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Simple Beam Theory
This is not the smallest beam in this page, but it is one of the simplest - a simple skew bridge over a stream. In spite of small scale, you can see two features - the netting to give a good grip in wet conditions, and the way the soil has eroded asymmetrically because of the skew. So even at this size, construction is non-trivial.
Books about bridges often begin by saying that early bridges were possibly tree trunks, perhaps later squared off after tools were invented, as in the second picture. The third example comprises two tree trunks and a large number of slats. Although the squaring off reduces the depth of the wood, making the beams thinner and weaker, the rectangular section is in fact more efficient than the round one, as we shall see later. The next picture shows a more spectacular example.
What could be simpler than a beam? Compared with a truss the solid beam looks easy to understand.
The opposite is true. A pinned truss made of narrow ties and struts, if not over-determined, can be worked out using a set of linear equations; tedious, but not difficult. But a solid beam has an almost infinite number of parts. Even if it is completely uniform in every way, and supported in a simple manner, calculating the stress and strain at each point is not easy. Using finite element analysis, the calculation can be made as accurate as we want, at the expense of computing time. But having done it, we don't necessarily "understand": all we have is a set of data for the forces at many points. But then, that is all we need for most purposes. Note the phrase "completely uniform". The fabrication of a structure may induce stresses and strains that persist after the work is complete. Examples are the production of a rolled steel joist and the welding of a box girder.
The bodies of many animals can be considered as beams between the legs, and their necks can be considered as cantilevers, though of course the internal structure is vastly more complicated than anything made by people Look at a giraffe, for example. Since animals move, they can be seen as both structures and mechanisms, and as both they can be elegant and efficient, and as a result, beautiful and elegant. People sometimes use a charging rhinoceros as a symbol of clumsiness, but look at a trotting rhinoceros - it is very graceful indeed, and you can almost see the centre of gravity moving with minimal energy loss. A picture may be painted by making an exact copy on the canvas or paper of what the eye sees, but many artists have realised the benefits of understanding the underlying structures. Leonardo da Vinci and George Stubbs were both fascinated by the structure and movement of animals, and it shows.
Understanding is a strange business. If we knew the position and velocity, and the chemical bonding and energy levels, of each atom in a living thing, would we understand? No more than a person viewing a cricket match or a soccer match without knowing the rules. Such a person views, but someone who knows the rules, sees, and understands.
The difficulty of calculation was dramatically demonstrated when several box-girder bridges collapsed during construction around 1970. These bridges are based on boxes, which are like hollow beams with stiffening diaphragms and flanges. Before building the Britannia bridge, comprising hollow beams, Robert Stephenson had extensive tests made, using scaled down models. He knew that current calculating techniques were not adequate as the sole guidance for building such a bridge.
One of the dangers in engineering arises when we use designs or methods in areas for which they have not been tested, and for which they have not been calculated.
One great advantage of the beam is its very simple appearance. In a town this can be very important. Another is the relative simplicity of construction.
This page tries to give some idea of what goes on inside a beam. Further information can be found in the page about pre-stressing. The idea of the simple log bridge is a little erroneous in the cases where the tree has generated stressing forces inside itself. Cutting up such a log and reassembling it to make a ship's mast or a bridge produces a structure with different forces in it, because the pre-stressing is lost.
In reality, building a small beam bridge may not require complicated calculations. Many types of small structures can be designed using codes of practice developed over many years of experience. But anyone who uses rules outside their areas of validity courts disaster.
As already stated, one of the simplest ways of spanning a gap is to put a long object across it, such as a flat stone, a log, or a plank. The first picture here shows a single stone slab over a small stream south of Naunton. It has a clear span of only about half a metre. The second bridge, Keble's bridge, with four stone slab spans, is at Eastleach. The other two are at Lower Slaughter, one stone and one wooden. All four are in the Cotswolds. Horizontal stone slab bridges are sometimes called clapper bridges.
The diagrams under the white panel below show what can happen as the span is increased, everything else remaining the same.
The loss of one support of this roof ridge, probably from the effects of age, has doubled the span - see first picture. The effect is dramatic. Removal of supports in a building is a bad idea, unless you understand the building and you know that you can safely make the changes. The pillars at the front are capped with short cantilevers that spread the load into the beams. The builders of Chinese buildings took this idea to an almost extravagant degree of development, producing complicated, though logical, series of cantilevered brackets to support roofs. This practice migrated to Japan. The second picture shows the result of completely rebuilding the roof. Perhaps as the years go by, the roof will acquire a new population of lichens, mosses and sedums.
These pictures, the second compressed laterally to clarify the effect, shows the effect of removing an intermediate support. These windows were built with a central mullion. Removing this and replacing the old frames by plastic ones has increased the area of glass, and improved the illumination in the rooms. The lintels look thick enough to span the gaps, but in fact they are not continuous: they are in two parts. The original mullions acted as piers at the joint. The pictures show how the lintels are sagging.
For a given material and cross-section, increasing the span beyond a certain point makes the sag unacceptable. Furthermore, adding a load would produce greater sag, and such a structure would be alarmingly flexible.
Galileo showed that for animals and structures, it is impossible to scale something up while keeping the same proportions. This is because different variables vary linearly, as the square, and as the cube of the dimension.
The crane fly could not be scaled up by even a factor of ten without some changes in relative dimensions. We instinctively recognize this when we notice the absurdity of the cruder type of science fiction film or horror film which depicts giant insects or other animals. Galileo explained this in masterly fashion. Small and large aircraft, small and large bridges, small and large mammals, do not resemble each other very closely in proportions, though the anatomy of mice and elephants, and many other mammals is in fact quite similar (homologous) in many ways. People even have the same number of toes as lizards. More about insects can be found in the page about tubes.
can see easily that the sag increases faster than the length, because
each span is a part of the one below (approximately). When we add a bit at each
end, we already adding it at a slope, which itself goes on increasing
towards the ends.
There is a hidden assumption, namely that the beam does not break. Stone would break before bending this much. It requires a force to bend a rigid object, and in applying this force and making a deflection, energy is put in. Any rigid object will break when a certain amount of energy has been put in, unless it fails by some kind of plastic deformation or creep.
Some natural substances have a great capacity for absorbing energy. Examples are tendons, spider threads and leather.
Glass is very rigid, and it breaks very suddenly. But if glass fibres are embedded in another substance, the composite material can be very strong indeed. Many gliders are made in this way.
The problems of sagging and of breaking can be solved in several quite different ways.
One way to solve the sagging problem would be to build a beam which is curved in the opposite direction, so that it would sag to a straight line when placed in position. Prestressed concrete beams are in fact bowed slightly upwards, if only because of the tension in the wires.
But for a large sag such as those shown above, this solution would not work, because beams that flexible would sag further under live load. The effects of heavy vehicles travelling at speed would be unpleasant, if not dangerous.
Perhaps this is as good a point as any to say what engineers are trying to do with beams, and indeed many other structures. We have to remember that no material is infinitely rigid, so every beam sags. The job of the engineer is to choose the type of material, the quantity of material, and the disposition of the material, so as to achieve the required amount of rigidity. What we are attempting is for a given deflection of our beam, the required input force or energy is as large as we can make it. The optimal disposition of the material means that we need to place it where bending the beam will stress as much of the material as much as possible. Conversely, if there are places in a beam which are unaffected by the deflection, they do not need any material at all. This will be discussed in more detail further down the page.
We have seen that scaling structures to bigger sizes is not straightforward, and that a longer beam needs to be made stiffer than a shorter one. But we must be careful not to generalise this idea too much. If we scale the Severn suspension bridge down to a 10 metre span, we would have a very thin structure. Indeed, a rope bridge would do the trick. But who would want to use it to cross a river in a city, with bags of shopping or a push-chair? So the need for rigidity seems to work the other way in this instance.
Katsushika Hokusai made a picture showing a funicular - Famous Bridges of Various Provinces: The suspended bridge between Hida and Etchu. He shows clearly the discontinuity in slope at the position of each of the two people, but he has made a bigger change of slope for the person with the smaller load. He has also assumed zero mass for the bridge. Most pedestrians would be happier with something more rigid.
What is the curve of a sagging uniform beam? It is tempting to think that it might be related to the curves of suspension bridges. But we can easily see that this is not the case. If we consider a suspended cable, and we imagine a longer cable, and towers further apart, we can superimpose the central part of the new cable on the old cable.
We cannot do that with beams. At the two supports, the bending of a beam is zero. We can see that this is the case because there is nothing outside the support to force the beam into a particular shape. So making a long beam cannot replicate the shape exactly. The curve contains a parabolic part and a quartic part, giving a point of inflexion at the supports, where the bending moment is zero.
Already we see that the behaviour of a beam is quite complicated, be also see that some understanding of a system can be obtained from simple principles, just by considering the forces and the boundary conditions. Just because we can't work out everything, it doesn't mean that we can't work out anything. The curves below represent a circle, a catenary, a parabola, and a beam, all having the same curvature at the lowest point.
effect of curvature on length is shown in the layout of a running track, in
which staggered starts are needed to allow for the varying radii. If
different layers of a beam could slide along each other, creating a stagger, the
beam would be much less stiff.
This idea is used when a support needs to be strong but flexible, as in a set of leaf-springs supporting a truck. By suitable shaping of the individual springs, as in the upper diagram below, the stiffness can be made progressive. Sometimes a constant force is needed, whatever the position. A weight can provide this, with a rope and pulley to change the direction.
|The next two diagrams show the beneficial effect of doubling the thickness of the beam - the sag has been reduced to one eighth of the original value. But twice as much material has to be made, transported, and erected.|
|In fact by taking a flat beam and setting it on edge, the
same amount of material can be used much more effectively. The pictures
below show some examples in which ribs and cellular construction are
used. Arches too can benefit from ribbed construction.
By doing the opposite from making ribs, that is, making grooves, one can make something weaker. Why would anyone wish to do that? To make it easier to break, like this chocolate, which is weak in two directions.
This picture shows a kozolec, a rack for drying hay, in Slovenia. Sometimes one of the wooden beam twists until the ends are relatively displaced by a right angle. At the centre of the beam, the cross section is at 45 degrees to the horizontal. Does this make the beam less rigid or more rigid? Can you work this out by thinking about it, rather than doing some mathematics? This design is good engineering, because wood is plentiful in Slovenia, and the weather is conducive to drying, without consumption of non-renewable energy. And little land is required. Would you place this arrangement along the direction of the prevailing wind or at right angles to it?
Here are two orientations of a square beam. How do we work out which is the more stiff? We have to work out the average of the square of the distance of every part from the neutral axis. This is a simple bit of integral calculus. Can we do it without calculus?
This picture shows the stress at each point by the colour density, using red for compression and blue for tension. The colours emphasise the fact that the usual orientation puts more material where it is more useful, and less where it is less useful, while the other way does the opposite, though it puts a small amount of material further from the axis than the usual beam can do. From the diagrams we cannot tell which is the better beam. By calculation we find that both cases have the same stiffness. Do you think that some other orientation would give a higher stiffness or a lower stiffness? It would have the disadvantage of asymmetry. Would a load in the middle make it sag vertically or in some other direction?
Imagine the beams divided into horizontal strips. The next diagram shows the effects of the strips on stiffness, and the way they add up, which is roughly equivalent to the integrals.
Does this result make sense? Let's look at other regular polygons. The simplest case is a triangle. Rotating it to replace vertices by sides makes the same shape as turning it upside down, which cannot affect the stiffness. Another simple case is a polygon with so many sides that they cannot make any measurable difference, or in the extreme limit, a rod of circular cross section. Again, the orientation cannot matter. So can we generalize to say that the orientation of a horizontal regular polygonal beam does not matter? Only calculation can decide, but calculation alone would not persuade someone to think of extending an idea beyond the original application. That is where imagination comes in. It turns out on calculation that the stiffness of a beam of regular polygonal cross-section does not depend on the orientation.
Engineering, mathematics and science require imagination and creativity, and aesthetic judgment as well. Some mathematical conjectures are so difficult to prove that many years can elapse before a proof is found, and in some cases they are so alluring and profound in their consequences that mathematicians sometimes publish proofs that are true "if the WXYZ conjecture is true". The Riemann Hypothesis awaits a proof. The Taniyama-Shimura conjecture (in itself a very deep idea) was proved by Andrew Wiles and Richard Taylor. By this time it had long been known that such a proof would imply that Fermat's Last Theorem was also true, and proven.
We got off the subject again. We were looking at beams. But these paragraphs were not completely irrelevant. If you can find general rules or formulae, you don't have to work everything out from first principles - you can set out rules and formulas in codes of practice and use those. The danger comes when you forget the original restrictions and assumptions under which the rule was proved. Severe consequences have sometimes occurred after unproven extrapolations into unknown territory.
In fact, you can even get into trouble by doing what you've done before. You could cross the road without looking at the traffic, and each success might be taken as more evidence that this is a good procedure. You could even walk a little slower each time, and again take your success as proof of reliability. In both cases, you would be repeating something - in the first, repeating the crossing with the same speed - in the second, repeating the crossing with the same reduction of speed. This is a an absurd example, but consider building many bridges with the same thickness, and building many bridges with the same reduction of thickness from the previous one. Other extrapolations have concerned using things at lower and lower temperatures, or higher and higher altitudes, speeds, pressures, etc.
To make a beam more stiff, instead of merely using more material to deepen a beam, it is better to split the existing material into several parallel beams which are deep but narrow; this is a better use of the material. Later in this page you can find more explanations about stresses in beams.
The same principles apply to small bridges. The large planks and the small planks both have to carry the weight of a person, but the short ones can be much thinner because they are so short. The two diagrams below illustrate the effectiveness of two beams with the same cross-sectional area. The horizontal value at each position shows the contribution to stiffness of the material at that point. We can see how the effectiveness of the material increases rapidly with the distance from the centre line. The stiffness is proportional to the area of the graphs, and is given by the number. The tall beam is four times as stiff as the square one. We cannot, of course, take this procedure too far, or the tall narrow beam could bend sideways. Later in these pages we will see the remedy for that.
The third set of bridges, below, has intermediate supports. This solution can be extremely difficult and expensive when there is deep water, unsuitable rock, a requirement for navigation, or simply something in the way.
The supports can be used much more effectively if the beam is continuous across them instead of being made in separate short sections. Even better, the variation of stress along the beam can be partially evened out by pre-bending the beam at the supports during construction.
A multi-span bridge can be made with all the spans completely separate, which is convenient for lifting them into place, or they can be joined together, usually after placement. The pictures at left, one compressed, shows the fences of a gallop. The fence in the foreground has come off the two supports at the left. If the sections had been independent, there would be a sharp dip at the third support. In fact, because the fence is continuous, the beam has been lifted off the fourth support, and the effects of the damage go far beyond the two left hand supports. Although this is an example of something that has been damaged, it shows clearly how loads can be spread through a structure by judicious connections.
In the Britannia bridge over the Menai Straits, Stephenson had the several beams joined in such a way as to even out the bending moments. This was an early form of prestressing, as well as being the forerunner of the box girder bridge.
Click here to see how forces at the supports can control the forces in a beam.