The Middle Third
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upright. Get someone to push your back, with slowly increasing
force. What happens down at your feet?
Most artificial structures rest on the ground all the time, and others, such as aircraft, that fly, do so for some of the time. Only ships and boats are immune from this requirement, and even they have to withstand the pressure of the water. In fact, because of the waves and the winds, the surface of the sea can be an exceptionally severe environment. Some people refer to the Atlantic Ocean as "the pond", which is easy to say if someone else is flying them at 30,000 feet above it. Were they to try sailing on it they would learn to show respect for it.
Structures on the ground, apart from vehicles, are generally required not to move around or fall over, and even vehicles are generally required not to turn over, except in films, where special techniques are used to make this happen in a predictable way. Sitting in an aircraft on the ground in a high wind, you can often feel it moving. If you have to park a glider in a high wind, you place it almost at right angles to the wind, with a slight tail wind component, and you place a weight, such as your parachute, on the windward wing-tip. Buildings, too, move as a result of winds, and may be damaged if the wind is strong enough. Another cause of damage occurs if the earth moves and the building stays where it is, as in an earthquake. This is a simplification: in practice, the building does move, and the problems occur when parts of it move differently from other parts. During a very small tremor, you may see all the leaves of a plant oscillating in phase for a short time, because the pot is being shaken along with the house. This can sometimes be observed even when the movement of the house is not felt by the occupants.
Let's look at a simple object, a wall, which is being pushed by the wind. In the diagrams below, these forces are represented by single arrows, though of course the force of a real wind would be spread out over the height of the wall.
The colour red has been used for the compressive force in the wall, which increases from top to bottom. In addition, the distribution is modified by the effect of the horizontal force. The little graphs at the bottom show the distribution of pressure on the ground. In the right hand example, the pressure just reaches zero at the right hand side. Any further increase in lateral force would create tension, both in the wall and at the connection. In the case of a gravity dam this could be very dangerous, as water at high pressure would creep under the dam and create uplift, thereby reducing the effective weight still further. No attempt has been made to show the forces in the ground.
The average position of the downward thrust is easy to calculate. In the case of the right angled triangle at the right its distance from one wall is one third of the thickness of the wall. And so we are led to a very simple rule - If you don't want tension, keep the average line of thrust inside the middle third of the section. This average line is rather like the centre of gravity: nobody imagines that all the force is actually concentrated there: it is simply a useful shorthand. So we could speak of the centre of force or centre of pressure or the centre of thrust, though these are not official names.
If you cannot control size of the lateral force, you can reduce its effect by making the structure heavier. Medieval builders did this by adding spires to their flying buttresses, and by adding decorative walls to the top of their main cathedral walls. Thus, all other things being equal, a heavy wall is harder to blow over than a light one.
It is not always convenient or desirable to increase the weight. Here is an alternative, at the base of a tall lamp post.
Another method is to continue the structure deep into the ground. In fact the foundation of the post shown above must do just that. Making the foundation wide and heavy helps as well. A wide or heavy base increases the moment available to counteract the moment caused by the lateral force.
The effect of the middle third is much stronger for a pillar than for a wall, because it acts in both directions. The middle third in length becomes the middle eighteenth in area, as in the diagram of the square below. For the circle, the kern, or safe area for the eccentricity of the forces, covers only one quarter of the diameter and one sixteenth of the area. Looking at these diagrams tends to increase respect for builders of older times, who had to discover this kind of thing without the mathematics or the science that are available now. But then, we should never underestimate the abilities of the people of other times and other places, though we sometimes do.
To experience this for yourself, stand upright, and get someone to push you carefully with a slowly increasing force, from front, back or side. Pushed from the back, you will eventually feel your heels lifitng off, and in fact you will probably do it instinctively to get the line of thrust through the balls of your feet. If pushed from the side, you may find yourself moving your feet apart in order to increase the area of the base, and therefore of the kern.
After looking at the previous diagram, can you guess the shape and size of the kern for a triangle, a pentagon and a hexagon? The general rule for a polygon must of course be such that with an infinite number of sides, the result is the formula for a circular kern. Given that the area of the kern of a circle is bigger than the area of the kern of a square that has the same area as the circle, you might guess that the kern of an equilateral triangle is smaller than the kern of a square, but this is not so. The triangular kern has one sixteenth (1/16) (0.03125) the area of its base, while that of the square has one eighteenth (1/18) (0.05556). The circular kern is the largest at one sixteenth (1/16) (0.0625) of the base area. Can you guess the general formula for a regular polygon of N sides? Perhaps there is no formula.
The graph below shows the ratio of base area to kern area for regular polygons with three to fifteen sides. The kern polygons are parallel to the base polygons for an odd number of sides, but for even numbers of sides the kerns are orientated so that the vertices of the kerns point to the mid points of the sides of the base.
Here are some cross sections showing the kerns. The area through which the centre of thrust has to pass is quite small, though all may not be lost if the thrust falls a little outside, because there are situations in which tension is allowed to occur, even in reinforced concrete and pre-stressed concrete.
The next four pictures show a very light slab of very rigid foam plastic which was subjected to a vertical force at four places - in the centre, at the edge of the kern, and well outside the kern. The base was a piece of rather soft foam plastic, and you can see at the edge of the hard block the regions where there was no pressure.
Can we discover the area over which the there is no pressure on the base? Yes - the object behaves as if the centre of pressure is on the edge of the kern of an imaginary object, shaded pale blue in the diagrams below, in which the pressure point is shown by the red circle. The imaginary kern is shown in blue.
Even when the centre of pressure is within the kern of the object, it creates a new kern, and a new imaginary object which is bigger than the real one. The problems occur when the imaginary object is smaller than the real one. The pressure in the shaded area is not uniform: it falls linearly with the distance from the edge of the real object, becoming zero at the boundary between the shaded and unshaded areas. The area of the real object that exerts no pressure is the area outside the imaginary object; this is unshaded in the diagrams.
The graphs shown below give the widths of the kerns for square and circular bases which are hollow. The width and area of the kern vary as the width of the hole changes from 100% to 0% of the width of the column. The width and area are shown as fractions of the width and area of the columns, for the cases of square and circle. By width for a square kern we mean across a diagonal. We see that a hollow column is better than a solid one.
Thus, as in trying to maximize the stiffness of a beam, getting the material to the outside is beneficial, so a tubular mast not only resists bending well, but resists toppling well also.
Now that we have seen the kerns for several shapes of base, we must ask a question. It is a question that we must ask whenever we come across a technical rule. The question is this - "What is the range of applicability of this rule?"
We have considered bases and substrates that are horizontal planes. What if the object is supported by narrow legs, as in a camera tripod or an electrical pylon. The kern is now the entire area bounded by the supports, not a small area in the middle. So thee middle third rule is not applicable. Can you see why this is the case? The pictures below show a pylon and a viaduct with legs that are intermediate between solid ones and points.
The viaduct shown above is curved: this affects the traffic that uses it. The trains are moving in a curve, as opposed to a straight line. Therefore they are accelerating towards the centre of curvature of the track. The necessary force is supplied by the track, and so the reaction of the train on the track is not vertical, but sloping outwards. In order to keep the line of thrust in the safe area at the bottom of the piers, the trains must not exceed a certain speed limit, though in this example, the speed limit is probably set for the very large spans to which the viaduct leads. Pylons too, can experience lateral forces, usually from the wind, but at places where the route of the wires changes direction, the pylon has to be wider than the normal ones to cope with the extra lateral force.
Here is an attempt to give a very rough idea of the way that pressure spreads beneath a load on the ground. The bands are artefacts of the display: in reality the pressure would vary smoothly.
Do you think that the spreading depends on the nature of the ground? For example, in the extreme case of a floating ship, the pressure in the water near the ship is completely unaffected by the presence of the ship. During an earthquake, poor ground may behave like a liquid, and lose almost all its rigidity. Suppose that you could build a very strong boat-shaped foundation under a house, and you built it in a sand filled pit. Would it survive an earthquake?
If the pressure under a heavy object varies somewhat as shown, does this mean that a wall built on soil would be easier to push over than one built on concrete? Given the apparent reduction in pressure, would a wall tend to sink into soil? Well - think about a footprint on soil. Why can you often walk on beach and barely leave a footprint? Why does treading on the beach seem to make the sand dry out around your foot?
The next picture shows a piece of foam that has been pushed by a block. You can see compression, shear and tension. A real substrate would experience much greater stress from a building, but would exhibit much lower strain.
Although the compression decreases rapidly away from the block, some effect is visible throughout the picture.
Here are some diagrams showing the behaviour of an arch with increasing loads. In the last one the limit has been reached. Any further loading will place a part of the soffit in tension.
For a simulation of the behaviour of an arch with a variable moving load, click here.
The discussion in this page has concentrated on compressive forces that are found in arches, columns and struts. In all these cases, it is desirable that the forces not be eccentric, that is, off axis. In the absence of sufficient stiffness in the structure, the system becomes unstable, because any bend exacerbates the eccentricity. This is in contrast with a tie or a cable, which tends to straighten under increasing load. The picture below shows one of the four columns in Salisbury cathedral that support the tower and spire.
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