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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: the word is derived from the Latin word catena, a chain.  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.

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.

To generate catenary arches with different ratios of height to span, click here to download program Brancat.  By pressing the PrintScreen key, you can copy the picture into the clipboard to use it as the basis of a model.

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 bending moment at any point in an arch is proportional to the vertical distance from the centre line to the thrust line.  It is also proportional to the horizontal thrust.  You can see in two of the pictures above that the arches are thickest where they are furthest from the funicular, for this reason.  The extra thickness adds weight, which actually moves the funicular nearer to the arch.  This is an interesting idea – that by adding weight, you can improve a structure.  Medieval cathedral builders discovered this (the hard way?), and they added spires to buttresses to improve the position of the thrust line.

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 taken into account.

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.

This picture has been expanded vertically by a factor of five times.  The black lines show the changes in direction at the hangers, slightly exaggerated, because the cables are in fact slightly curved between the attachments, as they are not weightless.  This corresponds nicely with the segmented arches by Menn which were mentioned in a previous paragraph.  So the curve of the cables is not only not a catenary or parabola – it is not a mathematically simple curve at all.

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?

Here are two pictures showing spider webs weighted down by drops of water.  In one, the distribution of drops is fairly uniform, and the curve is nearly a catenary.  In the other, we can see very clearly how the curvature responds to the sizes of the drops.

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 then 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 fifth cable in each set below seems to sag more than the rest.

This picture shows the jib and tie-bars of a crane, compressed by a factor of about six.  The line of the tie-bars is curved, as expected, but why do you think the jib is curved as well?

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 compared with the longitudinal forces 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 a space craft needs to deviate from the path, rockets are used.  The difficulty of deviating from free fall was well illustrated by the immense power of the Saturn V rockets.  Hovering aircraft, such as a Harrier or a helicopter, require great power as well, and yet they are barely moving.  Compare this with the performance of a high performance glider that in perfectly still air, if flown ideally, can achieve a glide angle of 1 in 60, using its own fall as a source of energy.

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, possibly 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.

An Unusual Curved Bridge

On the cover of "Ekiben – The art of the Japanese box lunch" (Kamekura, Bosker and Watanabe – Chronicle Books – San Francisco – ISBN 0-87701-490-6) there is a picture of a pontoon bridge.  The bridge makes a sweeping curve across a bay or estuary, pushed by the flow of the tide or the river.  Had the builders tried to make it straight, the tension would have been too great.  The curve is not a catenary because the speed of the water is greater near the middle than it is near the edges.  But it is a funicular, perhaps stretching the term a little.

The millenium footbridge near Lancaster has a deck which is not straight, as does the new rotating bridge in Newcastle.  They achieve stability in two quite different ways.

Having got this far, how about summarizing what we know about funiculars.  Firstly, the term is only applicable to a structure which pulls inwards or pushes outwards.  An arch shaped beam resting on ice is not a funicular, and never can be.  Secondly, the funicular is not a fixed mathematical shape such as a parabola.

And now for three dimensions . . . .

Why can’t we have funiculars in three dimensions?  The answer is – we can.  The next pictures show them in compression and in tension respectively.

The first picture shows Brunelleschi’s dome in Firenze.  The second shows a net, and the third a roof.  Actually, domes are not often exact funiculars, and thickness or bracing is required for stability.  The dome of St Paul’s cathedral in London actually has three concentric shells, of different shapes.  Do you think that power station cooling towers, in the form of hyperboloids of revolution, are funicular?  They are certainly thin.  The peculiar thing about these towers is that they can be generated by straight lines.  The shells of Sydney opera house, as originally envisaged, were not funicular, and the design had to be radically changed.  Are they funicular now?  If so, to what forces do the ridges at the top respond?

The four examples in tension have opposite curvatures in directions at right angles.  The total curvature at any point is small, and may even be zero.  On the principle that a funicular is a curve or a surface of minimum energy, we can find them in many phenomena, for example surface tension, which finds expression in the sticky blobs on spider webs.

How Buttresses Work

This section was copied from the page about Arches in religious architecture.

An orang utan hangs from a rope, distorting the natural curve.  The natural curve of the forces in an object is called the funicular, and a freely hanging flexible object follows it.  An applied force, as here, makes a discontinuity in the slope of the curve.  The animal could also make a kink by standing underneath and pushing upwards.  Turning the curve upside down forms the funicular of an arch, as in the Gateway Arch, St Louis.

The funicular is not necessarily a "mathematical" curve – it differs for each stucture.  But building something that does not follow the funicular leads to the need for extra stiffness to resist bending moments.

Not many buildings follow the funicular, the main reason being that we generally prefer vertical walls for rooms in which to live and work.  But large enclosures such as hangars have often been built with curved walls and roofs.  A huge example is the airdock at Akron.

The diagram below represents a cross section through an imaginary building, with a section drawn thick, to represent, roughly, the vault of a cathedral.

An actual cathedral usually has vertical walls, as we see below.  Just as the downward weight of the gibbon kinks the tensile funicular upwards, the upward force from the wall kinks the compressive funicular upwards also.  In fact, the funicular becomes horizontal, because the wall takes all the weight of the vault.  Where it goes after that depends on the distribution of mass in the supporting structure.

On the left is a solid buttress.  We see immediately why the builders needed to make their vaults as light as possible, not for artistic reasons, but to reduce the effort required of the buttresses.  On the right is a flying buttress.  Real buttresses sometimes have another arch further down to supply rigidity.

To make the flying buttress narrower at the base, we could make the funicular dip more steeply.  If the wall pushing up kinks it upwards, adding weight at the top kinks it downwards, as we see below.  We can also see that adding mass above can reduce the necessary mass by a larger amount down below.  The great medieval builders discovered all this without mathematics, and even succeeded in reaching the limits of the technology.  Sometimes they went beyond the limits, and some cathedrals collapsed, suddenly, and apparently without warning.   It is unlikely that there were no precursor movements, but the congregations would not have been looking for them.  The cracks and crumblings might well have been high up out of sight.

The foot of the buttress is narrower than before, and the line of thrust lies within it, not at the edge.  The funicular as drawn, is still at the edges near the top.  But this whole scheme is highly idealised.  Vaults do not necessarily connect to the walls at narrow points, and the walls do have a certain amount of rigidity.  This can be seen on the south side of the nave of Salisbury cathedral, where buttresses are provided only at alternate bays.  So don’t take these diagrams literally.  But the system works, and has worked for hundreds of years.

More peculiarly, on the north side of Tewkesbury Abbey, there is only one buttress, albeit a massive one.  It is hard to believe that the wall, however thick, provides beam action over such a long distance.  Perhaps the builders felt nervous about some feature of the building or the ground at that point, though there is nothing to see, either inside or outside.

Look at the flying buttresses of the cathedral of Notre Dame.  The daring of the arches is breathtaking – remember – the line of thrust must remain inside the stonework at all points.

A roof has been added to our imaginary cathedral above.  This could be supported on wooden beams.  As drawn, it would exert lateral force on the walls.  By raising the roof, space would be created for ties to hold the beams together, making a simple truss.

In Beauvais cathedral, another trick was tried.  Around the buttresses of the semicircular east end. tie-bars join all the buttresses.  These act like the chains that often serve to take the tension in domes.  They act like the metal tyre of a wagon wheel that is shrunk on after heating.  You might ask – If the bars contain the thrust, why did they build buttresses at all?  Why not just fix the bars around the walls?

Perhaps they were an afterthought – a belt and braces solution.  The full story of Beauvais may provide another clue.  Beauvais was intended to be the most stupendous cathedral ever built.  Its choir is almost 160 feet high.  But in 1284, 59 years after construction of the building began, it collapsed.  During the next forty years it was rebuilt, more robustly.  Over 200 years later, in 1569, Beauvais boasted a spire reaching 492 feet into the air.  It lasted less than four years.  When it crashed, it damaged the choir, which was then repaired for the second time: the nave was never started.  No wonder if the builders took precautions with the rump of a building that was left.

It is easy to say that these people did not know what they were doing.  To think that people of different times and of different places were necessarily inferior to ourselves is as absurd as to romanticise them or to place them on a pedestal.  Strangely, some people have thought that the great pyramids required the help of creatures from another planet.  You don’t hear this about cathedrals, which are much more ingenious and much more daring.  To have made a 492 foot spire that stayed up for well over three years was a tremendous achievement, and the structure was clearly only just outside the region of stability.  We shouldn’t necessarily blame the builders: it’s possible that they were browbeaten by an ambitious bishop – who knows?

That the builders did not use our mathematics and science does not mean that they did not understand what they were doing.  A cricketer does not understand the mechanics of a well-timed cover drive, but only a fool would say that he does not understand the game.  An arch, a beam, a cathedral, you name it – it requires heart as well as head.  Building is art as well as science.  There are several pictures in these pages of structures that have too little art, as opposed to pointless decoration.

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 visibly.  The wires didn’t seem to move appreciably when the birds flew away.  Attempts to detect the effect of the birds by measurement of the picture have been inconclusive.

At the other extreme we have the result of a gibbon or an orang-utan hanging on a rope.  Here the load is much heavier than the corresponding section of the rope, and we see kinks in a fairly straight rope.  Watching a gibbon brachiating along a rope shows very clearly why so few suspension bridges have been built to withstand the concentrated load of a railway locomotive and train.  Compare "Famous Bridges of Various Provinces: The suspended bridge between Hida and Etchu" by Katsushika Hokusai.

Here we see a gibbon picture upside down.  It illustrates the idea that if an arch changes curvature, in this case to an extreme, this ought to correspond to some change in the force per unit length.  Greater curvature should correspond to greater force.  The existence of many different shapes of masonry arches tells us that the funicular is not the whole story – the arches are stiffened by the walls above, just as suspension bridges are stiffened by the deck.

That this is not a silly idea is proved by the use of ropes and weights to simulate the behaviour of arched structures.  This was done, for example, by Antonio Gaudi, builder of the La Sagrada Familia in Barcelona.

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 pile of bricks 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 the pile of bricks also remind you of a cobra which has reared up, ready to strike?  Some cranes have a curved shape which is quite similar, though they are more often made with straight segments, for simplicity.  There is such a curved crane at the docks in Bristol.

The 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 throughout, because it cannot resist tension.  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 of the bricks must be reduced.  What is the new formula for piling up the bricks?  Can you still achieve an arbitrarily large overhang?  Can you make a tower at all?

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, much 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 of the 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 in a gravitational field 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.

To generalise the idea of the funicular, we can consider a skipping rope.  What is the curve of a rope which is whirled around so rapidly that we can ignore gravity?  For a hanging rope, we had two curves, the catenary (with derivative sinh x) when the rope is uniform, and the parabola (derivative x) when the load is uniform per horizontal length.  We can try this for a skipping rope.  Instead of a uniform gravitational field, the effective field is one which increases from the axis to the outside.  For the uniform axial load, the rope forms a sinusoid, whose derivative is a cosine.  For the realistic rope, the curve is like a series of near-catenaries that form an oscillating curve, whose derivative is like a series of linked sections similar to sinh x.

We can in fact look at the parabola as a part of a sine of infinite wavelength, and we can think of the catenary as a part of the skipping rope curve of infinite wavelength.  But how can a skipping rope have a wavelength?  If you tie a long rope to a fixed point, and swing the other end, you can make a shape with one loop, two loops, three loops, and so on.  Here is an example.  The blue curve represents the rope, and the red curve the tension, which is complemented by the curve of kinetic energy.

For comparison, here are the curves for a catenary.  Notice that as with the skipping rope, the tension is never zero.  In a suspension bridge, the tension would vary much less than this, because the cable would have a much smaller sag.

If we imagine that the amplitude and wavelength of the skipping rope rope are increased in a suitable manner, we can take smaller and smaller fractions of the curve.  To keep it looking about the same, we need in increase the amplitude by the square of the factor by which we increase the wavelength.  Very soon we find that the variation of the vertical force becomes negligible, which is exactly the condition for making a catenary.

Here is a completely different idea for a funicular – a drop of water hanging from a small circular object, such as a tube or a sphere.  The smallest drop in the scale diagram has a radius of 0.1 mm.  The surface tension of the water is the same at all points.  If you imagine the diagram upside down, you can see among the curves the general shape of a hot air balloon, which ideally has a uniform tension in the envelope.

Let’s look a bit more at a generalised idea of the funicular.  We already compared a funicular with paths in space time.  The funicular of a cable or an arch with a load that is uniformly distributed along the horizontal is a parabola.  So is the path of a projectile in a uniform field in a vacuum.  The horizontal component of its velocity is constant.

For a cable or an arch that is loaded uniformly along its length, the funicular is a catenary.  This is discussed in the page about suspension bridges.  How can we get a projectile to travel along a catenary?  It must travel with uniform speed along its path, rather than with a uniform horizontal component.  To do this it needs a thruster fore and aft, to produce the forces needed to deviate from the natural curve in space time.

The curve which gives the shortest time of fall between two points is a cycloid.  It can be achieved by making a bead slide on a wire.  As with the projectile on a catenary, the wire provides forces, in this case transverse, which guide the bead from the "natural" path.

Not Quite Funiculars

The idea of a natural path can be generalised.  For a small body in the gravitational field of a larger body, in Newtonian physics, the path is an ellipse, a parabola or a hyperbola.  In fact, the paths of all systems can be accounted for by a simple rule, which involves the minimisation of a function called the Lagrangian.  This can be applied to systems and forces of all types.

Another type of natural path is seen when light is reflected or refracted.  The path we see is not always the shortest one in distance: it is usually the shortest one in time, though it could in principle be the longest.  When an image is formed, for example by an ideal lens or a mirror, all the light that contributes arrives in phase, having taken exactly the same time to travel from the object.  The wavefronts are everywhere at right angles to the direction of travel.  In practice, this perfection of arrival times is very difficult to achieve with lenses.

The diagram below shows the wavefronts in a mirage over hot ground.  Both the curvature and the wavelength have been greatly exaggerated.

The resemblance to voussoirs in an arch is clear, but there is a fundamental difference.  Waves in space are not confined, and they do not exhibit the discontinuities shown here.  Only in finite structures such as tubes, fibre optical cables, and the like are waves confined with sharp edges.  And here we find another phenomenon that does not appear to occur in structures – the waves can add or subtract.  In fact, a structure can respond to transient or periodic excitation in the same way as a cavity containing light.  Pump energy into a medium in a cavity, and you have a laser, if you do it right.  Pump energy into a suspension bridge, and you have an oscillator, if you don’t do it right.  In both cases, a steady supply of energy can create the conditions for oscillation to occur.  You can find more information about waves in brantacan waves.

Other curves

This page has been mainly about funiculars.  Many other curves arise in the world of engineering and nature.  Watch the glide of a jackdaw between the towers of a castle or a cathedral.  Is it following a brachistochrone – the curve of minimum time?  Look at the path of a projectile.  What about the path of the centre of gravity of a brachiating gibbon?  Or that of a trotting rhinoceros?  From the ancient Greeks to Johannes Kepler, some people have tried to fit the world to elegant curves.  But the simplicity is more often in the laws or the rules than in the results.  Einstein’s idea of curved space-time was in itself simple, but he took a long time to make it work.  Hilbert did it much more quickly.  See Nature’s Maths.