Force lines in semicircular and segmental arch bridges

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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 (left) has a voussoir arch thickness about one-fourteenth of the span.  In the 15th-century, Alberti recommended a ratio of less than 1: 15, yet he knew nothing of lines of thrust; his recommendation was completely empirical. More recent analyses by Couplet, Milankovitch and then Heyman, all demonstrated, in fact,  that true lines of thrust in such arches required 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.  It underpins the application of  Heyman’s Safe Theory.  

 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.    Below left is a theoretic force line in which the upper part lies comfortably in the middle third of the arch.   However, the lower parts of the force line will not do at all. 

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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 to this. The arch could be very 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.   

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In fact, 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 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. If ashlar was used,  each voussoir needed a tailored template. It was cheaper to use loading and buttressing to manoeuvre the force lines to match the geometry.   Furthermore, masonry had ceased to be the only option: new reinforced materials with tensile strength had arrived, hugely reducing the weakness of non-optimal shapes.    Some would also say that pure structural integrity cannot be the only determinant of architectural design.  

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Dec. 2012                                      Site last updated  Feb 2019