Scotland’s Oldest Bridges.

A map-based catalogue of the oldest masonry bridges in Scotland. 

Force lines in arches.


If a semicircular arch were designed to be increasingly thinner, by increments, then eventually only one catenary curve would be able to fit within it. The arch on the left has a voussoir arch thickness about one-fourteenth of the span. In the 15th-century, Alberti recommended a ratio of 1: 15, yet he knew nothing of lines of thrust; his recommendation was empirical. Much later analyses by Couplet(R), Milankovitch(R) and then Heyman(R), all demonstrated that true lines of thrust in such perfect arches require a minimum ratio of around 1:18.(width/span). This permits a force line which lies entirely within the masonry and thereby implies that all stress will be compressive rather than tensile. ( A considerably larger margin of safety is sensible in practice; the‘middle-third  rule).   

In fact, many semicircular bridges with a greater ratio than this appear to be stable. For example, in Fife, Guard Bridge's semicircular arch has a ratio of almost 1:30, and yet it has been standing since the 16th century. As an explanation, a wider flatter parabola might be considered, to get around these limitations.  Above, left is a theoretic force line in which the upper part lies comfortably in the middle of the arch section.  However, the lower parts of the force line will not do at all.

The solution lies in moderately high abutments which will then contain the forces. (This whole proposition is really the wrong way round: abutments permit the masonry to contain a wider range of parabolas/ catenaries). There are further advantages here. The arch can remain slim but yet contain at least one suitable force line. Furthermore, the angle that the force line makes with each voussoir joint is much more right-angular which confers additional strength and stability.

The implication is that the parts of the voussoir arch below the top level of the abutments are really redundant and could be filled entirely by thicker abutments.  The result is a raised (or stilted) segmental arch as in the above right railway bridge. The equilibrium of forces is effectively the same. The difference is cosmetic. Curiously, because of the improved relationship between the line of force and the voussoir joints, it may be a more stable option than smaller semicircular arches which have less abutment support. 


Why do we see so few catenary arches on buildings and bridges? Since the early 20th century, when the Serbian engineer Milutin Milankovitch(R) fully described the geometry of force lines, it was quickly acknowledged that catenary or parabolic architecture was expensive compared with rounded or segmental arches. The wooden centering was difficult to optimise as the radius of curvature varied through the arc. Furthermore, each voussoir needed a tailored template. It was cheaper to use loading and buttressing to manoeuvre the force lines to the geometry.   In the early 20th century, masonry had ceased to be the only option: new reinforced materials with tensile strength had arrived, hugely reducing the weakness of non-optimal shapes. 


‍ Interestingly, Inuit igloos are usually catenary in shape.  Could it be that the ephemeral nature of the building-material fostered many rebuilds and a very early empirical solution?   


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Page last updated Nov..2020