Back to Home Page Back to Bridges 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. 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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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.
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. Some plastic garden hose-pipes have orthogonal helical windings around them to provide resistance to water pressure. |