Arched Railway Viaducts
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These pictures show a disused railway viaduct in Smardale. About a mile away another viaduct carries the Carlisle-Settle railway, which also passes across the famous Ribblehead viaduct. Note also the ancient stone wall. Thousands of miles of these walls exist in Britain, though many are in danger from poor maintenance. The pictures were kindly provided by Dr David Newton, who has also contributed very much to the ideas that have gone into this web-site.
The Smardale viaduct has two rows of five blocks below the springing of each arch, which could have been be supports for the centring. Roman bridges such as the Pont du Gard are liberally supplied with such blocks. But some other viaducts have smooth piers. How so you think the centring was supported in the smooth examples?
Here is a sketch showing a possible way of making centring, resting on protruding blocks in the piers.
Centring and other falsework have to be designed as carefully as any other structure to achieve the required strength economically. Safety, especially, is a very important consideration.
If we look closely at a structure, we may spot features that we didn't expect. We realise that their is no need for the courses between different arches to line up with each other. We see also various distortions - perhaps there has been some subsidence. We see also that the penultimate course is not even of constant thickness.
The Yorkshire viaducts mentioned above share features with many others in Britain - the tall narrow piers, the circular arches, construction in local stone, and a grand and fitting appearance in magnificent landscapes. Putting it another way, this viaduct is typical. Typical is an easy word to say, but how often do we ask why things are typical - why they are done this way and not some other way?
Now let's consider the next two pictures. By the way, these pictures are wrong in that real piers are often tapered. In the Smardale viaduct, the thickness of the piers at bottom and top is in the approximate ratio 3 to 4, the taper being along the length of the viaduct. The width is the same at all heights. Parallel piers can look very plain and stilted.
What could influence the choice between these two designs? Not to mention the many intermediate ones.
One consideration is the mass of masonry required. The first design has more piers, twentyeight in fact, though some are shorter than the standard length. The second design has six. If we ignore the variations in length, and also the width of the bridge we can calculate the volume of masonry. If the width of each of the N piers is W and the height to the crown of the arches is H, we can use the formula, volume is proportional to NHW.
The volume of the curved sections is simply the area of a semicircle subtracted from the area of a rectangle. If the radius of an arch is R, then the volume is proportional to 2NR2 - 0.5piNR2 = NR2(2 - 0.5pi) = 0.43NR2
N, of course, is related to the length, L, of the viaduct, the arch radius and the pier width. In fact N(2R + W) = L. Since L is fixed, we can eliminate either N or W. Since W has a physical meaning for the engineer, we will eliminate N, using N = L/(2R + W)
We now have -
Pier volume HLW/(2R + W)
Arch volume 0.43LR2/(2R + W)
Let's try an example. We will take L = 1000 feet, H = 100 feet, and W = 15 feet. We can now vary R and see how the volume varies with arch radius, which we will vary from 15 feet to 100 feet. The graph below shows the result. The horizontal scale runs from 0 to 100 feet.
The rise at the left side is the result of the many piers when the arches are narrow. The minimum occurs around a radius of fifty feet, which is about 3.3 times the width of the pier.
However, it is very likely that other engineering considerations play a more important role. These could include the cost of centring, which would rise rapidly with the span, as the centring is in fact a bridge in itself, made of timber, which has to support the entire weight of an arch until the mortar has set. It is very likely that the cost of a cubic metre of pier is less than the cost of a cubic metre of arch.
Note that all the arches must be completed before any centring can be removed, so that the thrust can be transmitted to the ends of the viaduct. This means that every arch must have its own centring.
In the next diagram the cost of centring, in red, has been assumed to be proportional to the radius of the arches, while the cost of masonry is in blue. The total is in black.
The relative cost per kg of wood and stone have been set at an arbitrary value, so the position of the minimum has no meaning. These calculations are grossly over-simplified. The idea was to point out that the engineers who built the viaducts must have made calculations to find the cheapest structure compatible with structural requirements.
Links about Centring
Links about Viaducts
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