Many Islamic buildings include rows of semicircular arches on very narrow columns. Some builders were not confident of the thrust being transmitted properly, and so they added tie rods between the tops of the columns.

The next picture shows a type of Islamic arch which covers more than a semicircle, though the designers realised that the thrust can never turn inwards, as the voussoirs show: their interfaces don’t go past the horizontal. The red line shows that a line of thrust can just about be contained within the voussoirs. The drawing is not good: actual arches curve inwards rather more than this and are more elegant. In practice these arches are often employed in series, each passing the thrust to its neighbours, with walls at the ends to transfer the thrust to the ground.

As well as the unconnected columns idea as an explanation of a masonry arch, we can try the equally idealised idea of masonry as a fluid without resistance to shear, which might be slightly more realistic if the spandrels were filled with rubble, as they often are. The curves are now rather different, as the next example shows. The actual curves in real arches probably fall between these extremes, but the main point is the ability of the masonry to cramp the voussoirs in their positions. Some builders have deliberately changed the mass distribution by leaving circular holes in the spandrels. These can also act as flood relief channels, though they are usually so high and so small that their effects are probably not very important.

After looking at all these diagrams, we are left with the feeling that the arch is not as simple as it looks. Nothing is, not even a vacuum, which is considered to be seething with virtual particles.

You might wonder what happens if we go on calculating the curve below the springing of the arch.  The next pictures show that.

Nobody would build these peculiar shapes, would they? But look at the photograph, which shows approximations to this type of shape, using circular arcs. Very often in engineering a much simpler and cheaper solution is preferred, because the result is almost as good as the ideal would achieve.

Very often also, science, mathematics and engineering, we find similar results from different fields. The curves shown below are the shapes of drops of liquid which rest on an unwetted surface. They are not the same as the ones above, because the drops are circular, whereas as the tunnels are cylindrical. To find out more about this, please click here. We must, however, beware of assuming that because two things look similar, they must be related.

Some of the shapes are flat at the bottom. This is not correct: all the shapes are flat at the bottom, but for the smaller ones it is not apparent. How big are the flat areas? The area of the flat part, multiplied by the pressure at that level, is equal to the weight of the liquid in the drop. In the same way, the four flat areas under the tyres of a car are related to the downward forces on the wheels. What about a railway train with steel wheels on steel rails, which are not flexible? They are flexible – everything is flexible. The wheels and rails deform just enough to create a contact footprint with the right area.

The profile of the terminal building in Charles de Gaulle airport which partially collapsed in May 2004 was not dissimilar to these shapes, and was definitely non-funicular. This in itself does not mean that the design was faulty: it only means that the concrete shell would have experienced bending, shear and tension, as well as compression. Concrete resists these stresses poorly, and forces are sustained in concrete by using steel pre-stressing wires or reinforcing bars.

The next two diagrams show the bending moment distributions in black for two bridges, and the bending moments in red for the same mass distributions but with the span shaped by the funicular. The bending moment is in different units from the mass, and so the vertical scales cannot be compared.

Varying the funicular

The funiculars we have seen so far don’t seem to vary very much, so let’s see how much variation we can create. The next diagrams represent three variations on an idealised three-pin arch, of which a half is shown.  The funiculars do not differ very strongly.

We see that the funicular does have a weak tendency to change in a similar way as the shape of the arch, but one thing you will not find, with any distributed mass, is a funicular with corners. The reason is that the curvature of the funicular at any point is closely related to the density of mass at that point in the structure.  Discontinuities in the mass distribution result in discontinuities in the curvature, which are not easy to see. To obtain a kink, you have to add an extra concentration of mass at a single point. The medieval builders who added heavy spires to flying buttresses must have realised this intuitively, or by trial and error.

In the next three diagrams, the mass distribution is horizontally the same, but the height of the corner is varied.

These funiculars vary much less than they did when we varied the mass distribution. You might even think that they don’t vary at all, and in fact they don’t.

So without knowing the exact shape of any particular funicular, we can confidently say that unless some heavy weights have been added, certain shapes are definitely not funicular. These include bents (rectangular frames), gothic arches, and the shells of Sydney Opera House. Gothic arches are usually held in place by the walls about them, and in the other two examples, large bending moments are set up that are resisted by the stiffness of the structures.

In the diagram below we see a funicular arch in the centre. Above it and below it are arches which deviate from the funicular enough for the edges of the middle third of the arch to reach the funicular, at which point tension is just avoided, while the outer two examples have twice as much deviation, showing clearly that beam action is required of the arch.  

Note that the areas of maximum stress do not necessarily occur at the places of maximum or minimum curvature: they occur near the points of maximum deviation from the funicular. Put simply, we are adding beam-like stresses to arch-like stresses. The beam-like ones are those that are governed by the deviations. Even if an arch is funicular, live loads will change the line of thrust, and the thickness of the arch must take the changes into account.

The three pictures below show a small archway of a type found in many thousands of gardens. This one differs from the majority of British examples, which are almost always based on circular arcs, usually semicircles. The pictures show, from left to right, a circle, a parabola and a catenary superimposed on the arch. Which best describes the actual arch? All are well within the middle third, so the exact shape is unimportant. The builder has provided neat abutment blocks to take the thrust. Although the details of such a thick arch scarcely matter, it is nevertheless a nice piece of work which reveals its workings clearly.

What detemines the curvature of a funicular at any point? The same rules apply as for any structure; the sum of the forces on any part of any structure in equilibrium must be zero in all directions. In the examples above, any segment experiences inward forces from its neighbours and the downward force from its own weight. It is the balance of these three forces that determines the curvature. For the plain arch supporting no load, the curve turns out to be a catenary, but if there are loads, as there usually are, the shape is changed. For example, if the inward forces are very great compared with the weight of the segment, the curvature will be small.  This is why the members of trusses are almost always straight. This is well seen in the Forth railway bridge. The designers were sometimes asked why the lower chord was not curved, and the reason is simple – the compressive forces on the chord are far greater than the weight of the members. But between the segments, there is a large input of forces from the struts and ties, and there we do see a change in direction.

The same is true of arches. The deck-stiffened ones often have straight segments, while those with heavier arches often have a smooth continuous curve. Ideally, all would have kinks at the vertical supports, and none would be quite straight, but in practice, the thickness required to prevent buckling includes the funicular. The prevention of buckling, in fact, is responsible for much of the appearance of many bridges. Christian Menn, in his bridges over the Valse-Rhine near Uors and over the Hinterrhine in the Viamala gorge, actually did provide slightly curved arch segments as theory requires. These bridges have only a few widely spaced supports, and the curvature does reduce the effect of the polygonal shape.

Next we see a series of arches with different weights of deck, with the ratio deck weight : arch weight as parameter. The eye notices the kinks in the arch above the curvature, and so the segments may look straighter than they are.

The last two cases illustrate deck stiffened arches.  

The diagrams below illustrate the changes in curvature for different loadings on a part of an arch.

Much has been written in these pages about the funicular with respect to arches and suspension cables. What about a shape that lies between the two – a straight member, or beam. It turns out that the funicular is not a useful idea for beams: it is really applicable only for members with one dominant stress – compression or tension. Nevertheless, if a theory is to be complete, it ought to apply as widely as possible.

An example this type of thinking is Bohr’s correspondence principle, invented to cope with the observation that quantum mechanics worked well for the smallest structures, but classical mechanics worked well for almost everything else. Bohr’s idea was that by considering a series of structures of successively greater size, the results from quantum mechanics should eventually and asymptotically merge with those of classical mechanics. In principle, this idea might even be useful in deciding on the exact formulation of quantum mechanics.

In the same way, let us imagine an arch that is held, as usual, between two abutments that generate vertical support and horizontal thrust. One of these abutments is provided with jacks which allow the horizontal thrust to be varied. The slope of the funicular at the abutment is related to the values of the vertical thrust V and the horizontal thrust H. If the funicular is at an angle A to the horizontal, we have the equation tan(A) = V/H.

If we can vary the H thrust, and hence the angle of the total thrust T, we are led to the idea that building an arch of "correct" shape does not necessarily result in a fit to the funicular. This idea is in fact correct. Imagine building a rigid arch, lying on its side, and then lifting it into position between two abutments that are not at the right distance apart. If they are too close, the arch will have to be curved a little more to get it in. If they are too far apart, the arch will have to straighten before it fits. Either way, bending forces are set up, and the funicular is not along the centre line.

With our variable jack, we can adjust the H thrust of our rigid arch from the true value right down to zero. The smaller H is, the bigger the angle A becomes, and as H tends to zero, and the arch tends to perfect beam behaviour, the height of the funicular becomes infinite. So that’s it – the funicular of a beam is at infinity.

If we use a straight beam, and vary H from positive to negative values, we can make the member behave a little like an arch or a cable. Passing through H = 0, the position of the funicular flips from infinitely above to infinitely below, which mimics the behavior of the function tan(A). So there is, in principle, a continuous gradation of behaviour from arch through beam to cable.

We can look at this another way, by examining the variation in stress across an arch, cable or beam. In an ideal beam, the stress is zero along the neutral axis, but the line of zero stress moves when live loads are added. The position of the zero is related to the funicular. The distance of the funicular is inversely related to the distance of the zero from the axis, and as we change the load from downward to upward through zero, the position of the funicular changes from above the beam to below, going from plus infinity to minus infinity at the crossover. In an arch that is not about to fall down, there is no line of zero stress, but we can extrapolate the graph of stress versus position within the arch to find a place in space where the graph reaches zero. Again, the position of the funicular is inversely related to the position of the zero line, exactly as for the beam. Thus the neutral axis is not the same as the funicular.

We see again that the difference between arch and beam does not reside in the shape, as beams may be curved. It resides in the distribution of stress from bottom to top of a vertical section. Any actual structure can exhibit arch-like or beam-like behaviour, or a superposition of both, depending on construction and live loads.

This was a purely academic exercise, with no practical value, done to demonstrate consistency and continuity of behavior among arch, cable and beam.

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 conflicting requirements 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 over the required width. 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 diagram below, constructed from a photograph, and including the funicular in red, represents New Hall footbridge on the M6. A photograph is given above, second picture down in the right hand column. Compare this with the top picture in the right hand column, showing a bridge that is much further from the funicular.

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 (first picture) 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 roof line in the second picture is also far from the funicular, in the same direction. Presumably the roof is supported front and back by the concrete walls, and so the shape is immaterial, but it is rather unsettling, nonetheless.

This window, too, is absurdly shaped: it is very hard to believe that bricks and mortar alone can sustain re-entrant corners like these. Presumably there are invisible supports.

The first picture here shows two windows with almost flat beam-like lintels, but with black brick "voussoirs" around them. What is going on? The brightness variations in the second picture have been exaggerated to show the joints between the stone blocks, which are sloping at about 45 degrees. Therefore an arch is intended, and it is just about possible to imagine a funicular contained within the total volume of bricks and stones. Were the joints horizontal, the longest stone would be a pure beam. Were they vertical, it would also be a beam, held in place only by shear stress in the mortar at each end – not a viable proposition. Such a structure is far from funicular, so beam-like behaviour is present as well as arch-like thrust. The third picture reveals that the higher windows have lintels which are thicker towards the middle, creating some arch action to relieve the stresses below. Indeed, as we see below, many buildings have arched lintels, or even a separate arch to relieve almost completely the weight on the beam.

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 perhaps more pleasing than the effect of an arch with straight segments.

There is, however, good reason to avoid curved segments when loads are very variable; it is that under greater stress, a curved strut will tend to bend more, when we want it to bend less, because the funicular becomes straighter under more force.  So a curved strut is in principle potentially unstable.

The next four pictures show examples in which the funicular is not a continuous smooth curve.

The first two pictures show how, between each large 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 third picture shows. The fourth picture is an enlargement of a tiny part of a slide of the Severn suspension bridge, showing the segmental curve of the main cable. The thin cables used as handrails do not have kinks, because nothing hangs from them.

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.  It might be a set of catenaries between the hangers, but near the hangers, the stiffness of the cable might well cause deviations from the mathematical curves.

Here is another example. Notice the difference between the angles in the cable at the empty seats and the angles at the occupied seats.

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. If someone hangs on a tyre and slides along the rope, the funicular changes continuously. This is true of any structure, but most are so rigid that we do not notice.  

This is the main reason that so few suspension bridges carry railways. The carrying of road vehicles is only made possible by the relative rigidity of the deck, which spreads the effects of a point load along a considerable length of the cables.

The first 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 second picture, showing the sag of the cables more clearly, is the result of rotating, and then squashing, the first picture. Later in this page you will see a similar picture of a much larger bridge. The suspension bridge, with its clear distinction between the dominant main cable and the thin hangers, does not suffer so much from the difficulty of achieving of an ordered appearance.

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 suspension bridges and cable-stayed bridges, 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. Nobody wants to walk or drive over a flexible bridge, either, and the varying stresses would soon break such a deck because of fatigue.

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.

This picture, a very small section of an entire negative, shows a line of buoys attached to a wire. The wire is one of several that mark of the lanes for a rowing event. This one had not yet been tensioned, and the curve is the result of small currents in the water and in the air. Even when pulled "straight", the cable would show similar, but very much smaller, deflections.  There are not many straight lines outside the textbooks.

Structures like flying buttresses are designed to compensate for the departure from the funicular in Gothic cathedrals, and in this glasshouse, in which the props with the roof is closer to a funicular than the walls and the roof. 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.

The roof of this glasshouse is nearer to a funicular than most, but you can see in the first picture that the rafters are trussed to obtain rigidity.

The rafters are trussed here too, enabling the external profile to be non-funicular while retaining arches inside.

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. At the left you can see two metal discs, which are connected to two similar discs on the right, via long steel rods. These discs and rods are all that stops the barn from collapsing. Such steel discs and crosses may be seen on many old buildings in Britain.

This topic, the funicular, 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 this web-site. In some places 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 many cultures 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? We must not look for funiculars where they have no meaning.

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. The greater the weight, the bigger the angle of deviation in the cable. Turning the diagram upside down, it is clear that for a gothic arch to be funicular, it must have a heavier load at the apex than anywhere else. This never happens in practice.

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. The placing of loads 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?

These diagrams with the weights give us a clue about designing structures. Antonio Gaudi used the method for some of his buildings, such as El Temple Expiatori de la Sagrada Familia.

Suppose we want to make a shape like this, a bridge shown earlier in the page. We could fix a lot of hooks on a rope, and then we could hang weights on the hooks until the shape of the rope would take up the required shape. The weights at each point would give us a very good idea of the weight distribution of the arch. People don’t often add weights to bridges to make them work. Antonio Gaudi made models in this way – for which building? (But think about the Forth railway bridge [Where are the weights?] and some bascule bridges.) Instead of adding weights, we could simply vary the thickness of the bridge. This raises an interesting question. The pictured bridge is thickest where it has the greatest bending moments, in order to resist that moment. But that is also where it deviates furthest from the funicular of a uniform arch. Remembering that the funicular is the line of zero bending moment, is this connection between deviation and bending moment a universal rule? This is another excuse to mention the beautiful bridges of Robert Maillart, beautiful, not because he added something, but beautiful because he took something away, and beautiful because appearance of the resulting structures follow the function.

In the bridge above, the two beams are very narrow at the central pin, even though there is an angle between them. What would happen if there were pins at the other two angles? Obviously the bridge would be unstable, because it would have five pins.  If it were to move symmetrically, would the central pin move up or down? Putting it another way, if we were to hang the five-pinned bridge upside down, what shape would result?

Does the presence of a pin force the funicular to pass through it? What would happen if it didn’t?  

If you use a material that can resist both tension and compression, and can therefore withstand bending, you are absolved from following the funicular. Your arch will take on some characteristics of a beam. Here is a bridge over the river Cam in Cambridge. If you set the bridge up so that there is no horizontal thrust at the abutments, then you have a beam, not an arch, whatever it actually looks like. The cost is greatly increased stress in the structure.

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. But in a completed bridge, the deck is very often not connected to the towers. 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 by 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.

Most of us never see a partly built bridge, but these pictures show the same effect. The first picture shows high voltage cables attached to heavy insulators. The second shows the same picture compressed horizontally by a factor of six, revealing the change in curvature very clearly. The small inset shows one section of the cable, compressed by a further factor of six. The curvature is just visible.  The second pair of pictures show similar effects. nverting the pictures, and replacing tension by compression, we see very clearly how the curvature of an arch must change with the load per unit length, if it is to be funicular.

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.

Here, someone had the amusing idea of making arches to offset the chains. Brunel used a similar idea, but in a structurally meaningful way, in his Royal Albert Bridge.

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. They actually have small stone struts that make sure that they do not sag towards the tower, so we can feel sure that the builders realised that straightness had its drawbacks.

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 first picture shows a very flat arch, with ends that suddenly curve down into the supports at the ends. Can we really believe that the funicular does that, bearing in mind the small amount of masonry above the arch? How far should builders go in preferring design over function? Or function over design?  The second picture has been adjusted to remove the curved ends of the arch. Is the result better or worse than the original? What about the arch in the third picture?

This arch deviates from the funicular in the opposite way, but not enough to fall down, though the halves look perilously close to beam action.

In this arch, few of the voussoirs are at right angles to the curve, demonstrating beyond all doubt that the funicular does not always go through the voussoirs in an arch in a wall. The yellow lines show that the arch sagged before the opening was filled in.

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 narrow for the camera lens, and the camera was tilted, leading to converging verticals in the image. The truss is reminiscent of the trussed chains of the Tower Bridge in London.

A lot has been said in this page about funiculars, even though most structures don’t follow them, for practical and economical reasons. Often the designer is willing to pay the price of deviation to get a bigger payback.

Where else might we look for funiculars? We haven’t mentioned pre-stressed concrete structures, which contain wires.  Is there a funicular for them, buried as they are in masses of concrete? In a way, there is. Adding a pre-stressing tendon may induce secondary moments in the structure, and the tendon will not then be along the pressure line. The tendon can be made to lie on the right line, and then it is called a concordant tendon. But like the funicular, the pressure line is an ideal that is not often followed, for good practical reasons.

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 your "natural" path in space-time is 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 a platform.

Summary

A compression funicular is a perfect strut. A suspension funicular is a perfect tie.

A bent or frame deviates far from the funicular. A gothic arch deviates in the opposite way.

Curves in space and curves on earth

Imagine a pair of space vehicles, a billion kilometres from any star or planet, floating freely in space. A long flexible cable joining them would float loosely as well.  But suppose that the rockets of both craft were turned on, making them accelerate equally along parallel courses. The cable would take up the form of a catenary. Einstein’s general theory of relativity says that the local effects of a uniform gravitational field are indistinguishable from those of a uniform acceleration. The pull on the cable would in fact make the vehicles accelerate towards each other, so maintaining parallel courses would require the rockets to points inwards as well as backwards.

Imagine a model suspension bridge, lying on its side on a flat-bed truck, with the piece of wood representing the earth fixed to the truck, and the towers pointing toward the front of the truck. If the truck accelerates, the cable takes up the same shape as if the model were stationary and standing up.

A skipping rope is accelerated, and it forms a funicular, but not a catenary, because the acceleration is not the same at different distances from the axis or revolution.

An electrically charged insulating cable, with uniformly distributed charge, fixed at two points in a uniform electric field, forms a catenary in the absence of gravity. But a current carrying wire in a uniform magnetic field forms a circular arc (more generally a helix), because the force is everywhere at right angles to the current.

A long fishing net, towed between two ships on parallel courses, forms a curve, but it isn’t a catenary. We can’t say what it is unless we know the way the drag of the net in the water varies with the angle between the net and the direction of motion.

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