Some engineering/ structural perspectives.
An arch embodies strength from compression; tensile stress is avoided and stone is uncompressible, or at least it appeared to be so to the early masons. The voussoirs have facings which are usually at right angles to the force lines which channel the gravitational forces laterally, to be contained by the abutments. It is important that the lines of these forces are contained within the structural masonry itself. To best achieve this, the optimal shape of an arch should be 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 differently to be loose or tight.
Although the perfect masonry arch may be the shape of a catenary, this design is not essential as long as a catenary line can be drawn down through the interfacing edges of the voussoirs themselves, preferably through the middle third of each cross-section. If the best-fit catenary force line lies outside these interfaces,(i.e. the arch shape cannot contain it), then the structure is unreliable. Clearly, the wider the voussoirs, the easier it is for a catenary line to lie within them all; a second tier ( or order) of voussoirs is helpful; and since a catenary is not a unique curve, any one of many options will be adequate. This permits a range of different arch shapes, some of them more optimal than others. If a masonry arch is a precise catenary shape then the voussoirs may be very slim whilst still retaining strength.
A bridge arch is a little different from a roof-arch because it has the added burden of a dead-load: ramps, parapets, spandrels and infill. The weight changes the shape of the force-line: the optimal shape approaches a parabola rather than a catenary. More modern bridges may also have huge live-loads to stress them, such as locomotives, but this was not a problem before the 19th century. Catenaries and parabolas appear almost indistinguishable in shape but mathematically they are very distinct families of curves. Bridges usually require abutments at each side to further contain the forces.
In 1675 Robert Hooke provided the theory of these principles but the practice of arch building had long preceded him. The Etruscans, the Romans, the Persians and the Chinese Dynasties had all built masonry arched bridges. Many Roman bridges remain intact in Europe, today, but in Scotland there is room for no more than speculation. In Europe, arch building became a lost art for 600 years but re-emerged in the 11th century. The Romanesque semicircular arch (above)then became the fundamental solution for ecclesiastical architecture as well as for bridges; a semi-circle had been the Roman mainstay for arches, and it was this that they copied. However, there are only a few Romanesque bridges in Europe; in this period most were made of wood and those few that were masonry probably had a late Roman origin.
Romanesque architecture gave way to Gothic in the early 13th century; an ogive or gothic arch has a geometry which is much closer to a catenary/parabola. Consequently, cathedral arches and rib-vaults could be given slimline voussoirs and bridges built in the gothic shape could be slimmer yet stronger.
The mid 15th century Renaissance saw a return to semicircular, elliptical and segmental arches. Long before Robert Hooke’s time, segmental arches had been used for bridges. This shape has a flattened profile akin to a small segment of a circle. Again, a parabola or a catenary can be drawn within the voussoirs, particularly if they are wide enough, but the angle of force lines in such a flattened arch always implies the need for substantial abutments. In terms of structural equilibrium there may be less difference than might be expected, between a segmental and a semicircular arch.
We don’t know if any catenary theory was known or understood by the early master masons. Some, perhaps in China, may have understood the significance of the hanging chain. Most European historical writing was in Latin, and largely the preserve of clerics who seldom had an interest in engineering so on the whole there is very poor documentation on the practical detail of construction. Our only recourse is to extrapolate from existing buildings, but this has been compared to looking at a finished crossword and trying to guess the clues. Renaissance architects certainly applied principles of geometrical equilibrium and it seems there were some prescriptions or rules of thumb, and occasionally some complex formulae, empirically recommending maximum ratio size. Palladio(Venice c.1460) and Serlio(Bologna c.1540) both recorded the average arch thickness/ span of the Roman antiquities. The most prolific author and architect was Leon Battista Alberti, from Genoa, around 1460: he recommended that pier width should be one quarter of span, and that voussoir width should be one fiftheenth of span. He also recognised that segmental arches were important and for these he proposed a voussoir crown thickness of one fifth of the intrados radius. In the light of modern engineering design, informed by elastic theory and limit analysis, these ancient empirical solutions may seem simplistic, yet Roman arches are still standing today and medieval cathedrals remain safe despite their complexity of vaults and flying buttresses. The Chiao Shui river in China was spanned by a segmental arch bridge in 610 AD and in Florence, the Ponte Vecchio’s segmental design was conceived in 1345.
In earlier medieval times the masons were builders, masters of work, architects and engineers, rolled into one. Master-masons were the consultants of the period: prosperous middle class professionals, in charge of other craftsmen with support from qualified ‘journeyman’ masons (day-wage or journée) and apprentices. The Master Mason was the architect and master of works. Well-known names were in great demand, even internationally. They were very well rewarded and their names are recorded in the archives. Their journeymen colleagues were divided into quarrymen who extracted the stone, sawyers who cut the fairly undressed blocks, bankers who worked on site dressing the stone and sizing it, carvers who applied art and design and fixers who raised the blocks and put them in place.
In 1716 Henri Gautier, a french doctor, mathematician and architect wrote the first book on the construction of bridges. His Traité de Ponts addressed all aspects of design, choice of site, construction and contemporary regulations. He reported on the great 17th Century Italian masters and using examples both wooden and masonry, provided operational instructions for the size and shape of piers, cutwaters, arches, spandrels and parapets.
Even a simple semi-circular voussoir arch cannot be built without faux-works or centering because the structure will not stand up until the keystone is in place at the top. Temporary centering structures were made of wood, and the carpenters often erected, dismantled and re-erected the same centering on different rings of the arch, across the barrel, until each keystone was installed. Even today masonry voussoir arches may need centering when repairs are necessary.
The construction of multi-arch segmental bridges posed a particular problem. Here, each arch provides the lateral resistance for its immediate neighbour, but the obvious implication is that false-work centering is required for the entire structure, all at the same time, to avoid complete collapse. Once completed, however, the advantage lies in reduced height for a given span, so that fewer piers are needed, implying less risk of being washed away by floodwaters. There is also more flexibility and room for diversity with a slimmer more elegant profile.
Multispan bridges had always presented the extra challenge of constructing piers in mid-stream. Gautier discusses this in broad principles. Ruddock provides a modern detailed analysis. When possible a rock foundation was chosen and a site without tidal range was preferred. A small tidal range might permit the Roman approach of constructing a wooden cofferdam on the river bed(Old Scots: bulwark. Old French: bâtardeau) In this case, a gin and ram was employed to drive piles into the mud adjacent to each other until a full semi-wartertight circle was completed, perhaps 50ft. in diameter; then the central pool was emptied by chains of men with ladders and scoops. A wider tidal range usually required 'starlings': these were artificial islands. The construction was similar to a cofferdam, but rubble was poured in to the chamber on each receding tide. The stone piers were built on top of the starling. This arrangement was more commonly seen below English bridges and was clearly recorded at Berwick. A more common approach, seen more often in Scotland, and also useful for a significant tidal height, was to sink a large wooden frame or 'brander' made up of longditudinal planks ; this frame would would be filled directly with rubble and sunk on the required spot. In this way a different form of starling was created which had no vertical piles. The French called it a "creche," the English, a "grating". Finally, Gautier described one further approach: that of diverting the whole river while foundations were being built. He tells us that Trajan employed this last approach on his famous bridge over the Danube in 105AD.
Multispan bridges need piers with triangular cutwaters, facing the current, to provide the best protection against scouring ( undermining of the foundations by the current) and it was only discovered rather late in the evolution, that these were required on the downstream side as well.
On top of the piers the centering for all the arches would then be built, and this would be followed by the voussoir arches, themselves. The spandrels were last to be constructed with infill rubble to provide a plane for the cobbled surface. Parapets were an optional extra, sometimes deliberately omitted: for example, the old packhorse bridges required that nothing should impede the large panniers on each side of the horse.
Today, many of these old masonry arch bridges are expected to carry modern traffic. This implies a live load way beyond the expectations of the original builders. Plastic Limit Analysis appears to be the mainstay of modern safety specification. This approach, which is now hugely enhanced by modern computer programs, leans on line of thrust theory and computations around expected end-point 'hinge' failures of the old arches. This is not the only approach; recently the important strengthening role played by the infill has led to the development of some new modelling strategies.