Masonry Arches

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The masonry arch is one of the oldest types of bridge. The first known ones were probably built in the middle east around 5500 years ago. Books often refer to something called a corbelled arch, which is not an arch under the usual definition.

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Corbelled Arches

TinternCorbelled414.jpg (190230 bytes)A corbel is a stone which projects from masonry. It is a simple example of what has come to be called a cantilever. A corbelled arch looks something like the diagram at left. Working up from the base it looks like a series of cantilevered projections, but at the top, where the two sides meet, they presumably press against each other, creating horizontal thrust that must be restrained somewhere down in the base. The photograph shows a sort of corbelled opening that has been created by the disappearance of a set of voussoirs. It appears that two or more openings of different size and shape have been employed at different times.

An example of a corbelled structure is the traditional Ossetian tomb, which has four walls that are stepped inwards towards the top, making a shape rather like a gothic arch. These walls are closer to the funicular than is the simple corbelled arch, but the mortar between the blocks is still partly in shear, though the steepness and thickness of the walls makes this comment rather academic. The funicular is the line of action of the forces in a structure: in an arch it is the line of thrust. The further a structure departs from the funicular, the greater the penalty in terms of bending moment and shear stress. An ideal arch is one which follows the funicular exactly.

We can see from the diagram at left that the weight of the red area is not inside the supporting base, and so there would probably have been falsework to support the blocks. When the falsework was removed, there would have been some tendency for the two overhanging sections to press against each other.

When the work is finished, the tier of blocks above the red section is complete, but the red section is prevented from sagging only by adhesion to the tier above, and by adhesion within own volume and to the side. It is clear that this construction will be stable only for small spans and steeply inclined sides.

William Hogarth's satirical frontispiece to Dr Brook Taylor's Methods of Perspective includes not only ridiculous examples of wrong perspective, but a bridge that could not stand up, because it is apparently built from parallel courses of blocks. David Hockney copies the bridge in Kerby (After Hogarth) Useful Knowledge. Hockney has pointed out that the errors in perspective can actually create space, just as correct perspective does, but differently. So both Hogarth (with the bridge) and Hockney (with the perspective) remind us how very different an artist and an engineer may look at the world.

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True Arches

Most of us know what an arch looks like, but how do we define what makes something an arch? Perhaps that will become apparent if we look at the properties and behaviour of arches. One advantage of the masonry arch is its ability to withstand considerable deformation caused by earth movement. It can be made with blocks that are not accurately made, provided the gaps are filled with adequately strong mortar or cement.

The arch is the only bridge that can be made entirely with masonry, because it is the only bridge that is not subject to tensile stresses, which masonry cannot withstand very well. Yet no structure can be made which includes only tension or only compression, because the net force on any stable object must be zero. The answer is that the tensile stresses are in the ground under the arch, though they are relatively weak, except near the abutments, because they are diffused over a volume that is bigger than the arch itself.  

Over1.jpg (28255 bytes) We speak of normal arches and tied arches, but every arch is a tied arch - it's just that in most arches the earth is the tie. The ground must never be forgotten in designing a structure. The picture at left shows an arch that has sagged by about 25 cm at the crown because of earth movement.

The ruined arch we saw above shows that masonry does have some ability to withstand tension and shear. Sometimes you see a portion of a collapsed arch that projects out like a bracket. The position of the break shows the point at which the masonry could not withstand the tension. Once the remainder of the arch has fallen away, the remainder experiences much smaller forces, and will often remain for hundreds of years if undisturbed.

The masonry arch is one of the types of bridge that is built less often than in the past, because there are cheaper and simpler ways of spanning the gaps. Therefore we should consider the possibility of conserving at least a proportion of these bridges, some of which are very elegant indeed, whether in brick or stone. Here are pictures of a few examples, the first of which was taken by David Newton.

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SmardaleBlocksSS.jpg (321508 bytes)This picture by David Newton shows a short section of a 19th century railway viaduct. It exhibits the main features that we associate with masonry arches. The superstructure, which rests on piers, comprises two parts - firstly, sets of stone blocks called voussoirs, arranged in curved shapes, and secondly, normal masonry filling the volumes, called spandrels, between the curves. The first shape used, about 5500 years ago, was the semicircle; it was still in use 3500 years later in the days of the Roman empire.

It was long after that before people began to discover that other shapes would work as well, and in some cases better, than the semicircle. Pointed arches, segmental arches and elliptical arches are just a few of the shapes that have been employed. Islamic mosques and Christian churches are among the buildings which have made imaginative use of the arch and its three-dimensional equivalent, the dome.

BredwardineAMirror2.jpg (382046 bytes)BredwardineElliptic.jpg (279330 bytes)BredwardineEllipticB.jpg (298649 bytes)CulhamEllipse216.jpg (148645 bytes)Elliptical arches have been used to keep height reasonable, to reduce approach gradients, and to increase the width of gauge clearance below. The first three pictures show a river bridge, and what it could look like with elliptical arches. In practice tall narrow elliptical arches are not built: it is cheaper and simpler to put semicircular arches on taller piers. The last picture shows an elliptical arch used to gain width over gauge without increasing approach road gradients.  I K Brunel built two extremely flat masonry arches over the Thames near Maindenhead; they are still in use.

What is the essence of the arch? Whether we look at a Roman arch across the Tiber, or a mighty railway viaduct, we see the massive strength and solidity. We might think that rigidity is the essence of the arch, but the Pont du Gard tells us otherwise: it contains no cement or mortar, except in the top tier: it is literally a pile of stones.

Having used a few technical terms, we had better provide a picture that shows how they are applied. The drawing was done badly in one place, but luckily that shows a type of distortion that is not unusual in arches.  In fact, if you look carefully at a lot of masonry arches, you will see distortions in a significant proportion of them.

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What do we mean by arch action? Consider the staves of a barrel. They are compressed together by the iron hoops. The voussoirs of an arch are held together by the pressure of the masonry, which is derived from the weight. Actually, some voussoir sets can stand on their own, pressed together solely by their own weight. One shape that works in that case is the catenary, the same shape that works for a uniform hanging cable. If the masonry were replaced by single monolithic blocks, one each side, the voussoirs would not experience the right forces unless every single part had been cut very accurately to shape. The total length of the voussoirs could be slightly too big or too small, for example. Of course, the forces created by the weight would change the dimensions of every part, and they might then fit together, but the forces would not be as designed. And if the original fit were too poor, the forces would not be enough. The actual bridge works because the masonry is not rigid until the mortar or cement has cured, and the blocks or bricks bed down on each other while the bonding is fluid.

Why are there two kinds of blocks, voussoirs and the rest? If the bonding between mortar and blocks were as strong as the blocks, and if the resistance of the mortar to shear were also as strong, you wouldn't need voussoirs at all. What the voussoirs do is to provided a load path for the thrust in which all the joints are at right angles to the thrust. All that is needed is for the line of thrust to lie at all points within in the voussoirs, and preferably in the middle third of the section. However, since the masonry isn't infinitely weak, the thrust can leak into the spandrel region, within reason, provided that the abutments are capable of receiving it.

The next two pictures illustrate the forces in a barrel, and the third illustrates a cartwheel - red for compression, blue for tension.

These diagrams do not do justice to the skills of the cooper or the wheelwright.

CartwheelUD.jpg (91126 bytes)Note that the cartwheel spokes are not shown as being either in tension or in compression. They could well be protected from compression by the rim. It is even possible to employ springy spokes. In use, the forces in the spokes vary as the wheel goes round. Some ancient people stored their chariot wheels off the ground to prevent deformation by creep. Note that the rim of a bicycle wheel is in compression, and the spokes are in tension at all times, though the forces vary as the wheel rotates on the ground.

GrindstoneJuneA.jpg (245269 bytes)The picture shows a grindstone which is held together by the tension in the iron rim. The behaviour of the compressed blocks is similar to that of the voussoirs in an arch. Both the grindstone and the cartwheel rely on maintenance of the stressing produced by the rim.

It is the separateness of the masonry blocks that enables them to press against the voussoirs correctly, though the forces remain about the same after the mortar has set. In fact, the space between the faces of the spandrels could be, and sometimes is, filled with rubble. If the deck were stiff enough, the space could even be filled with water. The masonry fits the arch properly because the voussoirs are placed first, on centering, and then the masonry is added. For the Pont du Gard, the masonry blocks must have been made very accurately, because there is no mortar to take up any errors, except in the third tier.

VintgarArch.jpg (432065 bytes)Sometimes the spandrels of large arches are open, except for walls which reach up to support the deck. These walls may support smaller arches.  It is possible that this example may be a concrete arch faced with masonry.  This is perhaps one of those examples, like the railway viaducts depicted earlier, which seem to enhance the landscape. A splendid site containing an unworthy bridge is sad to see, for there is little possibility of change unless the bridge becomes unsafe.

The next deal with semicircular arches, segmental arches, and pointed arches.

Here, the forces in the masonry increase from top to bottom in a smooth manner. The voussoirs transmit the force to the foundations at the springing.

In the segmental arch, the forces in the voussoirs are injected back into the layered masonry at the springings. A larger block at those points helps to spread the load. Calculating the forces in any masonry arch is complicated, and in fact was probably never done in detail. The sums are made more difficult by the presence of both stone and mortar, which may have different elastic properties. In the case of mortarless arches, friction is the only force which can transmit shear between blocks, though in a heavy structure, the friction can be very great.

Aelius.jpg (63926 bytes)Bristol531.jpg (140663 bytes)Here are two pictures of masonry arches. One shows an ancient Roman bridge, the Pons Aurelius, with wide piers and and semicircular arches, a very conservative and long-lasting design, though in fact it has been rebuilt in the middle ages. The other shows a 19th century segmental arch with narrow piers, giving good clearance for the loading gauge of the railway. The Romans may not have realised that piers can be used only to take the weight, leaving the thrust to be transmitted through the arches from end to end. But perhaps they did know, and chose to build conservatively, so that in case of arches being damaged water or war, the remainder would survive.

Mortar or cement may differ from stone, not only in their elastic properties, but also in thermal properties and in ability to absorb water. These differences can produce damaging effects with during weather cycles; if the stone is pervious to water, special care may be needed in choosing the bonding material. Limestone is a good example of a pervious rock.

Finally, here is a pointed arch, much used in buildings with religious purposes, as well as in other buildings, and also in medieval bridges.

You can find more explanations about the forces in a masonry arch in the page about funiculars, but we must remember that if we regard the whole structure of blocks and mortar as monolithic with no cracks, then the idea of the funicular may not be very useful in describing the action.

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Building a masonry arch

Many bridges can be built by bracketing out from either end until the halves meet in the middle. Sydney Harbour bridge and the Mississippi Eads bridge are examples.  Cantilever bridges are built in this way, and so are many box girders. Cable-stayed bridges are built out from the towers in stages.

But masonry arches exist either complete or not at all. The secret is that the entire weight during assembly is taken by falsework, or centring as it is called in the case of an arch. Traditionally it took the form of a wooden truss in the shape of an arch. It has to be strong enough to hold the weight of the structure without deflecting unduly. The diagram at left hints at the type of thing that can be made.

SmardaleBlocksSS.jpg (321508 bytes)Sometimes the centring could rest on the ground, with suitable foundations to spread the load, but in the case of tall piers it could rest on corbels - blocks which project from the main mass of masonry, as the picture shows. Once the centring was complete, the vousoirs could be laid and cemented into place. Then the spandrel masonry was placed, and the structure was left alone while the cement or mortar was curing. Eventually the centring could be struck.  Sometimes it was eased in several stages.

Over1.jpg (28255 bytes)CheltArch2.jpg (47381 bytes)In any case, the arch would always deflect slightly when the centring was removed, because the forces were completely different. The examples show this clearly. It was usually the result of poor understanding of the ground.

Click here to see photographs of centring for the Nicholson bridge.

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More about the way arches work

This picture shows an opening spanned by a lintel, which is supported at the ends and loaded throughout its length. It is a beam, and is in compression along the top and in tension along the bottom. These forces are developed as the beam bends under the load. The beam bends until the internal forces are in equilibrium with the external ones.

PointArch3588.jpg (133819 bytes)Next we look at an adaptation of the corbelled opening. The "voussoirs" are so long and narrow that they must be acting as beams. The overall behaviour can be thought of approximately as a three-pinned arch. Now there are two beams, but this time, in addition to bending forces, they experience forces from the ends, the weight from above, and the support from the piers underneath. So there is some strut action as well as the beam action. 

We will take this idea one stage further in the diagram at left  We now have three strut/beams. They are shorter than before, and so they bend less, but they still experience the compressive forces. Even the lintel is pushed inwards from each end, because it no longer simply sits on the blocks below: the angled ends ensure that. If we go on with this process, using smaller and smaller wedges, the bending becomes negligible, and the compression remains as the only force. More mathematically, the voussoirs become better matched to the funicular.

This picture is like the previous one, but with each strut/beam replaced by two smaller ones.

The practice of using voussoirs arose, of course, because of the limitations imposed by the nature of rock and the cost of cutting it. In theory, there is no need for voussoirs to be short if they are carved to the correct curved arch shape. But a long curved block means a waste of stone, especially if a block is spoiled during the shaping process.

The advent of concrete was what made small voussoirs unnecessary: large concrete arches can be made by pouring concrete into moulds made with strong wooden shuttering. The pictures below are of concrete arches.

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We have moved away from the subject of masonry bridges.  That's how it is with structures - they are all related to each other, however tenuously.

Finally, a diagram to show the forces acting on one part of an arch. Every part of every structure, big or small, has to be in equilibrium, except of course, when variable live loads are applied.

 

Here we see the two forces from the neighbouring parts of the arch, the inward force from the spandrel, and the weight of the arch section itself, in green. The forces as drawn clearly don't balance: the forces into the ends of the voussoir have to be much bigger for this to happen. This is not surprising, since the whole weight of the bridge acts through the voussoirs. Even at the crown of the arch, the force is still great, although there is little weight above. This is the case because the horizontal thrust runs through all the voussoirs. Fortunately, stone and brick are immensely strong in compression.

The curvature of an arch is a direct consequence of the forces that push inwards towards its centre. Any structure in which the forces act inwards will be in compression. Examples include submarines, submersibles and bathyscaphes, arches and domes. And if the pressure inside is greater than the pressure outside, the structure is in tension. Examples include pressurised aircraft and spacecraft, children's balloons, rubber tyres. The tyre is exceptional in that it is not topologically equivalent to a sphere, having a hole through it. Nevertheless, the net curvature of a tyre is always of the same polarity, because the curvature around the axle is less than the curvature around the narrow dimension.

glosfanb.jpg (78210 bytes)OgeeArchXTS.jpg (118115 bytes)OgeeArchXTSX.jpg (119205 bytes)So what are we to make of the arch shown here, which changes the direction of its curvature?  It's a fake. The idea is to convey the feeling of thrust coming down from fan vaulting overhead, and splitting around the arched opening. But the wrong-way curvature at the top cannot work, because the pressure is at all times inward. Note, however, the great thickness of the voussoirs - much thicker than in other parts of the building. The effect is that they contain a true arch within them, as the third picture shows. You can make an arch of many different shapes, as long as the correct shape is contained well within the one you make.

That's not to say that you cannot make things curve as you wish. The rule only says that if you change the direction of curvature you don't have an arch: what you have is something that has bending forces within it, which implies tension as well as compression, and that rules out masonry. The more your structure deviates from the natural path of the forces, the greater those forces become. Unfortunately, such deviations are often unavoidable. If we supported the floors of a building by arches, there would be great intrusion into the space below. The same applies to suspension by cables, and so we are more or less obliged to use wall-and-beam construction, often known as trabeate architecture. Where, however, roof and ceiling are one and the same, we can create a dome, which is an ideal engineering shape, whether we are building an igloo, a tent or an enormous building. Even then, some fakery may be applied: the dome of St Paul's cathedral in London that you see from the outside is not the one that you see from the inside. And between the two is yet another dome that bears most of the load.

In the example shown above, the engineering solution would look something like the diagram at left. This might have been rejected because of the low headroom available.  Note the way that the pointed arch bears the force from above. Many pointed arches bear no such concentrated load, and are employed purely for architectural reasons. The ridges of Sydney Opera house are among the largest examples. Compare these with the shape of the Gateway Arch in St Louis, which is an engineering shape.

 

Of all the kinds of masonry bridge that exist, one fascinates people perhaps more than any other; it is the skew bridge, occasionally (and wrongly) called the "screw bridge", but not inaccurately, because it is one of the few instances of the helix in structural design. You can read about it here - Skew Arches.

For pictures of masonry arches please see Masonry Arch Photographs.

Concrete arches are described in a separate page.

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Stress in a Masonry Arch

There are not enough calculations in this website, so let's put one in here. We will calculate the maximum stress in a masonry arch.  Here is the diagram.

In a sense, we don't need to calculate the stress in a masonry arch, because we know that in a mine which is hundreds of metres deep, the rock is not crushed, so the rock in a mere bridge cannot be anywhere near its limit. Nevertheless, the calculation may be interesting in showing up the important variables. The stress will be very high at the springing of the voussoirs, because all the weight of the span comes down to those areas. From these two points, the forces in the abutments spread out, and the stress becomes less.

The calculation will only be approximate, and we will use a parabolic curve for simplicity.

The area of masonry above the span is then S(C + H/3), and if the width of the bridge is W, the volume is SW(C + H/3).

If the density of the stone is D and the acceleration due to gravity is g, the weight of the supported stone is SW(C + H/3)Dg.

One half of this weight is supported at each end, and if the angle of the arch from the horizontal is A at the springing, the force F at the abutments is given by

F=SW(C + H/3)Dg/(2sinA).

The area for support is TW, and so the stress at the abutment is

SW(C + H/3)Dg/(2TWsinA).

This simplifies to

S(C + H/3)Dg/(2TsinA).

Rearranging, we get

0.5(S/T)[(C + H/3)Dg]/sinA

The first brackets enclose the ratio span : voussoir thickness, and the square brackets enclose the weight. Finally, the sine of the angle is important too.  So large stress is associated with a high ratio of S : T, a high weight, and a small angle, that is, a flat arch.

If we insert some typical values, we find that the stress is far below the limit of the material.

Failure of Masonry Arches

CirenArchesL.jpg (114627 bytes)The stress being low, failure of of the material is rare in masonry arches.  A more likely mode is through the formation of hinges. In the page about trusses we see that a triangle of members (with three hinges) is stable, but a quadrilateral (with four) is not. Under a heavy concentrated load, an arch may develop a downward deflection that is mirrored by an upward one in the other half, if the line of thrust moves too far from the centre-line of the voussoirs. To some extent, the material between the roadway and the voussoirs will spread the load. The formation of four hinges is shown diagrammatically at left. The second picture shows a real example.

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A Bridge Needing Repair

 

Skew825.jpg (185876 bytes)Skew826.jpg (197854 bytes)Skew822.jpg (155768 bytes)These pictures show an arch bridge after some voussoirs had fallen off. If even one voussoir falls out, the compressive force on all the others in its row are to some extent relieved, and those nearest to the gap are held in place only by the adhesion of the mortar. The release of stress means a release of strain, and that implies that the relieved row of voussoirs tends to become slightly longer than the neighbouring row. There will thus be shear stress between the broken row and the unbroken row. This shear stress will in fact tend to share the compressive stress between the two rows, especially far from the gap.

Another repair job?

VoussoirBath5307.JPG (186620 bytes)VoussoirBath5308.JPG (163518 bytes)These two pictures apparently show a voussoir that has been cracked and displaced. But given the huge pressure exerted on voussoirs by their neighbours, it is not easy to understand how this could have happened. Looking under the bridge, however, reveals that the arch is in fact built in concrete, and the lovely Cotswold limestone blocks are only a facade, cemented on the concrete. The blocks that look like voussoirs are attached to the concrete like all the others. The cracked block might have fallen right out, but for the fact that it is resting on the ground.

There are implications for a repair. If any structure contains more than the minimal set of parts that will keep it standing, loss of a part may not be disastrous: the stresses are simply shared among other parts. But when a repair is attempted, simply fitting a new part is not enough: in theory, the gap should be jacked until the stresses are similar to their values before the break. Sometimes the loss of stress is irreparable. When the Britannia bridge over the Menai Straits caught fire, the internal stressing put in by Stephenson was lost, and the spans now have arches to support them. In the example of the skew bridge, any one row of voussoirs is so lightly stressed that simple cementing new bricks is probably an adequate repair. An example where repair is difficult is a large piece of pottery that breaks, and the pieces are found not to match when held together. This happens when internal stresses and strains were locked in during the cooling process after firing. They are released by the breaking, and the parts take slightly different shapes.

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Types of masonry arches and their construction

Interesting page about arches and domes

Notes about arch design

Interesting article about masonry arches

Analysis of masonry arches

Interesting catenoid roof

Analysis of masonry arch failure

Some ancient arches, including a very flat elliptical arch

Masonry arch glossary