A masonry arch embodies strength from compression and stone is practically incompressible. Tensile stress must be avoided as ordinary masonry has poor tensile strength. Because of this, structural stability becomes a function of geometry rather than the nature of the material. The forces of gravity are directed sideways and downwards through the masonry voussoirs towards the ground. A freestanding arch, with no weight on it, has gravity force lines in the shape of a catenary: a complex hyperbolic function represented by the shape of an inverted hanging chain. In fact, catenaries are a family of curves, just as a hanging chain may be configured to be loose or tight. A catenary physically equalises the stress at every point and an inverted catenary, forming a stand-alone arch without loading, will comply with Leibnitz's principles of least-action; it will be self-supporting without buttressing. Although the need for solid foundations cannot be avoided, it would seem sensible, generally, to build arches in a catenary shape.

However, to respect the principles of limit analysis, it is only necessary that a single catenary line can be found that lies wholly within the masonry (between in the intrados and the extrados of the arch stones). If such a line exists, then the structure, before collapse, will find it and remain safe. If such a line cannot be found then tension can be expected and stress fractures will be followed by collapse of the arch. For safety, it is usually proposed that this line should be within the middle third of the stonework. Clearly, the wider the voussoirs of a circular arch, the easier it is for catenary lines to fit within it. A range of different arch shapes is therefore possible, some of them more optimal than others, but if a pure catenary shape were chosen, the voussoirs could be very slim without compromising safety.

All this applies mathematically to a theoretical arch. However, a real masonry bridge is different because it has weight and substance- the additional burden of dead-load: decking, parapets, spandrels and infill. This weight on the arch alters the shape of the force-lines a little; they change their profiles to approach that of a parabola, which is different mathematically as well as having a little less spread. The heavier the whole structure, the more parabolic the force lines become; the implication is that weighting an arch, especially on the haunches, brings the force lines in a little. Compare above and left. Recent technologies such as finite element analysis also seem to suggest something else: that large volumes of infill tend to nudge the force-lines a little nearer to a semicircular profile. Of course, unlike a simple arch, a bridge has more material to contain the force lines, but in practice, the lines of thrust are best contained by solid quality, anchored masonry- voussoirs, abutments or buttressing, perhaps ideally in ashlar.

Today, live-loads such as heavy lorries or locomotives apply point pressures with quite different force line profiles; however, before 1750 the maximum live load was dwarfed by the weight of the masonry itself.

In summary, most old masonry bridges have shapes which are less than optimal for containing the forces of gravity; consequently, they must have either wide voussoirs or solid abutments on both sides.

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