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Funiculi  Funicula

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.  There are some songs which refer to bridges, such as "Sur le pont d'Avignon", "Under the bridges of Paris with you" and "Bridge over troubled water", but the emphasis of these is not really on engineering matters.

But the title of this page refers to a topic which is extremely important in engineering - the funicular.  

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 case a catenary.  If we make a uniform arch, it should also follow a funicular if we want it to stay up.  If we make a uniform 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.

It is perhaps in the design of foot-bridges that the engineer has most freedom, for in general he or she can include gradients (or even steps) that would be unacceptable for wheeled traffic.  The greatest challenge here is found when a foot-bridge has to cross a very wide road or river in flat ground.  

The pictures below show some attempts to solve the problem, along with a parabola and a catenary to give some idea of a funicular.  Near Patchway, Bristol, the designer of a footbridge made a brave attempt to remain close to the funicular.  The resulting curve looks like a composite of several curves, and appears a little uneasy.  The page on foot-bridges gives many more examples.  Click on the thumbnails to see larger pictures.

Greater departure from the funicular does not necessarily mean inferior design.  In the top left example, on the M6 motorway, the design quite honestly reveals that it is a three-pin arch, with steps at each side.  It is an elegant solution to the problem.  Whatever else a motorway bridge has to do, it must clear the height specification over the whole width of the road, including any hard shoulders.  The designer's job may be made easier if the road is in a cutting.  This topic is discussed more fully in the pages about Footbridges and Arches.

The top of the lower window in this brick wall has been shaped to match the circular window above it.  This leaves the profile of the bricks precariously far from the funicular, which is of course arched upwards in this situation.  Geometry, or perhaps art, has clearly been given ascendancy over engineering.

The further the parts of a structure depart from their funiculars, the greater the internal forces they have to resist.  In theory, every strut and tie in a truss ought to be slightly curved, to follow the funiculars.  In practice, the spans of such members are so small that they may as well be straight, and in any case, the funicular would vary with the live load.  But Christian Menn has built an open spandrel arch with segments that really do follow the funiculars.  This bridge, the Viamala bridge on the Bernardino Pass road, is shown in Figures 12.20 and 12.21 of Fritz Leonhardt's book "Bridges".  Whether or not you consciously notice the slight curves, the effect is distinctly more pleasing than the effect of an arch with straight segments.

The first picture shows how, between each dew-drop and the next on a spider's thread, a different funicular is formed.  The cables of a suspension bridge show exactly the same effect, as the second picture shows.

Leonhardt also shows a bridge in his Figure 9.34 in which the arch fails completely to follow the funicular.  The shape of the arch totally ignores the two spandrel walls which spring from it, resulting in an absurd effect.

An extreme departure from the funicular is the rectangular portal frame.   Many footbridges across main roads are compromises between this shape and a funicular arch.  The problem to be solved is to obtain the specified clearance across the road and any hard shoulders, while providing a reasonable design for the ramps or steps, and also obtaining an economic and good looking design.  The same goes for entrances under buildings, through which delivery vehicles have to go.  This topic is discussed under Footbridges and Arches.

In a suspension bridge and a cable-stayed bridge, all the cables by definition follow the funiculars, because they are flexible.  The rigid deck does not, but the spans between the hangers is so small that this is irrelevant.  and nobody wants to see an undulating deck that appears to hang limply from the cables.  Besides, some rigidity in the deck helps to spread the load, reducing fatigue-inducing strains.  

A splendid example of an array of funiculars is a spider's orb web, like the one at the top of this page, and the one at left.  On a dewy morning the weight of the drops produces a set of deep curves, which demonstrate the extensibility of the threads.

Structures like flying buttresses are designed to compensate for the departure from the funicular in Gothic cathedrals.  The weight of the statues or apparently superfluous height of piers was used to move the line of thrust towards the centre line of the masonry.

These branches look as though they might be imitating a suspension bridge.  But a funicular is not a curve like a circle, ellipse or parabola, defined by an equation.  It is defined by the path of the forces.  For example, if you hang a 1 kg weight from a clothes line and a heavy chain, or move it along one of them, you will get different shapes.  But all will be funiculars.

Like other ideas, the funicular must be servant and not master.  It would be silly to build houses with funicular floors and walls: floors must be flat, and walls must be vertical.  But where there are degrees of freedom to play with, ideas like the funicular should be kept in mind.  In fact, rooms near the tops of buildings often have ceilings which are in part sloping; the effect can be quite comfortable, especially if the ceiling is covered with wood.  Why do you think this is the case?

In the world of physics the analogy to the funicular is the geodesic in space-time.  Einstein's idea to describe gravitation was to replace the "force" of gravity by the curvature of space-time.  Thus the discomfort of a hard seat, experienced during a long lecture on general relativity is caused by the fact that you really want to follow a "natural" path in space-time, not the one that the seat is pushing you along.  In an orbiting space-craft you would not feel this force, because nothing would be pushing you along an "unnatural" path.

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