Scotland’s Oldest Bridges.

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

Masonry Arches, Shapes and Forces

 

Natural building masonry has immense compressive strength but very low tensile strength, so stretching or bending must be avoided.  Limit Analysis tells us that if it is possible to draw a line of thrust within the stones of an arch  this will avoid tension and the arch will not collapse; the design must be such that the force-line of gravity is directed outwards and downwards within the voussoirs.  If this necessary condition is  met, there is no need to consider the inherent strength of the material; it is the shape of the structure that governs its stability. However,  it is the shape of the body of the arch itself that matters, not the loose surrounding infill material, spandrels or walls.   The arch stones and the buttresses must contain the force line.   

 

An evenly loaded  freestanding arch  has a gravity force-line which follows the shape of a catenary:  a mathematical function represented by an inverted hanging chain.   Catenaries are a family of curves  just as a hanging chain may be configured to be loose or tight. If the force-line has to lie within the arch, it would generally seem very sensible to build arches in a catenary shape.  However, any shape is possible as long as the arch thickness  (between in the intrados and the extrados of the arch stones) is wide  enough to allow at least one catenary line to lie within it.  The force line need not be real or actual- it only needs to be possible. If it exists, the structure will find it and remain safe.   Should the force-line encroach on the edges or stray outside the masonry  then tension would be be expected and this would be followed by collapse of the arch.  

 

 

All this applies to a pure theoretical arch.  A real masonry bridge, however,  is a little different because it has weight and substance: decking, parapets, spandrels and infill.   These cause pressures, which along with any irregularity, may cause the force-line to deviate a little from a pure catenary shape, but as long as the arch is thick enough, with a decent margin of error, the structure should be sound:  within the middle third of the thickness is recommended. Today, of course, live-loads such as heavy lorries or locomotives apply point pressures with different  force-line profiles, but before 1750 the maximum live-load was dwarfed by the weight of the masonry itself.   

 

The semicircular arch in this sketch has been made increasingly thinner by increments until a point is reached where only one catenary curve would fit within it. The arch has a voussoir arch thickness about one-eighteenth 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 mathematical analyses by Couplet , Milankovitch , Heyman  and others confirmed that true lines of thrust in such perfect arches require a minimum ratio of  0.055 or 1:18(width/span).  This is the limit-arch, and permits a force line which lies entirely within the masonry thereby ensuring that stress will be compressive rather than tensile. (A considerably larger safety margin is recommended  in practice; the‘middle-third rule’).  

 

In fact, many semicircular bridges with a greater ratio than this are quite 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.    The best fit catenary curve lies comfortably in the middle of the arch section, but it can be seen that the lower parts of the force line profile will not do at all. The solution lies in large abutments .  The arch can remain slim but yet contain at least one suitable force line. 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 shown in the railway bridge. The equilibrium of forces is effectively the same. The difference is cosmetic.  Interestingly, the flatter the segmental shape the more a catenary coincides with a circle segment. 

 

Medieval masons had useful empirical geometric rules of thumb for relating  arch and abutment thickness to  span.   The requirements are  simple in the case of catenary or parabolic shaped arches.  Pointed arches are also quite well shaped to contain the force-line.  Semicircular, segmental and elliptical arches are less suitable.    

 

 

 

In 1675, Robert Hooke provided the principles of catenary theory, but the practice of arch building had long preceded him.  The Etruscans, the Romans, the Persians and the Chinese Dynasties all built masonry arched bridges. Today, many Roman bridges remain intact in Europe, but in Scotland and England none remain standing.  Arch building became a lost art for 600 years but reemerged in the 12th century when vaulted stone bridges appeared in Britain for the first time.  The Romanesque semicircular arch was the fundamental solution for Norman ecclesiastical architecture as well as for bridges. A semicircle had been the Roman mainstay for arches, and it was this that they copied. Some Romanesque vaulted bridges can be seen in Europe, but in Britain, in this period, most bridges were made of wood, perhaps with masonry piers.  Fountains Abbey Infirmary Bridge in Yorkshire and Clattern Bridge in Kingston are rare examples of  English  Norman (Romanesque) vaulted arched bridges.   Exeter Bridge (above) was recently uncovered. There are no Norman bridges in Scotland.    Romanesque architecture gave way to pointed Gothic in the 12th and 13th centuries.   Pointed arches have a geometry that is slightly closer in shape to a catenary,  so bridges built this way could be slimmer yet strong and need less buttressing.   There is an odd additional stratagem here :  a weight on the pointed tip (crown) of the arch further alters the geometry of the thrust line,  bringing it even more into line with an ogive shape.  This was known about in medieval times, often seen on domes,  but seldom seen on bridges.   Above is the Puente del Diablo near Barcelona, a 13th century 37m arch on the River Liobregat . Note the remarkably slim voussoirs. 

Not all medieval bridge arches from the Gothic period  are of ogive shape; there are some semicircular and segmental arches, too, possibly remnants of a bygone style. Around 1500, the Renaissance arrived in Britain and brought an end to the building of pointed bridge arches. Rounded shapes predominated thereafter  From the mid-eighteenth century segmental shapes became most common and, in Scotland, after 1800, ellipses appeared.    

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Why do we see so few catenary-shaped stones arches on buildings and bridges?  In the early 20th century, when the geometry of catenary shapes was fully described, it was quickly acknowledged that this architecture was expensive compared with rounded or segmental arches: the radius of curvature of the arch varies smoothly through the arc, and each voussoir required  an individually tailored template. It was cheaper to use loading and buttressing of abutments to manoeuvre the geometry to match an estimated limit force-line.  Furthermore,  in the early 20th century ordinary masonry had ceased to be the only option: new reinforced materials with high tensile strength had arrived which reduced the inherent weakness of the less optimal shapes. 

 

The most suitable shape for a dome is also a catenary, and it is interesting that Inuit igloos are usually of that shape.   Could it be that the ephemeral nature of the building-material fostered many rebuilds and a very early empirical solution?  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Semicircular arch
Chain and catenary
Semi-circ and catenary
semi-circ and force line
Guard bridge andd small voussoirs
Abutments and segmental arch
abutments and forceline
Puente del Diablo
Igloo

Last updated Sept. 24