Force lines in semicircular and segmental arch bridges

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The  semicircular 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.  It can be seen that the best fit parabola is only just contained within the voussoirs, yet Alberti knew nothing of lines of thrust and his recommendation was empirical. Modern investigations by Couplet, Milankovitch and more recently, Heyman, all  demonstrated, in fact,  that true lines of thrust in such arches required a minimum ratio of around 1:18.(width/span). This is the limit arch according to 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 might be considered, to get round 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 in this. The arch could be even slimmer but still contain the 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 left 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?  In fact, catenary or parabolic architecture is expensive compared with pointed, semicircular or segmental arches.  The wooden centering is difficult to optimise, as the radius of curvature constantly changes through the arc. Furthermore, if ashlar is to be used,  each voussoir must then have a tailored template. It is cheaper to use loading and buttressing to manoeuvre the force lines to match the geometry. Some would also say that pure structural integrity cannot be the only determinant of architectural design.   

Dec. 2012                                      Site last updated  Feb.2016