Shells are forms that base their behaviour in their geometry. From antiquity the art of building has been conceived as the ability to span and grew with little pieces, using bricks and stones, until there were great closed spaces. Arches, vaults and domes work because of their geometrical disposition more than because of their intrinsic strengthening. From their invention, domes have been subjected to a process of optimization that arises slowly to diminish thickness at the same time that it increases covered areas. How the thick Roman Pantheon dome is converted, almost with the same span in a thin skin in the Hagia Sophia, explains with clarity the path from heavy to light buildings. Maybe Saint Vital in Ravenna is the end of this process, not surpassed for twelve centuries by Renaissance nor Ottoman domes. Saint Vital was the culmination of lightness thanks to the use of a special kind of hollow brick. When some architects were impelled to build in the Baroque style with short budgets and pour technological advice, they thought that the solution was to be found in the geometry and not in the materials. Architects such as Borromini, Guarini and Vittone were the masters who taught a new young generation headed by Juvara, Fischer and the Dienzenhofer family. Their proposals for double curved single layer domes were really shells, thinner than ever executed before. Spherical dome caps realised at this time were as thin as twenty centimetres, with or without ribs. Vaults composed of sectors of spheres and cylinders were varied and showed a complexity not seen since the Muslim domes. The complexity of Baroque domes has not be conveniently studied until now, may be because engineers think more in terms of mathematical analysis than in terms of geometrical concepts. This is the reason we have planned a geometrical approach by using the immense potential that computer design programs such as CAT1A and ABAQUS brings to the architectonical analysis. The series of churches and other constructions studied are extensive and in this paper we will include only some of the more surprising Balthazar Neuman designs such as the Chapel in the Banz Monastery, Residence Chapel in Wiirzburg Palace and Fourteen Saints Basilica. The program of the advanced research includes the geometrical definition of the architectonical models, most of them checked in situ, the graphical determination, the mathematical FEM definition, and the final analysis and dimensioning. The main objective is to demonstrate that the basis of the optimal behaviour is the special geometry considered for each design that considers only membrane stresses instead of shell stresses. Bending is really acting in borders and edges and not in the general surface as occurs in concrete and steel shells. Baroque was the inventor of membranes as conceived, actually with such complexity that none of the great concrete builders tried to build. It is a thesis well founded in the research that we present in this paper.
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