Arches - Part Two
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Continued from Arches One - Arches Three - Arches Four
The diagrams above represent two concrete bridges, one flatter than the other. The lines at the right abutment represent the force provided by it. The vertical line represents the weight, and the horizontal line represents the thrust. The flatter the arch, the greater the horizontal thrust.
A simple way to see that there must be outward thrust is to imagine a simple arch of two loosely hinged rods, standing on ice. Obviously it will collapse, because the ice cannot oppose the outward thrust.You can try this by standing with your legs wide apart on an ice rink. On second thoughts, don't try it. When you put a ladder against a wall, the wall provides a force as if there were a second ladder leaning against the first, like a tall narrow arch. The ladder must not be too far from the vertical, otherwise friction at the ground cannot balance the outward thrust. If someone stands on the bottom of the ladder, this adds weight, and increases the available friction.
If we look at the two pictures above we can see why a circular set of voussoirs cannot be the correct shape. We assume for this purpose that the joints between the voussoirs cannot sustain a shear force.
In the upper diagram of a semi-circle, the thrust at the bottom can only be vertical, and so it cannot counteract the outward thrust of the upper voussoirs.
In the lower picture the situation is even worse. The thrust into the bottom of the ring is outwards, when it needs to be inwards. Finally, as a set of voussoirs must be like an inverted cable, it is immediately obvious that these shapes cannot work.
The inverse example of a hanging cable works in the same way. The two ends of a hanging cable are never vertical, unless they are at the same place. Perhaps the purest analogue of the cable is the Gateway Arch in St Louis, which is 192 metres wide and 192 metres high. From the point of view of resisting transverse wind forces, this arch is a huge cantilever.
In a real masonry arch, the volume above the voussoirs is generally filled with massive material, which changes the line of thrust to a steeper line. This material also help to rigidise the bridge.
As soon as we introduce masonry into the spandrels, the volume between the voussoirs and the roadway, all the considerations so far discussed needed to be rethought. The masonry presses in the voussoirs in a way that is not easy to calculate. We could imagine masonry consisting of separate columns of bricks pressing vertically on the voussoirs. At the other extreme we could imagine liquid contained between walls. The liquid would exert pressure radially on the voussoirs. Rubble might approximate to this condition, while normal masonry would behave in an intermediate manner between liquid and solid columns. The diagrams below illustrate these three possibilities.
The fact is that when a masonry arch covers an arc that is a large fraction of a semicircle, we must abandon the idea of an inverted suspension bridge and funicular, and accept that horizontal thrust enters not only at the springing, but higher as well. We should not always be too influenced by what we see on the outside. For elliptical arches in particular, the masonry inside the arch may be distributed in such a way as to direct the forces in the optimal directions, as in the idealised example below.
These pictures show a tunnel under a wide road embankment. The tunnel is lined with corrugated metal. The effective thickness can be no greater than the depth of the corrugations, but the tube has a rigidity that is denied to a voussoir arch because of its continuity. Soil is not a fluid, but neither is it an elastic solid, and soil mechanics is complicated. For our purposes it can be assumed that the soil presses inwards on the tube all round. Tunnels are very often in the form of complete circular tubes, of which we see only the parts above the flat road or railway. The remainder may be used for services or drainage.
Here are diagrams of a barrel with wooden staves compressed together by metal hoops in tension. Wooden cartwheels are made in the same way. Compression is represented by red: tension by blue.
The builder of this magnificent barn evidently did not believe that the line of thrust could remain in an arch which was almost a semicircle, even with a heavy wall above it, for he has provided a buttress that pushes well above the arch. At any rate, if he did believe it at the start, he was forced to think again, and added the buttresses afterwards. The lines in the second picture hint at the reason for the building of the buttresses. See buttresses for more on this subject.
Here is another Cotswold building, with the usual roof of Cotswold limestone tiles. This stone does not split neatly into thin layers like slate, and these roofs are very heavy. This barn has substantial buttresses to resist the thrust of the roof, which can be thought of as a three pinned arch resting on the walls.
This barn was definitely built in complete defiance of the funicular. Before the two short and almost vertical struts were added (far too late) the roof acted as a rather flat four-hinged arch. Their was no hope of sustaining the enormous weight of the standard Cotswold stone roof over a long period. The distress of the walls is obvious: they are failing in shear and by leaning out.
Building in local materials, in this case wood and stone, is advantageous in terms of transport costs and long practical experience, not to mention aesthetics (something that would not have been considered at the time of building). Here, however, the poor technique has nullified those advantages completely. Given the flatness of the roof timbers, no practical wall thickness would have sufficed: the only solution would have been to incorporate buttresses from the start. The rear part of the building also shows a severe crack, presumably also the result of spreading.
These three pictures show an arch which is bowing outwards. In the third picture, two cracks are indicated by red arrows, and a surveying target by a yellow arrow. Evidently this arch is causing concern. The buttress looks very substantial, and there is little weight on the arch, so perhaps the foundation is not adequate.
Another arch nearby is in an equally bad way. This arch is actually double: someone in the distant past has built a new arch against the older one, with complete disregard for style. But we must remember that old abbeys were life support systems, not works of art. They were perhaps like space stations, supporting a small number of people with some supplies of food from the outside world, though they had their own gardens as well.
The bridge shown below is a propped beam, but it could be looked at as a three segment arch, and indeed the sagging span and the reflections in the water show that the thrust has slowly pushed the piers outwards.
The diagrams below this panel suggest how an arch can behave as a five-pinned structure, which is unstable. Two suitably placed and well-founded buttresses can reduce this to a stable three-pinned arch. Click on the diagram to see a picture of a foam plastic model. This material exaggerates the strains to make them more obvious. In a real structural material they are so small that strain gauges and amplifiers are usually needed to measure them. Measuring stress within a material is usually very difficult, so it has to be inferred from strain.
This row of arches illustrates very nicely some of the problems with arches. At the two ends, the thrust has pushed the piers outwards, and on the left, two incipient hinges have formed. The other piers remain upright, because the forces are more or less balanced.
One way to avoid problems like this is to provide heavy buttresses at the ends. Another is to have no ends. How can we have no ends? The Colosseum in Rome has no ends, because it curves round and joins up. That is not very useful for a bridge, of course, and it raises a couple of other questions. If a viaduct of arches is curved, the thrust of each arch is not parallel to the arches on either side, and so the balancing is not perfect. The piers must be wide enough to take the asymmetric load, or you might consider batter on the outside faces. The other consideration in a curved bridge is the effect of fast moving heavy loads. In order to accelerate these loads inwards, the bridge must generate forces. The reaction on the bridge is to push it outwards. Do you think that these forces would be significant in a real bridge? For a mass M, moving with a speed V on a circle of radius R, the outward force is MV2/R. Try some suitable values for a heavy vehicle or locomotive. Even on a straight bridge, cars may swerve quite sharply when overtaking.
We got off the subject there, as a result of thinking about the row of arches. But that is a good way to learn: given a structure, extrapolate it or change it in some way, and think about the consequences.
To look at a possible consequence of the formation of hinges, which is where we were before the deviation, we go back many years to the invention of an early electronic device, the triode valve or tube. The curve below shows roughly how the current might vary with the potential difference across the valve.
A great problem with the triode was the capacitance between the anode (output) and the grid (input), which had a severe effect on the gain at high frequencies. A solution was found by the use of a second grid, between the first grid and the anode, to act as an electrical shield. This solution to the capacitance problem introduced a new problem, shown by the curve below.
There is a region in which the curve slopes downwards instead of upwards. The consequence of this "negative resistance" is that the device can oscillate, which may or may not be desirable. The first solution to this problem was to insert a third grid. A second solution was the "beam grid" tetrode. This type of curve is found in some more modern electronic devices such as Gunn diodes, IMPATT diodes, and tunnel diodes. In all cases they can be used in oscillator circuits.
Strangely enough, such a curve can arise in the case of an arch which does not fit together correctly, and the arch, too, can oscillate.
can look at this a little more mathematically. The diagram at
right shows three forces in equilibrium, acting at a point. This
is possible only if the vectors can be joined up to form a
triangle. This is only a calculational tool. Three forces
acting like that would actually tend to rotate an object, although it
would not be translated to another position.
An example would be the forces at some piece an arch. We have the thrusts from the rest of the arch, and the weight of the piece. These must balance at all points, unless the structure has enough stiffness to cope with the induce bending moment.
The picture shows the changes of angle where larger dew-drops hang on a thread made by a spider. At every point where a drop hangs, the tensions in the visible thread and the invisible weight of the drop are in balance, leading to a change in angle.
the rule given above, we can deduce the transverse force on a curved structure such
as an arch or a cable sustaining a force F, and having a radius of
curvature R. If the transverse force per unit length is f, then
f = F / R
As an arch becomes steeper towards the ends, the transverse component of the weight becomes less and less, and is zero if the arch becomes vertical. If f is decreasing, either R must increase, that is the curve becomes straighter, or F must increase, in other words the arch must become thicker.
For a masonry arch, a part of f is provided by the masonry between the voussoirs and the deck, which provide both weight and sideways force.
In medieval cathedrals, buttresses, often spectacular flying buttresses, were deployed in order to control the forces generated by large and heavy roofs and vaults. The piers of the buttresses were often higher than the buttresses themselves in order to add weight, thus getting the forces into the right alignments. The towers at the left are provided for this purpose.
The transverse force is equivalent to the pressure on an airliner fuselage. A three-dimensional analogue is the excess pressure inside a water-drop.
In the picture below, we can see the way the curvature of a spider web is related to the weight per unit length.
Notice, in the buttresses above, that the builders did not just build the arch part - they, like almost everyone else, used the arch to support a straight row of blocks. Are these blocks there to add weight, helping to steepen the line of thrust, or do they take some thrust themselves?
In practice, the actual forces might well be distributed throughout the buttress, but these structures illustrate the idea that the line of thrust does not necessarily remain within an arch which is a circular arc. If the thrust lies within the straight row of blocks, the buttress has merely pushed the problem further out, forcing the pier to do what the wall of the cathedral cannot do, that is, resist the transverse force. To find out exactly what goes on, we would need to calculate everything from the dimensions of the structure. In practice, the builders probably built up empirical knowledge from years of experience.
The upper line of blocks in a flying buttress looks a bit like a sloping beam. But what is a beam? It is usually a more-or-less horizontal, more-or-less straight object that rests on supports and does not take any thrust. And it is an object that does not fall apart when moved. So these blocks do not form a beam. What is a pier? A more-or-less vertical rigid object that takes thrust. And what about an arch? A more-or-less curved thing that takes thrust. Our straight line of blocks illustrates the fact that you can always find things that cannot be placed into simple categories. Perhaps we should call it a sloping pier, prevented from bending by the arch below.
We cannot manage without categories: the time taken to analyze everything from first principles would make life impossible. The use of categories appears to be almost instinctive, but when we categorise people or things wrongly, or we fail to take account of the vagueness of the boundaries, we can create problems. Ancient people correctly placed the planets and the stars into two distinct categories, based on apparent movement, but few realised that the earth was a moving planet, that the sun was a star, that neither was special or at the centre of the universe.
In England, but not in France, people have separate words for butterflies and moths, and jam and marmalade. In both countries there are separate words for frog and toad, but are these really biological categories above the species level? In some species of fish, individuals can even change sex. The simplest example of the problem in biology expresses itself in the distinction between "lumpers" and splitters". No doubt there is a vague boundary between these categories as well. The supreme example of splitting is to deny that evolution took place, and to deny that species are connected at all.
In engineering, these ideas may not matter as much as they do in other fields, because the important thing is to find out what works. Nevertheless, generalisation is a powerful tool, which enables the same calculational techniques to be used again and again. But if we extrapolate further than the regime in which a technique has been tested, we can find problems. For example, a short stout pillar works well, and we can build taller and taller examples, until we suddenly find that buckling occurs, at a width-to-height ratio calculated by Euler. An important part of a code of practice is its range of validity.
Enough of this uncertainty: here is a structure that is definitely not an arch. It comprises a set of planks, some of which act as beams, others partly as cantilevers as well. If there is no friction between the planks, there can be no horizontal force. The rule is very simple - no thrust: no arch. In a real construction, there would be friction, and some lateral forces would be created. But in the frictionless case, what carries the weight of the unsupported parts? See beams and cantilevers.
You could probably say that this is an arch. At any rate, it generates lateral thrust. Like a gothic arch, its apex does not correspond to extra load: quite the reverse in fact. As we saw earlier, the curvature of an arch or a cable is closely related to the load that pushes or pulls it. So a sharp corner ought to be resisting a localised force.