steady wind producing vortices gusts of wind turbulent water flow any other transient phenomenon
Periodic forces Steady wind producing vortices |
