Click here to hear the delightful melody.
Who wrote the music? Who wrote the words? Where was the funicular?
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Not many pieces of music are concerned with engineering matters. There is, of course, Honegger's "Pacific 231", which brilliantly conveys the power of a great steam locomotive. Another example is "Pavaroz", by the Kyrgyz composer Tumanov, telling a similar story using only two instruments - a chopo cho'or and a comuz. There are some songs which refer to bridges, such as "Sur le pont d'Avignon", "Under the bridges of Paris with you", "Underneath the arches", and "Bridge over troubled water", but the emphasis of these is not really on engineering. The same goes for "Bridge over the river Kwai", "A bridge too far", and "The bridges of Madison County." Some poems by Rudyard Kipling, however, show appreciation of the forces in structures, and in one poem he says that if we ignore the laws of physics, we, or someone else, will die.
The funicular that is discussed in this page is not the one that the song is about, which was a newly built funicular railway. The words were written by Luigi Denza: the words by Peppino Turco, to celebrate the construction of the first funicular railway up Mount Vesuvius in 1880. Mount Vesuvius being an active volcano, it could be expected to affect the railway, and it did, in 1906. Several replacements have been built since then, as the result of eruptions. Only that of 1953 to 1984, a chairlift, is closely related to this page.
The word funicular is derived from the Latin word funiculus, a thin rope or cord, which is related to funis, a rope. From this we get the 18th century word funambulist, a tight-rope walker. A funicular railway is always on a steep slope, the rolling stock often being pulled by a steel cable, driven by a stationary engine, though some mountain railways use a rack-and-pinion drive, using a motor on the train.
The title of this page refers to a topic which is extremely important in engineering, namely the funicular. In a great many cases where the concept is useful, no cable is involved, as we shall shortly see.
What is a funicular? Consider a flexible cable or chain, hanging between two points. The forces in the cable run right through the middle, because it has no stiffness. The curve it follows is a funicular, in this example a catenary: the word is derived from the Latin word catena, a chain. The funicular, in fact, may be thought of as the average path of the forces through a cross-section of a structure, though only for the purposes of certain calculations. In this it is rather analogous to the centre of gravity (CG), which may be thought of as a kind of average point where the weight of an object can be considered to act, though only for the purposes of certain calculations.
This should in principle be distinguished from the centre of mass (CM), but in a uniform gravitational field the CG and the CM are coincident. Real gravitational fields are non-uniform, and the result is tidal forces. Other such ideas are centre of lift and metacentre. There are many such abstract ideas in mathematics, science and engineering. Their value is that they make explanations very much shorter. The danger in their use arises when we become so familiar with them that we forget the limitations that were used in defining them, and use them where they are inapplicable.
If we make a uniform arch, it should also follow a funicular if we want it to stay up. If we make a rigid beam, and if we give it a shape which differs from the funicular, it will experience bending moments, in other words, competing internal forces, which it will have to be stiff enough to withstand. The word funicular is derived from the Latin word, funiculus, a small rope: funis meant a larger rope.
Some mountain railways are referred to as funicular railways because the tracks are inclined. The word is then being used in a different sense from the way it is used in this page, where it refers to the curve of a cable or and arch in free space. A flexible cable supporting a set of passenger cars does follow funicular curves, by definition. Here is an example assuming a weightless cable.
Note also that the word catenary is used in this page to denote a particular mathematical curve, whereas some people use it more generally for any suspended system, an example being the overhead wires of an electric railway.
diagram shows an arch based on a catenary, the red curve.
To generate catenary arches with different ratios of height to span, click here to download program Brancat, which makes pictures like this. By pressing the PrintScreen key in this program, you can copy the pictures into the clipboard to use it as the basis of a model. You can also download Branpar, which draws parabolic arches, and Brancirc, which draws circular ones.
The vault of the banqueting hall at Ctesiphon, Iraq, looks fairly close to a catenary. The walls are thickened towards the ground, to make sure that the funicular remains inside them. This is shown clearly in Figure 20 of "The Story of Architecture" by Patrick Nuttgens, Phaidon, ISBN 0-7148-3616-8. Click here for a picture which does not show the side walls quite so well. Built in about 550 AD, this building survived until 1987, when most of it was destroyed by a flood.
Click here to skip some maths.
so special about the catenary? We are told that it's the curve
that a hanging flexible cable adopts. But that doesn't tell us
anything at all. All we have is a sentence that is equivalent to
"catenary". It's like saying "Roald Amundsen was
the first man at the South Pole.": "the first man at the South
Pole" and "Roald Amundsen" are merely synonyms, if that
is the only fact we know about Amundsen. The usefulness of knowledge
must therefore reside in the relationships among a number of facts. Is
this really the case?
We could look at the equation, y = cosh(x), which isn't very helpful, even if we write it as y = 0.5 * (exp(x) + exp(-x)). This is the wrong kind of information if you are not mathematically minded. What really matters from the engineering point of view is that any part of the cable, big or small, is in equilibrium, because the three forces on it are exactly balanced. These three forces are the forces pulling the ends and the weight pulling down.
If you don't like maths, you can skip the next section, but then you won't find out so much about the catenary.
To see something more physical, here is a diagram, showing a parabola and a catenary. The hanging cables are assumed to be weightless, and pulled into shape by a series of heavy vertical rods, which are evenly spaced. For clarity, only a half of the rods are shown in each case.
To make a parabola, we make all the rods equal, so that there is equal loading per unit horizontal distance. For the catenary, the rods all line up at the bottom. Strangely, enough, this is related to the reason that the catenary is the natural curve for a cable. The load at each point of a cable is related to the slope of the cable and to its curvature. Mathematically, the load at each point is related to the second derivative of the curve. For the cosh function, the second derivative is equal to the function itself, which is why the rods line up.
This can be written mathematical notation like this -
y = 0.5 * (ex + e-x) = cosh x
dy/dx = 0.5 * (ex - e-x) = sinh x = slope of cosh x
d2y/dx2 = 0.5 * (ex + e-x) = cosh x = y = slope of sinh x
However, the rods would line up for an exponential curve also, and that curve is not the natural curve of a cable.
We must remember that the curve of a cable is a catenary only if the cable has no stiffness and is uniform throughout its length. Many formulas in maths, physics and engineering are correct only over a range of parameter values or circumstances: using them outside the applicable range will cause errors - often large errors. These pictures of cables show the effects of finite flexibility, especially near the supports, where the curvature is great. "Flexibility", like many others, is a term that requires some knowledge of the system to be useful. A mile of suspension bridge cable can fairly be called flexible, but a piece of the same cable two metres long would be quite stiff. The relevant parameters are the length, diameter, density and Young's modulus of the material, and the curvature of the shape when hung. Similarly, a column or a strut may be considered stiff or not, depending on whether it is "stocky" or "slender", terms which depend on the dimensions and the material.
You may object that arches are stiff, and yet they may be funicular. Yes indeed, but they have to be stiff to achieve stability. Such arches are built in the required shape. Were a "cable" to be made of stiff material in exactly the right shape and hung up, it would, like an arch, experience no bending moment. The flexible cable experiences no bending moment simply because it cannot sustain any - it simply moves until the bending moment is zero throughout.
The next diagram compares a catenary and circle.
Here, the loading needed to create a circle increases rapidly with distance from the centre, and the lines go off the bottom of the graph. In fact, for a semicircle, the loading would need to be infinite, because no finite vertical load can make the funicular vertical. Turning this upside down, why are semicircular arches apparently so good, some having survived since the time of ancient Rome? The reason is that the stone arch includes much other masonry besides the semicircular array of voussoirs. This masonry changes the weight distribution and the force distribution.
The next diagram shows the slopes, derivatives, or differential coefficients, graphically. On the right we see part of a circle, in red, Its slope in green, and the slope of the slope in blue. On the left the catenary and its slope are superimposed, making a purple curve.
Finally, we compare the catenary with the parabola, using the same colours as for the circle.
Having shown that the second derivative is related to the distribution of weight, we can now show that the catenary is the natural curve of a cable. The next diagram shows a minute portion of a cable.
By Pythagoras' theorem, ds2 = dx2 + dy2.
But dy = dx (dy/dx),
so ds2 = dx2 + dy2 = dx2 (1 + (dy/dx)2).
The weight of the section ds is proportional to its length, and from the earlier calculation we want the weight to be proportional to the second differential coefficient. That means -
ds/dx = (1 + (dy/dx)2)0.5 = d2y/dx2.
For a catenary, dy/dx = sinh x, and d2y/dx2 = cosh x.
The next three pictures show the funiculars in red for three shapes of arches built from voussoirs.
Note that the three funiculars seem to differ less than the actual arches. The funicular for the catenary is not exactly central in the diagram because of the crude method of iterating in the calculation. The parabola errs away from the crown in having too little curvature. The semicircular arch diverges in the opposite direction, and is unstable because the funicular goes outside it. Normally, of course, masonry would be present to hold the voussoirs in place. If you look at masonry arches, you will always find that even if the spandrels are partly open, the section nearest the springing is always filled with masonry.
The behaviour of a solid masonry arch is not easy to understand. We could, for example, treat the masonry as isolated columns exerting only vertical forces on the voussoirs. This is unrealistic. Equally unrealistic is to assume that the masonry is unbonded and frictionless and behaving like a liquid, pressing radially on the voussoirs without friction.
If we treat the masonry as perfectly bonded, it is then so rigid that it could stand alone without the voussoirs, as a corbelled arch. It might do that if the mortar were strong enough, but in practice, the tension and shear would break some of the mortar. We can therefore regard the voussoirs as changing the stresses in the masonry in such a way as to keep them below their critical values. And even if these values are exceeded and cracks appear, the voussoirs keep everything in place. The voussoirs are, in a sense, acting like the hull of a submarine holding the pressure of the water.
That this discussion is not purely academic is shown by Maillart's discovery of cracks in a concrete arch. Rather than strengthen the walls, he removed large areas, realising that the cracked concrete was not doing anything useful. And so began the series of elegant designs that keep the name of Maillart in our minds.
The next diagram shows an arch which is just stable, with the funicular only just contained within it.
Note that the funicular is not an actual physical entity. The forces are not concentrated along it.
The next picture shows a gateway which is so far from funicular that the two slabs are definitely acting as beams. They would be doing so, even without the masonry above them. The builder seems to have included a keystone.
So far, we have looked only at arches with no load. To see what happens when a load is moved across an arch, click here to download a simulation.
Earlier, we mentioned an over-simplified way of treating the mass of masonry above an arch - treating it as a set of unconnected columns. The resulting funicular depends on the depth of masonry above the voussoirs at the crown of the arch. The diagram below shows an example.
This is all very well, but we are only drawing curves. The horizontal thrust that permeates any arch has not been reduced. It cannot be reduced by any amount of weight, which can only act vertically. So the bridge still needs abutments to contain the horizontal forces. The conjoined pairs of arches in the picture hint nicely at the leakage of the line of thrust from the voussoirs.
The Romans knew this, and they made their piers so wide that the thrust reached the ground before reaching the edge of any pier. Here is an example, where the next arch will be added on the right.
That meant that they could build arches one by one, and it also meant that arches could be destroyed by flood or enemy. The next picture shows an example with three arches.
The Romans never realised that very narrow piers could be employed if all the arches were built at once, or if they did realise it, they were not interested. Of course, the piers cannot be reduced to very thin slabs, because the effects of live loads have to be contained within them, and these are not balanced like the self-weights. In a sense, the piers are like the neutral wire of a three-phase AC electrical supply, which only has to carry unbalanced currents, and is much thinner than the phase wires. The examples below show how the horizontal component of the thrust is carried right across the bridge, while the vertical components are taken down the piers. You can make up your own multi-arch bridge with any number of spans using these three pictures - Left - Inner - Right, or here to see all three together.
Two significant developments are associated with Jean Perronet, a great engineer of the 18th century. One of these was the segmental arches, which used only a part of a semicircle: the other was the greatly increased ratio of span to pier width. You could argue that until this happened, the potential of the masonry arch, in spite of the passage of many centuries, had never been realised. A huge advantage of the segmental arch is the small ratio of height to span, which allowed the use of much shallower and shorter approaches. The navigable fraction of the span is also greatly increased. The picture with this paragraph shows how tall narrow piers and segmental arches give good clearance for a railway loading gauge.