These structures didn’t require mortar and thus their stability is based purely on self-weight, which might amount to several tons, and on friction between the stones. The characteristic features, which would allow to consider these vaulted spaces predecessors of the aforementioned Roman shell structures are
- that the corbel vaults are surface structures, where the surface is, at the same time, the main load-bearing element and the enclosure;
- that loads, predominantly the dead load due to self-weight and the additional weight of the surrounding earth or rubble, are transferred inside the thickness of the surface to the ground, essentially following a similar principle as contemporary unreinforced masonry structures designed using Thrust-Network-Analysis (see Section 2.X);
- that the loadbearing surface is single- or double-curved thereby avoiding bending stresses and buckling of the surface; and
- that they create curved and vaulted spaces.
These criteria apply to the megalithic unreinforced shells of the late Bronze age and also to the cylindrical and hemispherical vaults from Roman times, which were built using opus caementitium, an early form of concrete. These criteria also apply to the pointed arches and vaults in late medieval Gothic cathedrals, which were formed by two circular arcs in order to vary the height and span of a vault independently (\citealt{Addis2014}, 33), and to the dome structures built during the Renaissance, most notably the double-layer dome of the Basilica di Santa Maria del Fiore in Florence by Filippo Brunelleschi (1377-1466).
Renaissance to 19th Century
(TODO: Describe how the dome works in principle; structurally completed in 1436.)
Brunelleschi’s dome, in turn, became a role model for Michelangelo (1475-1564) when he took over the building site of St. Peter’s in 1547 and with it the design of its dome. In the original design, Donato Bramante (1444-1514) proposed a solid hemispherical dome similar to the Pantheon, but in contrast to the Pantheon, where the dome is supported all around its perimeter, here it was supposed to rest only on the four columns at the intersection of the nave and the transept. Michelangelo, seemingly aware of the misalignment of form, forces and support conditions, tried to reduce the weight of the dome by following Brunelleschi’s approach of hundred years before and proposed a double-layered design. In the final design, Giambattista della Porta (1535–1615) increased the height of the dome in order to further reduce the outward thrusts at its base and he also introduced several iron chains to carry the tensile stresses in the lower part of the 41.9m diameter dome (\citealt{Addis2014}, 36).
In a parallel historical development, the Anglican Church tried to assert its independence from Rome under Henry VII with the act of Supremacy of 1534, the first Brexit, and under Elisabeth I who reasserted royal supremacy of the Church of England in 1554. A redesign of St. Paul's, originally founded in 604 CE, was proposed after mistreatment and defacing during the Civil War – part of the aftermath of the split from Rome - and became necessary after the Great Fire of London 1666 where many churches were destroyed, including St. Paul's.
Sir Christopher Wren (1632-1723), a scientist by training and one of the founders of the Royal Society in London, was commissioned to design and oversee the execution of the many churches, 52 in total, including in 1669 the redesign of St. Pauls (\citealt{Pevsner1994}, 291). Having been influenced by French representational architecture, such as Perraults Louvre-Facade for Louis XIV in Paris, and by Dutch classicist architecture through Vingboons's engraved publications, he designed a protestant cathedral, in a style blending the classical and the baroque (\citealt{Pevsner1994}, 281 & 288).
Many comparisons have been made between St. Paul’s and St. Peter’s with regards to the former vying with the latter. While Wren’s intention was not necessarily to rival the design of St. Peter’s, it was certainly sporting one of the tallest domes at the time and was widely received as asserting the power and independence of the Anglican Church (Saint 2004, 453).
Robert Hooke (1635-1703), a scientist and engineer, worked with Wren during the design of the 33m diameter dome of St. Paul’s, which was topped out in 1708. Hooke is most known for his formulation of the law of elasticity, which states that the force needed to extend or compress an elastic material, such as a spring, is proportional to the distance. He published the law in 1676, which since bears his name, as part of his ten ‘Inventions’ in from of a Latin anagram, which when deciphered reads:
"Ut tensio, sic vis."
(As the extension, so the force).
As part of the ten ‘Inventions’, he also published another important finding in the form of an anagram:
"Ut pendet continuum flexile, sic stabit contiguum rigidum inversum"
(As hangs the flexible line, so but inverted will stand the rigid arch.)
Based on his law, which is commonly called Hooke’s law of inversion, he proposed the use of a hanging chain model to determine if the shape of St. Paul’s dome was stable, that is if the shape of the catenary arch was inside the masonry, which is shown in one of Wren’s sketches (\citealt{Addis2014}, 35-36).
For a similar purpose, the method was later used by Giovanni Poleni (1638-1761) in the 1740s to assess the stability of the hundred-year-old dome of St. Peter’s in Rome, which had developed cracks. However, Poleni added weights to the chain to more accurately represent the effect of the voussoirs’ weight. Again a century later, Heinrich Hübsch (1795-1863) used the method as a design tool to determine the weights of voussoirs needed to achieve the desired shape of an arch or vault (\citealt{Addis2014}, 36).
The concern with the relationship between form and stability makes Hübsch’s form-finding approach for a compressive vault more akin to the aforementioned nuraghe than to the realisation of geometric primitives as was prevalent from antiquity to the Renaissance, which becomes apparent when comparing his sketches circa 1835 to a nuraghe section (Fig. \ref{565349}).