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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", "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 try to break the laws of physics, we will die, or at least someone else will.

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 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.

Aiguille du Midi – beautiful Alpine photograph showing funicular cables

Some mountain railways are referred to as funicular railways because the tracks are inclined.  The word is being used in a different sense from the way it is used in this page.  A flexible cable supporting a set of passenger cars does follow funicular curves, by definition.  Here is an example assuming a weightless cable.

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.  On the other hand, in a housing area or a park, or near a popular river bank, there may be requirements, often conflicting, from various sources, as to what is acceptable.  The greatest challenge here is found when a foot-bridge has to cross a very wide road or river in flat ground, because the height to be scaled can be considerable.  

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 this footbridge made a brave attempt to remain close to the funicular.  The resulting curve looks as though it might be a slightly uneasy composite of several curves, but is nevertheless an elegant solution to the problem of obtaining the required clearance.  The page on foot-bridges gives many more examples of footbridge designs.  Click on the thumbnails to see some larger pictures.

Greater departure from the funicular does not necessarily mean inferior design.  In the right hand example in the second row, 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 diagrams below show some three pin arches and some two pin arches, deviating from the funicular by different amounts.  Are any of these designs usable?  Are any of them unusable?  The shapes were made using equations of the form  X^N + Y^N = R^N, with values of N from 1 to 3.5.  The depth in the second column was varied according only to the horizontal position, but in the third column both vertical and horizontal positions were used.

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 or design, has clearly been given ascendancy over engineering.  The bricks are almost certainly supported by a piece of metal.

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.  

These pictures show the catenaries and the discontinuities in the cables in a children’s play area.  Note the sloping struts that are required because there are no backstays, such as a those of suspension bridge.  In the left hand picture the wide A-frames of the swing in the background are designed to make sure that all the legs remain in compression, thus ensuring that they never pull out of the ground.

The picture below shows the catenary sag in the cables of the Sabrina bridge in Worcester.   Although this is an elegant bridge, this view shows the difficulty of maintaining a tidy and ordered appearance from all directions.  The suspension bridge, with its clear distinction between the dominant main cable and the thin hangers, does not suffer so much from this problem.

Katsushika Hokusai made a picture showing a funicular – Famous Bridges of Various Provinces: The suspended bridge between Hida and Etchu.  He shows clearly the discontinuity in slope at the position of each of the two people, but he has made a bigger change of slope for the person with the smaller load.  He has also assumed zero mass for the bridge.

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 ability of the threads to stretch and absorb energy.

In principle, any force field can generate a funicular, but the only other ones that have significant extent are electric fields and magnetic fields.  There are no magnetic monopoles to place in a magnetic field, and strings of electric charges are not commonly found either.  However, this picture shows bunting being pulled out into curves by the wind, which is a sort of dynamic equivalent of a funicular.  The curve is a compromise between the vertical effect of gravity and the horizontal effect of the wind.

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.  You can read more about this in the page about arches in religious buildings.  In a building with a pitched roof that isn’t held in shape by a truss, the sloping thrust of the roof meets the vertical line of the wall, and is liable to push the wall outwards.  This illustrates the difficulty of dividing engineering into discrete subjects – many of the topics that are in other pages could have been in this one.  In fact some paragraphs are indeed used more than once.  In fact the arrangement of this web-site has been made deliberately less tidy than it could be, to emphasise the fact that the divisions into types and topics are not hard-edged. 

You can’t make a force turn a sharp corner unless you provide another force to provide the difference in the two vectors, or you provide rigidity at the junction.  Think of it like driving a fast car in a race – you have to take the racing line to minimise the curvature of your path, thus reducing the transverse forces required of the tyres.  In a large open space such as a supermarket or an exhibtion hall, if the roof is not flat you may see gusset plates at the junctions of the rafters and the pillars to provide rigidity.

In our culture we are so used to seeing straight lines, rectangles and grids that we may lose sight of their artificiality.  In nature, forces flow more naturally, and an almost infinite variety of curves is seen.  Nature doesn’t have to compute, things are selected over time by survival.  In our own constructions, we need great computing power to analyse even quite simple structures, and we cannot always make the shapes we want because of the expense.  

But wait – what we call modern will one day be old-fashioned:  even as this is being written, people are using evolutionary computing to select designs.  And what will happen when more is understood about genetics and growth.  Imagine being able to grow a tree as fast as a bamboo, and imagine being able grow it in predictable shapes, with as few branches as we want, and with required stress distributions, possibly with the aid of computer controlled external restraints.

Having thought about buttresses, let’s think about a big tent with vertical walls and a sloping roof.  It doesn’t have buttresses – it has guy ropes.  A tent is to a cathedral as a suspension bridge is to an arch.

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, you will get different sets of shapes in the two cases.  But all the shapes will be funiculars.  These branches are not funiculars.  Why do you think they have these funny shapes?

Where is the funicular here?  Is it at the left hand side, where the beam looks like an arch?  Or is it at the right hand side, where the beam looks like a chain?  Or neither?

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 usually be flat, and walls must usually 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?

Let’s now look at some examples in which the distribution of weight is not always uniform, to see what happens to the funicular.

We see here the effect of a localised weight.  If the cable were a beam we would see where the bending moment is greatest.  We can see also the effect of live loads, which distort the funicular.  This is one reason to distribute the weight of an airliner by placing the engines, which are very heavy, on the wings, thus reducing the bending moment at the roots.  For the same reason, the hollow main wing spar is a very good place to keep the fuel, provided it is protected from engine break-up.  As the fuel in the wings is used up, and the aircraft becomes lighter, would you expect the wings to straighten, curve more, or stay the same shape?

Using a calculation like this we can see how the funicular changes during the building of a suspension bridge, only now we have to allow for the weight of the cables as well as the deck sections.  This calculation is unrealistic on two counts.  Firstly, the vertical scale has been exaggerated, and secondly, the deck sections have not been joined together.  In practice, they would be, for several reasons, and this would reduce the curvature.  The vertically exaggerated diagrams below show three cases – a complete bridge, an incomplete bridge with weightless cables, and a more realistic case in which both deck and main cables have weight.

We can also start building from the middle, but this is not necessarily a good way to proceed.  The unconnected deck is a big mass, waiting to swing in the wind.  By starting at the towers we can anchor the deck at that place, and so reduce the amplitude of oscillations.  The Severn bridge was built from the middle outwards.  Its aerodynamic deck, with low drag and slight downward lift reduced its susceptibility to oscillation caused the wind.

Pictures of suspension bridges under construction often show a curved deck.  As the deck is extended, the forces gradually straighten the deck towards the shape that it would have without the join.  The curve may look alarming, but the radius of curvature is very large compared with the depth of the deck, and so the strain is actually very small.  The diagrams above are vertically exaggerated.

The next picture shows a small part of the big Severn cable-stayed bridge.  The picture has been tilted and compressed horizontally to show that, although the cables look straight, they sag.  There are few perfectly straight lines in engineering, with the possible exception of verticals.  Every part that is not vertical will sag a little, though of course "rigid" struts will not deflect visibly.  The truth is that there are no rigid bodies.

The behaviour of an arch is in some ways the inverse of that of the suspension bridge.  Download a Simulation of Live Loads in an Arch.  In a truss, there is a funicular within each member, though the weights are usually so small that the members can be safely and more cheaply built straight.

Where is the funicular in a thick beam?  Does the term have any meaning in such a member?

The beautiful tower of Cirencester church has buttresses which go right down into the ground in the west wall.  Because of difficulties with the ground, threatening the tower, this wall had to be taken down on the south side in order to add the buttresses.  The NW and SW corners of the tower are also well buttressed.  Should you climb the helical stairway to the top to see the splendid view of the town and country, you can be quite sure that the tower will stay upright.

There is actually something peculiar about these buttresses – they are straight.  They should logically curve towards the ground.  In the case of Cirencester we can imagine that the builders wanted to anchor the buttresses as far from the tower as possible, in the hope of finding better ground. 

Perhaps appearance entered into the design, since a window was required, and curving the buttress around the window would have looked rather strange.  On the other hand, the builders might not have understood the flow of the forces at all, as everything seems to have been done empirically in the middle ages.  In the first picture, we see that the buttress actually reaches the ground perilously near the corner of the building, but in fact there probably isn’t much thrust left in it by that point.

The exciting Wildwalk at Bristol includes a tropical area with a tented roof and trussed ends.  Here we see at the bottom right how the support leans inwards, because the funicular can never become vertical.  Next to Wildwalk is Explore At Bristol, which is a magnificent exploratory of engineering and science.  Unfortunately, the road was too wide for the camera lens, and the camera was tilted, leading to converging verticals in the image.

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.

If you build something like a beam, which is not along the funicular, it had better be rigid enough and strong enough to stay that way.  In this example, the roof has sagged, probably because a support has failed, thus doubling the span.  Note also the small supports between the pillars and the roof beam, which reduce the brutal transition between the vertical forces in the pillars and the forces in the beam.  Allowing forces to "flow" smoothly can reduce local stresses very significantly.  Turn this upside down and you have the principle of a foot or platform.  

Bird  Funicular

The picture above is a small part of a photograph of starlings on electric power lines.  A bird watcher might be interested in the way they flock together, and might ask questions such as – "Do they squabble over position?", "Do they have a minimum acceptable distance apart?", "Is there a typical size of flock?", "Are they talking to each other?", "Are they all adults?", "How close can I get before they fly away?", "Will they all fly away together?".

Let’s ask a different question?  Are the birds heavy enough to make a detectable difference to the funicular?  The next picture shows the whole width of the picture, rotated to make the wires level, squashed horizontally by a factor of eight, and doubled in height.

How would you try to measure the effect of the birds on the wire?  One way would be to take a similar photograph after the birds had flown, and then make a comparison.  This was not done.  So another way must be found.  It doesn’t look as though the the birds are heavy enough to distort the curves.  The wires didn’t seem to move appreciably when the birds flew away.

A  Puzzle

An old mathematical problem concerns the maximum amount by which a pile of bricks can overhang before falling over.  It turns out that the critical overlap of the Nth brick from the top is 1/Nth of the length of a brick.  In theory the overhang can be made as large as you want, by using enough bricks, but in practice the sum of the harmonic series grows exceedingly slowly.  The proof neglects the compressibility of the bricks and of the foundation, and has no practical application.

Although this is an impractical object, a set of diving boards at Coate Water, south of Swindon, is supported by a concrete tower which has a shape which looks rather like a smoothed version of this pile.  Does it also remind you of a cobra which has reared up, ready to strike?

This pile of bricks violates the rule of the middle third, which states that the centre of gravity should lie over the middle third in both dimensions, that is, it must lie in the middle ninth of the area.  Only then can we be sure that tension will not develop anywhere.  In masonry, the rule should be followed from top to bottom.  If there are transverse forces, as in a buttress or a retaining wall,  it is the funicular which must lie within the middle third. 

Under this rule, the overhang must be reduced.  What is the new formula for piling up the bricks?  Can you still achieve an arbitrarily large overhang?

In a real tower, the funicular is a vertical line, isn’t it.  Not if the wind blows, as the builders of the first Tay bridge realised, too late.  They probably knew in it principle, but did not allow for enough force.  These variations in the funicular force designers to make towers wider at the base.  The iron Tour Eiffel was an early example.  Its steel counterpart in Tokyo weighs a lot less, but follows the same general design.  Electricity pylons often use similar designs, though the styles vary widely in many countries: many designs are very elegant.

If you are standing in a swaying bus or train, you may find yourself automatically placing your feet further apart, if you have nothing to hold on to.

Another solution is a parallel tower or mast with guys.  An extreme case is the mast of a high performance yacht, which is subject, along with the wires, to severe stresses during heavy weather.  Catastrophic failures are not uncommon.

Ignore the funicular and you will have work to do.  The shells of Sydney opera house look impressive, but they had to be rigidized because they are not funicular.

Summary

A funicular exists only in an external force field (gravitational, electrostatic or magnetic), or in a flow field (wind, water or solar wind).  There is no funicular in space, unless you consider some huge light flexible object in the solar wind.  But what about a large rotating space craft?  Does the Coriolis "force" imply the possibility of a funicular.  What is the shape of the funicular between two points on a turntable?  What is the curve of a skipping rope rotating at high speed?

The funicular can never be vertical at any point.

Deviations from the funicular need beam like behaviour, that is, a measure of rigidity.  The bigger the deviation, the bigger the bending moment.

The funicular is a universal physical idea, not a universal equation.

The funicular for a uniform arch or cable is a catenary, the sum of two exponentials.