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Arched Railway Viaducts

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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 2NR 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 0.43LR 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. |

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