Searching for admissible thrust surfaces in axial-symmetric masonry domes: Some first explicit solutions

Determining the internal forces acting within a masonry dome or vault loaded by an assigned distribution of external actions is a problem that is still open, as evidenced by even recent contributions to the scientific literature on the topic. The present work intends to address this issue by proposing a method for determining admissible distributions of stresses in vaults and masonry domes. The problem is solved analytically with the aim of obtaining, when possible, explicit expressions for the internal forces. Although applicable in principle to arbitrarily distributed loads, the procedure adopted herein for searching for statically admissible internal forces is described in detail for the case of distributed and concentrated vertical loads. The analysis is performed by building suitable analytical solutions to the so-called “direct” and “inverse” problems of a thin shell in which bending forces are nil and only membrane forces are present. The solutions thusly obtained are applied to vaults and domes by making use of the so-called “thrust surface” concept, which represents a natural generalization of the thrust line for masonry arches. According to Heyman's hypothesis for masonry, when the thrust surface is entirely contained within the vault thickness, a corresponding statically admissible stress field can be found. Thrust surfaces corresponding to admissible stress fields are determined by means of an expressly developed iterative procedure that begins by assigning an initial shape to the thrust surface. Then, by suitably using the solutions of the inverse problem, the shape of the thrust surface is modified so that the corresponding stresses become, when possible, statically admissible. By using the well-known theorems of limit analysis, both the mechanical and geometric safety coefficients are assessed for vaults and existing domes. As an example, the proposed procedure is applied to three practical case studies: a hemispherical dome of constant thickness, the dome of the Rome Pantheon, and the dome of Bernini's Santa Maria Assunta Church in Ariccia (Rome).