Skew Arches

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Railway2488RS.jpg (111324 bytes)Motorway1.jpg (72612 bytes)Railways (first picture), like motorways (second picture), do not allow of sharp bends. Acts of Parliament were often needed to allow purchase of land to allow the line to sweep gracefully across the country. And on a map of any town centre that has a railway station, you can see that the long curves or straights of the tracks cut right across the street plan, meeting streets at all kinds of angles. Some of these streets may themselves be based on ancient straight routes created by the Romans. And so we find the kind of bridge that produces wonder or admiration in anyone who sees it. This is the skew bridge made in brick.

Streets that were built after the railway tend to fit in with the existing routes, and may cross the railway at right angles. This is discussed in the page on cracks.  Nowadays, beam bridges are far more often used than brick arches.  So skew arches tend to be fairly old.

In principle, a road could be curved to fit the railway, but in an existing town plan this is often impossible. In the country there are indeed numerous examples of roads being kinked to go over or under a railway, introducing double blind bends. Railways over canals are often skewed, because canals, like railways, should not be sharply curved, as boats and barges can only turn slowly. 

Small arches in towns are often semicircular, and are therefore sections of cylinders. To cross a road at an angle other than ninety degrees, the span of an arch could be made longer than the width of the road, as shown below. 

Unfortunately, long spans are much more expensive than short ones. And if the cross-sections use a given fraction of a circle, they are higher, so they would have to be elliptical to keep the height down. This would add complication in terms of building centring.

 

The obvious solution is to make the cylinder parallel to the road, as the next diagram shows.

The next diagram shows that another problem now arises. If we consider one row of voussoirs, we see that they do not go right across the span, and therefore cannot transmit the thrust. With a smaller skew, some, but not all, of the rows would reach across the span. We are neglecting here the alternation that is used to bond bricks together, as this bonding does not affect the line of thrust.

We can use another solution; make the voussoirs parallel with the bridge and not with the road, as below, where two sets are shown.

 

If we now make many parallel sets of voussoirs we find three new problems.

Firstly, there will be an increasingly zig-zag surface under the bridge towards the springing, so each set of voussoirs will require separate centring to build it. This will be expensive and complicated.

Secondly, there will no means of bonding the sets of voussoirs together in the usual manner of bricklaying.

Thirdly, if the centring is semicircular, the resulting arch will have an elliptical profile, taller than a circular arc. This will be inconvenient, requiring higher, longer, and more expensive approaches, especially for railways. By making the centring elliptical, the sets of voussoirs can be flattened, to achieve a semicircular arch, but the woodwork would be more expensive to make than circular forms.

The actual solution adopted is to build normal cylindrical centring, and to lay the courses of bricks in the normal alternating way, but setting them to line up with the thrust-line at the crown. The inner surface of the arch is then smooth all over, but as the springing as approached, the lines of bricks become increasingly tilted. You can see how this works by cutting a tilted rectangle from a piece of graph-paper and rolling it into a half-cylinder. The diagram below shows a flat layer of bricks.

In the next picture the bricks have been wrapped around a cylinder.

 

We have too many bricks, so in the next picture the superfluous ones have been left out, leaving exactly half a cylinder, seen in plan and elevation (one half) below. Why are the rows of bricks tilted at the springing? The reason is that they are at the same angle to the axis of the cylinder at all points, but we don't notice this except near the bottom. The edges of the bricks lie on orthogonal helices: there are no straight lines.

We still have problems. In this plan view we would like to see straight edges to the bridge, and we need the arch to be vertical in both planes at the springing. The two problems are of course directly related by geometry.

The rectangular layer of bricks has been mapped on to a helical layer. In order to get a plane surface cutting a cylinder, we need an ellipse, not a helix. The only way out is to cheat: we need to add some bricks in some places and remove them in others.  

Our difficulty is one that cartographers will recognize: to map one surface on to another can only be done in certain ways without distortion. For a sphere on to a plane there are no ways at all to do it without distortion. At least with a cylinder we can actually succeed.

The final result is shown in the diagram below.

 

The diagram above shows clearly that whole bricks cannot match the entire arch. We have not completely solved the problem of incomplete rows of voussoirs, but for any reasonable skew, the effect is marginal.

The next picture shows the curved layer of bricks opened out to a flat surface. The scale has been reduced by a factor of two to get the picture in a reasonable space.

The next diagram is a side view (perpendicular to the under-road) of one half of a practical skew arch.

In practice, skew arches are seldom semicircular, an arc of about one quarter of a circle being much more common. This greatly reduces the visual effects of the skew geometry, as we see below.

Just to show that these shapes work - here are some photographs of a model made by printing the pattern on a sheet of plastic film.

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If we imagine a skew bridge that is very narrow, for example, carrying a single railway track, we can imagine that the stresses are more or less parallel to the faces of the bridge. But if we imagine a very wide bridge, carrying four tracks, this becomes less obvious, and if we imagine a bridge that is so wide that going under it is like going through a tunnel, then it becomes reasonable to suppose that the stresses in the middle will not be closely related to the faces: they will tend to be at right angles to the axis.

This means that if the stresses near the faces of a skew bridge are parallel to those faces, and the stresses in the middle are at right angles to the axis, then the stresses are not uniform. There will be stress concentrations one springing at each end of the bridge. In a plan view, the arch is a parallelogram, and the stress concentrations will be at the ends of the short diagonal. The effects will be stronger with greater angle of skew.  This effect may have been responsible for the collapse of the King's Bridge at Bendigo.

 

At the ends of the bridge the builder has either to cut bricks into odd shapes, or to allow parts of bricks to protrude from the surface in a zig-zag manner. Both methods have been used. Sometimes the builders place complete sets of whole bricks at the ends, parallel to the faces of the bridge, to achieve a tidy appearance.

If the bridge were to cross the road at a suitable angle, the use of hexagonal bricks would allow the pattern to match in every way, with the numbers of bricks in the two sets of helices being consecutive Fibonacci numbers. The page on Nature's Maths explains this.  The arrangement of parts around a pineapple gives some idea of the required pattern.

The photograph below shows how the bricks look near the springing of an arch near Catherine Street, Gloucester. The horizontal lines are the result of vehicles that were too high, being driven too near the side. The old bridge has suffered from the effects of water percolating through and leaving deposits which obscure the unconformity between the piers and the vault. It is indicated by a dashed white line near the bottom of the picture.

The bridge has a peculiar asymmetrical chamfer, shown by the dashed white line in the first picture below. The other pictures show this bridge and two other bridges with the same feature. The lighting is critical in making this kind of detail clear.

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Here are more pictures of the same bridge. One shows places (arrowed) where new bricks have been added to repair eroded ones. The new ones are ordinary bricks, which break the line of the chamfer. This is a simple example the effect of specialist construction on the difficulty and cost of maintenance.

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A Bridge Needing Repair

Skew825.jpg (185876 bytes)Skew826.jpg (197854 bytes)Skew822.jpg (155768 bytes)These pictures show the same bridge at a later date, 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.

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.

SkewTY.jpg (63670 bytes)This picture shows that the bridge pier was originally pierced by two arches, which have been filled in. Behind this pier is another arch of the bridge, which is occupied, as is common, by a small workshop.

A38SkewNA.jpg (193429 bytes)A38SkewSA.jpg (137088 bytes)Between Bristol and Gloucester, the railway line and the A38 road have to cross, because the railway curves around high ground. The crossing angle is about 48 degrees from the perpendicular in both places, resulting in extremely skewed bridges. The chamfer mentioned above is here much wider, to avoid the very sharp 42 degree corners that would otherwise have been built. The pictures were taken on a very dull day, and are consequently very poor. The first picture shoes the angle of skew, while the second shows the chamfered soffit, and also huge amounts of distortion, though the bridge still carries much heavier traffic than that for which it was designed. Why is the chamfer employed?  The next diagrams show horizontal sections through a bridge with a skew of 45 degrees, near the springing. The chamfer eliminates a narrow nose of brick that would be vulnerable to weathering and crumbling, which would be accelerated if the bricks were in compression. If they were not in compression they would be redundant in any case. Such a small area can contribute little to the strength of the bridge.

The pictures below show an unusual bridge near the Cotswold Water Park. It crosses the line of a disused canal, and has a skew of around 10º. The builders actually set the masonry courses parallel to the piers, and so they make slight zig-zags at the surfaces, as the fourth picture shows. The steps are of about the same size throughout, though if the arch were a complete semicircle, the steps would decrease to nothing at the springings. In a genuine skew arch construction the steps would vary from zero at the crown to a maximum at the springings. Like the previous bridge this one has arches that are only a fraction of a semicircle.

The holes in the piers were built with skew, which would seem to be unnecessary.  This makes their surfaces all a part of a single long tubular shape.

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CWPark5.jpg (91629 bytes)This picture shows clearly how the holes are parallel with the road above, and not with the piers. It also shows that, as with the main arches, their bricks are not built on a skew basis, but parallel with the holes, leading to the same uniform zig-zags as in the main arches. The middle picture below shows that subsidence has occurred. It is a tilted and squashed version of the left hand picture to show the dislocation in the line of the bridge. Within the same area there are two other bridges of the same construction. 

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Near Andoversford, Gloucestershire, two genuine skew arches with a skew of about 35° mark the route of a dismantled railway line. The bridges are similar, and have zig-zags at the ends. The one nearer to Andoversford has been repaired and pointed, but the other is showing signs of distress, with numerous broken and missing bricks, cracks, and even growing plants. Here are some pictures.

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All the bridges shown so far encompass only about a quarter of a circle. The next set of pictures show a bridge near Swindon, carrying a disused railway track (now a footpath) over a disused canal (now a very long pond). Much of the original brickwork has been replaced, and this has not been done with complete regard for the alignment along the faces of the arch. This bridge is located at a sharp bend in the canal. Had it been built only a few yards along the canal, it need not have been built on a skew.

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Note that the angle at which the bricks of arch meet the bricks of the side walls is the same as the angle between the line of the bridge and the perpendicular to the railway or road below. If the bricks meet at an angle of 35º, the bridge is at an angle of 65º to the road.

SwindonSkewA2X.jpg (110643 bytes)SwindonSkewA2BY.jpg (436573 bytes)Zoons3.jpg (51456 bytes)The bridge has one more surprise for us. The first picture at left, squeezed horizontally, shows that the bridge is distorted. Remember that as a railway bridge, it should be dead straight. The vertical yellow lines show two kinks in the masonry, and the two others show the misalignment in the courses. The second picture has circles added to show the distortion of the soffit from a single smooth curve. We have to be careful with this kind of thing to be sure that our lens has not caused some or all of the distortion. Using a wide-angle lens can also change the shapes of curves very significantly, as shown by the third picture, a poor example of a bridge picture.

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The way we tried to show the distortion of the skew arch was very crude. A better way would be to use a suitable mathematical comparison with the correct curve.  What if you don't know the mathematical method? In the diagram below, we have taken a lot of points on the soffit of the arch, and we have moved them towards the centre by various fixed amounts. This gradually exaggerates any distortion, though in fact the distortion is great enough to be seen in the original curve, the outermost one, as a slight flattening, to the right of the crown. The advantage of mathematics is that we would be able to put a figure to the distortion, but if we only want to see that it is there, a pictorial method will do.

What is the first thing that you should do after doing a calculation? Distrust it. Try to check that you get the same result by a completely different method. Failing that, use the method on an example for which you know the answer. Let's do it. Let's begin with a perfect circular arc and see what happens.

OK - that's not a mathematical proof, but it's encouraging, so let's put a small error in two points of the circle.

This process has the opposite effect from the inflation of a balloon, which tends to make the balloon more spherical.

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This page has been mainly about geometry, but a bridge has to resist static and live loads. The geometry is only a means of optimising the structure for that purpose. A brick structure can resist compression rather well, but it is poor against shear.

 

CarParkY.JPG (94262 bytes)As brickwork is a composite structure, cracking could occur, though this is not often seen in an arch, except locally, because of the compressive forces. The picture at left shows how brick walls can crack.

Suppose we have a skew of 45 degrees, and a very long bridge, that is, a tunnel.  Here is what we would see with square bricks.

With a long tunnel, the skew becomes irrelevant, and we might as well build in the normal manner for tunnels. But this diagonal pattern has actually been used in structures.  

It looks rather like a truss, for example, except for the lack of triangles. An aircraft hanger in reinforced concrete by Nervi, at Orbetello, consisted of a quarter of a cylinder based on this pattern, making a long arched roof. Portcullis House, Westminster, includes a glass and metal structure like this. The Great Court of the British Museum has a new toroidal roof with a similar structure, designed by Norman Foster.

During the 1939-1945 war, Barnes-Wallis used a similar pattern, which he called a geodetic framework, in the Wellington bomber. In fact, the usual arrangement of frames and longerons also consists of geodesics, which are shortest paths between points. The Wellington could sustain considerable structural damage and still keep flying. Click here for a web-page including a picture of a Kirghiz tent, to compare with the geodesic fuselage.

Once the stressed metal skin had been adopted, particularly after airliners were pressurised, there was no point in the helical structure. Longerons joining circular or oval frames makes for a simpler and cheaper structure.

These pages are mainly about the large and heavy artefacts that people have made to assist in living and travelling. But people who travel all the time, if they have dwellings at all, need ones that can be quickly built and taken down, and easily carried if re-used. Click A, B, C, and D for web-pages including a picture of a Kirghiz or Mongolian tent.

The walls of a Kirghiz tent are made using a large number of sloping strips of wood, with some horizontal hoops to help with rigidity. The roof is a dome. The diagram below shows roughly how the walls are built. The diagram is made with straight lines for simplicity of programming, but the real thing uses smooth helices.

In these examples, a great many identical parts could be used, with benefit for cost.

PipeHelix.jpg (44384 bytes)Some plastic garden hose-pipes have orthogonal helical windings around them to provide resistance to water pressure.

 

Links

http://mulder.umist.ac.uk/civil/research/historic/sources/rees_pl39.htm

http://penninewaterways.co.uk/huddersfield/hnc37a.htm

http://www.wbcanal.demon.co.uk/sect26.htm

http://www.media.uwe.ac.uk/masoud/projects/brick/images/legende/bridge.htm

http://www.surrey.ac.uk/CivEng/pgprog/modbridg.htm

http://fces.udg.es/english/girona/agullana.html

http://www.aardvark-world.org.uk/canals/sapperton/stg001.htm

http://www.wbcanal.demon.co.uk/sect26.htm

http://www.media.uwe.ac.uk/masoud/projects/brick/images/legende/bridge.htm

http://www.kielder.org/walking.htm

http://rsnz.natlib.govt.nz/volume/rsnz_09/rsnz_09_00_001380.html

http://home.vicnet.net.au/~aholgate/jm/texts/kingshist.html

http://www.skewarch.com/index.htm

 

Barnes-Wallis - with good diagram of geodetic fuselage

Wellington bomber - note structure of wings and fuselage

Great Court of the British Museum

Great Court of the British Museum

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