How  Arches  Work

The arch is one of the older forms of bridge. It is rather like an inverted suspension bridge, with all the tensions replaced by compressions, and vice versa.  The other great difference is in the stability of the system. You can hang a rope across a gap, and it will return to its original position, after some oscillation, if disturbed.

But you cannot hang it in the shape of an arch. Even if it could be positioned correctly, the slightest disturbance would send it flying. The diagram below shows a very simple system of hinged rods, to explain this fact, obvious though it seems.

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The blue lines in the upper diagram represent three rods, hinged at their ends, hanging from a fourth rod, drawn in black. The red lines represent a different position of the rods. We see that the centre of gravity of the rod on the right has moved up, while that on the left has moved down, but by a smaller amount. The central rod has moved upwards, so the net result is a higher centre of gravity of the whole system.

Therefore the symmetrical position was more stable than the unsymmetrical one, because a system is most stable in the position of lowest energy. If the rods were to be released at the red position they would swing back to the symmetrical position, and oscillate about that position until friction would have removed all the energy.

Because the hanging cable is intrinsically stable, suspension bridges can be made very light and slender, which led to a number of collapses before the dynamics were understood. Even so, the Millennium bridge in London provided a surprise.

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But the situation is reversed in the lower diagram - any movement, however small, lowers the centre of gravity of the system. So the symmetrical position is unstable. If the rods were to be released at the red position they would diverge even further from symmetry, until the right-hand rod would be resting on the line at the bottom.

We can use the same argument for more than three rods, and the limit of an large number of small rods, for a chain or cable. Any hinged polygon with more than three rods is not rigid, and is unstable if it is above the points of support.

Yet a stone or concrete arch looks very solid and robust. What is the secret?

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One reason for the stability of many arches is that the blocks (voussoirs) are quite thick, and they are rigid. Compared with a suspension cable, any arch looks very bulky.

Another reason for the stability of many arches is that the volume between road and arch is filled in with masonry, which adds rigidity. In fact the masonry spreads a point load in such a way that its effects reach several voussoirs of the arch. The masonry holds the voussoirs together much as the hoops of a wooden barrel hold the staves.  

A similar effect is seen in the brick walls of a small house. The opening for a small window needs no special treatment, even though there may be five metres of brickwork above. But the opening for French windows or patio doors needs a concrete or metal beam at the top, because the brickwork cannot spread the load across such a span.

WindowSmall.jpg (16294 bytes)We see from the bricks above the small opening that the bricks do not have to take the shape of an arch, as long as the line of thrust is within them, or there is enough support around them.

WindowBig.jpg (19203 bytes)But here the opening is so wide that a reinforced concrete beam is needed to support the wall above.

Some houses have a shallow arch above windows and doors. The wall each side has to be able to absorb the side-thrust. A beam contains the tension and the compression within it, rather like an enclosed tied arch or self-anchored suspension bridge.   arch bridges 15

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The other reason for the stability of an arch is that an arch has substantial thickness, so that even with variation in the load, the line of thrust passes through the voussoirs. In other words, the hinges mentioned above do not exist in a stone arch. But we also see that in principle an arch can be stable with up to three hinges, and such arches have been built. In some cases the voussoirs could in principle stand alone if the centring were removed.  "Packhorse bridges" often consisted almost entirely of voussoirs.

For a semicircular arch this cannot be true, because the outward thrust cannot be made to vanish just by curving the arch. If you don't believe this, imagine continuing an arch to more than a semicircle, in which case you are asking the thrust to turn inwards. In fact some Islamic buildings do contain arches that are greater than semicircular, but they always bear against something else, such as other arches.

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Why build arches with hinges?  If a system of parts, such as a truss, is assembled in such a way that it would be inherently rigid if hinged, and all the parts are then rigidly fixed together, then if the fixing is not done exactly, there will be unwanted stresses in the parts. The system is in fact over-determined, or over-constrained, and at least some of the stresses cannot be calculated, so it is statically indeterminate.

For an arch, adding three hinges removes these effects completely, leaving two freely mobile halves propped against each other. Arches have actually been constructed with no hinges, one hinge, two hinges, or three hinges. Some indeterminate structures are provided with jacks, so that the stresses can be controlled. In some cases the foundations are then set in concrete, but in other there remains the possibility of later jacking to correct for subsidence.

The behaviour of arches with changes of temperature depends on the number of hingers. Building large arches such as Sydney Harbour bridge presents a final difficulty when the arch is to be closed. The top and bottom chords may not have the same gap between them if the temperature is not that for which the calculations were done. Heating or cooling may be applied in order to achieve the required closure.

The diagrams below show schematically the possibility of hinges or pins in arches. The one-pin cases are not useful in practice. It should be noted that the word "hinge" should not be taken too literally: if a part can be made narrow enough, the small angular movements it makes will not cause it to crack or crush. In such a case, no actual hinge mechanism need be constructed. Thus, a concrete hinge may not have any parts that look like those of a door hinge.

 

Eiffel's Garabit bridge is a magnificent two-pinned arch. The Hell Gate, Bayonne and Sydney Harbour trusses do not taper to the springings like that of Garabit, but nevertheless the thrust reaches the abutment through the lower chord only, the upper one being only for stiffening at these points. At the crown, however, by closing the truss with suitably dimensioned pieces, the forces can be shared between the upper and lower chords. Therefore, between springing and crown, thrust must migrate between the chords through the bracing struts and ties. See also the page called Arch or Beam.

The next diagram shows a set of designs for three pinned arches and another set for two pinned arches.  Which ones do you think are bad designs? Which, if any, are potentially useful?   arch bridges 17

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M5RailXP.jpg (67131 bytes)Why is this beam bridge resting on such wide frames? The designers could have placed vertical columns directly under the beams. And why were the frames given those shapes? Do you think that the apparently thin walls could hold the pressure of the embankments? So what do you think the walls are like further down?

Transition from beam to arch

In the diagram below we see a transition from a simple beam at the bottom to something like an arch at the top, by adding more voussoirs and making them smaller, with thickness for stability.

The pictures below illustrate some arches with a small number of straight segments.

1 WindowsSmall0018.JPG (21432 bytes)   2 ArchTudor643.jpg (94370 bytes) PointArch3588.jpg (133819 bytes) OddArchB.jpg (75643 bytes)   3 ArchTrapezoid.jpg (76656 bytes)   4 ArchMultiSeg2407.JPG (58697 bytes)

The sloping ends of the slab in the first picture makes clear that arch action is intended. Vertical ends would require great shear strength in the mortar, while horizontal gaps would denote a beam. 

In the third picture, the two "voussoirs" are so thin that they are acting as beams, unlike normal voussoirs, which are much thicker, relative to their lengths. What is meant here by "acting as beams."? It means that the slabs are so long and thin that their undersides are probably in tension, because of the loads pushing on their upper surfaces. Properly designed voussoirs are never in tension, because the mean line of thrust passes through the middle third of their cross-section. This is discussed in other pages.

In a diagram we looked at earlier, we now replace all the straight parts by circular arcs, we get a second set of diagrams.

Note that with an even number of segments, there is no keystone. The keystone of an arch has no significance in engineering (it is not even where the thrust is greatest), though it is sometimes made quite prominent as in these photographs below.

Keystone1.jpg (32841 bytes)  EveshamY1.jpg (63034 bytes)

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ArchTXS.jpg (90980 bytes)This wall has two openings for beams, below, and two for ventilation, above. The latter are primitive arches, such we saw in one of the diagrams above.

ArchNeat.jpg (112194 bytes)This neat arch has two rows of voussoirs, and the designer has chosen the proportions so that 15 voussoirs in the upper row match 14 in the lower row. A massive block at each end carries the vertical and horizontal components of the thrust into the wall. Although keystones are strictly unnecessary, this one lifts what might have been a rather ordinary entrance.

Voussoirs are not actually necessary - they are used in masonry arches for the obvious reason - it makes no sense to carve an arch from solid mass of stone, even if you could carry it to the site. But natural rock arches do exist, carved by natural forces, which will one day go on to reduce these arches to rubble.

NaturalArch.jpg (131337 bytes)The next picture shows a natural arch in the valley of the Mostnica river in Slovenia. This region of the Julian Alps is made of limestone, in which the fast flowing rivers carve all sorts of curves and holes, often by means of whirling stones in cavities. This arch may have been made where two cavities have become large enough to cut into each other. The hole would then have been enlarged by water flowing by and through.

FootBSea.jpg (64217 bytes)RockBridge.jpg (87409 bytes)The wind is responsible for many natural arches, and here are two made by the sea. The sea cuts many more arches that do not lead anywhere; these are caves. But if two of these should join up, a through arch is created.

CrevasseKG.jpg (48963 bytes)DiabIcicle3.jpg (246231 bytes)CrevZV.jpg (117031 bytes)Here are three snow bridges in the Alps. You might not think of them as arches, but their spans are so small compared with their thickness that some arch action may be occurring when they are loaded by a walker. In any case, if a snow bridge is so slender that it is definitely a beam, you do not cross it without great care, because snow is not known for its tensile strength. In the third picture, a large section of the bridge has collapsed into the crevasse. Snow bridges are most safe in the early morning before the sun has had time to start its destructive work. By the afternoon some of these bridges may become very dangerous. Even if you are roped to someone else, getting out if you fall in is not a simple business. One solution is to use jumars, that is, if you are attached by a rope to someone else who hasn't fallen in..

How can we suggest that a snow bridge acts as an arch? Nobody actually designed and built it, arch or no. Well, if it is to act as a beam, it has to take tension near the bottom surface. Snow is not well known for its tensile strength. In the same way, although children of all ages build structures from damp sand on beaches, any openings they make are in the form of arches or tunnels, and not in the form of beams: damp sand, like snow and like masonry, has little tensile strength.

CamSagArch.jpg (211355 bytes)CamSagArch2A.jpg (51082 bytes)People sometimes wonder how the roof of a tunnel can hold up when it is thousands of feet underground. The second of these two pictures shows how little the distortion of even a very flat arch can penetrate into brickwork. Given that many types of rock are far better bonded than bricks and mortar, we can see that the depth is not very important.

In steel, as in stone, it is convenient to build the arch in sections, but a concrete arch may be poured complete, if required.

The way that the line of thrust behaves in a simple unstiffened arch can be seen by looking at this download. (Choose Run from Current Location.)  It simulates loads with random weights moving with random speeds.  In principle, for a masonry arch or concrete arch without reinforcement, the bridge will survive if the line of the thrust remains within the arch. To prevent disappearance of compression at any point, with the possibility of cracks, the line of thrust should stay within the middle third of the arch. With reinforcing or pre-stressing, or with steel arches, the line of thrust is not quite so critical.

The diagram at left is a frame from the download, showing the case of a heavy load, indicated by the arrow. In such a case, the arch would be in tension around the line of the load. See also the page about the funicular. You could say that what distinguishes an arch from other structures is that if it is perfectly funicular, it will experience no bending moments. In practice, for economy of construction, there may be slight deviations from the funicular. And for reasons of practicality, such as headroom, there may be large deviations.   arch bridges 19

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Note how the line of thrust moves towards the force. The bending moment thus created is proportional to the vertical distance of the thrust line from the centre line of the arch. It is also proportional to the horizontal thrust, which is the same throughout an arch. Deliberate moving of the line was used by medieval builders of cathedrals - they added weight to walls and to buttresses in order to change the shape of the line of thrust to better fit the shape of the structures.

If the line of thrust passes through the centre line, there is no bending moment.  Therefore, if at some place an arch comes to a point, such as a hinge, there cannot be a bending moment there. That makes sense, as a hinge cannot resist such a moment.

The next picture shows an even more extreme case, with the line of thrust outside the arch. The position of a crack is shown. If the arch were to crack at two other points, thus producing three extra hinges, the risk of collapse would be very real.

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At this point you will be asking why the line of thrust is so thin. In fact you should ask questions all the time. What you read, hear or see in any medium - including, of course, this web-site, could just be wrong . . . . .

The answer in this case is that the line of thrust is not thin. It was drawn like that for simplicity. The force is really spread throughout the arch. The "line of thrust" is just the average line, just as the "centre of gravity" of an object is the average of all the places where the weight acts. Now we will do it more exactly, and we will understand a little more. Meanwhile, if you don't believe in lines of thrust, stand up straight and get someone to push you in the back with steadily increasing force. You will find yourself rising on to your toes when the line of thrust goes too far forward. And if you carry an extremely heavy object in one hand, you automatically lean over.

Getting back to arches, look at the diagrams below, in which the compression is indicated by the brightness of the red hue. Going down the diagram, the point force at the crown of the arch is increasing in size.

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In the lowest diagram the compressive stress just reaches zero at the top of the soffit (the underside of the arch). Any more weight on the arch, and tension would occur. The single line we saw before is actually the average of the red paths. The density of the red, that is, the compressive stress, varies linearly through the arch, and in the case where it runs from zero to a maximum, the mean line is one third of the distance from the top surface to the bottom.

This is an example of the rule of the middle third, which states that if the line of thrust goes outside the middle third of a section, tension will occur. The implications of this are striking. To get a sumo wrestler's foot off the ground you only have to get his weight outside the middle third of his baseline, and that's if you aren't using his momentum. No wonder they place their feet so far apart. We see also the implication for the first Tay bridge and for Eiffel's arches and tower.

Click here to download a simulation showing the behaviour of the thrustline when a load moves across an arch.

And for a two dimensional section, such as that of a tower or a column, the middle third in two directions means the middle ninth in area. Now look at a picture of the leaning tower of Pisa. And think about the skills of medieval builders of religious buildings, which are described in another page of this web-site.

What is the correct shape for an arch? If the cross section is uniform, and no loads are present, apart from the self-weight, the shape should be a catenary, the shape taken by a uniform flexible cable, hanging between two points.

This diagram shows an arch based on a catenary, the red curve.

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

Real arches, of course, carry loads, and they very often carry weight between the voussoirs and the road. The distribution of these loads means that many different shapes have been successfully used. And if the bridge is made of steel and not masonry, its rigidity enables it to deviate hugely from the funicular, the ideal line of thrust, for example to obtain headroom over a wide road.

Because all arches generate thrust at all points in the arc, they cannot stand up until they are complete. One solution is to build falsework, called centring.  Another, less common, is to hold back the two halves using cables, until they are ready to meet. This is done with steel bridges, but not with concrete or masonry arches. The two URLs given here link to a picture by Canaletto, "London seen through an arch of Westminster bridge". The bridge had been built, but the centring had not yet been removed. A bucket hanging from the timber gives some idea of the scale.   Canaletto 1   Canaletto 2    arch bridges 22

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Arches are found in many shapes, but one shape is special - the catenary, because this shape is the same as its own differential coefficient. Here are some shapes for catenary arches, based on the assumptions that all the strength is in the arch, and that the masonry on top is comprises piles of blocks which contribute no horizontal force.

Few catenary arches are actually built, for several reasons. Firstly, the shape is more difficult to make than a circular arc, and secondly, the masonry contributes some stiffness, which enables a variety of shapes and weight distributions to be used. By making an arch using metal, especially if it is trussed, more freedom is obtained in creating shapes. A third, and important, reason is that most arches carry something besides themselves. The weight distribution of those loads will change the shape of the funicular. Thus the ideal shape of an arch, like the ideal curve of a suspension bridge, is rarely a catenary. The exceptions are monuments such as the Gateway arch in St Louis, which comprise only an arch. More about this in funicular and in the next section.  arch bridges 24

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3PinM6.jpg (20340 bytes)3PinGlos1.jpg (79053 bytes)No law of physics forbids the line of thrust to be outside the structure. What is true is that in such a case, you cannot use materials that can only resist compression, such as masonry and unreinforced concrete. You need materials like reinforced concrete or steel, which can sustain tension, and therefore withstand bending moments, as in the two examples shown. Notice how the bridges are thickest where the deviation is greatest. This has two benefits. Firstly, the stiffness is increased where needed, and secondly, the extra weight moves the thrust line closer to the structure, which reduces the bending moment. This is reminiscent of the spires that were often placed on the piers of flying buttresses to move the funicular.

The line of thrust is always raised by the load, mainly near the centre. So we might think that the arch should be built with the dead-load thrust near the lower edge of the arch. We might also expect that some arches might taper almost to a point at the abutment, in view of the behaviour of the computer simulation mentioned above. This is not practical with masonry, but it is with iron and steel, as Gustav Eiffel showed.  Maillart tapered a number of reinforced concrete arches early in the 20th century.

The next diagram shows a voussoir arch that is overloaded at the crown, making the line of thrust fall outside the arch. Three hinges have formed around this point, as well as two more at the abutments. Stability is impossible with more than three hinges. Is this diagram correct? Would hinges really form at the abutments, or would they form elsewhere? A similar effect could be produced by an overload at any point in the arch.  

This diagram demonstrates two points - firstly that extra load does not necessarily increase the stress at all points in a structure, and secondly, that reduced stress is not always desirable. If you don't balance your trailer correctly, you can reduce the load on the front wheels of your car, but this doesn't do much for the steering.

If you look at the pages about beams and pre-stressing, you will see that beams made of concrete cannot withstand tension, a failing that is overcome by compressing the beams using pre-stressing wires. We could imagine that the arch is a type of beam which is pre-stressed by curving and letting its own weight do the compressing.

Strange  Arches

FunnyArchFY.jpg (150631 bytes)Here is a gateway with, apparently, an arch above it. But it the two ends of the arch, the bricks have been tapered the wrong way, so that the junctions with the walls are vertical. So the weight of the lintel has to be borne entirely by shear stress within the mortar. The flatness of the arch produces a large thrust into the mortar. Perhaps this helps to hold the thing together. But it is not an ideal way of doing things. In fact it is possible for such a structure to hold up even if the mortar loses its adhesion. If the angle from the horizontal is small enough, in relation to the coefficient of friction at the ends, the arch will stay up. This principle is used by climbers in the method known as bridging, in which they can climb a vertical chimney or a vertical crack by bridging across it, using the legs or the legs and body.

ArchFunnyAA.jpg (102682 bytes)And here is one that errs the other way. None of the voussoirs are at right angles to the curve. The yellow lines show the slight sag which so often occurs. The archway has been bricked up.  arch bridges 25

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