The Middle Third
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| Stand
upright. Get someone to push your back, with slowly increasing
force. What happens down at your feet?
Most artificial structures rest on the ground, and others, such as aircraft, that fly, do so for some of the time. Structures on the ground are generally required not to move around or fall over, and even vehicle are generally required not to turn over, except in films, where special techniques are used to make this happen. Sitting in an aircraft on the ground in a high wind, you can 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. Let’s look at a simple object, a wall, which is being pushed, or blown by the wind. In the diagrams below, these forces are represented by arrows, though of course the force of the wind would be spread out. |
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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 increase in lateral force would create tension, both in the wall and at the connection. In the case of a 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 it is at one third of the width 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. If you cannot control 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. Its 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. Effectively, a wide 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 ninth in area, as in the diagrams below. Looking at these diagrams tends to increase respect for builders of older times, who had to discover this kind of thing wthout the maths or science that are available now. But then, we should not underestimate the abilities of the people of other times and other places. 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. Although the compression decreases rapidly away from the block, some effect is visible throughout the picture. |