, Volume 54, Issue 3, pp 525–547 | Cite as

A non-smooth-contact-dynamics analysis of Brunelleschi’s cupola: an octagonal vault or a circular dome?

  • Valentina Beatini
  • Gianni Royer-CarfagniEmail author
  • Alessandro Tasora


The cupola (dome) of Santa Maria del Fiore in Florence was ingeniously constructed by Brunelleschi using a conical bricklaying, radial-oriented toward a focus point on the central axis. Therefore, the dome is built as a surface of revolution but with parts cut away to leave the octagonal cluster vault form. This circular arrangement is compared with an octagonal horizontal corbelling in models where the dome is schematized as an assembly of rigid-blocks in frictional contact, analyzed with a Non-Smooth-Contact-Dynamics approach. The high indeterminacy of the contact reactions implies considerable difficulties in their determination, which are faced via a regularization procedure by adding a compliance at the contact points in representation of the deformability of the mortar joints. Numerical experiments, performed with a custom software, highlight the uniform flow of forces in the Brunelleschi arrangement, but evidence the disturbances induced by the herringbone spirals, mainly used for construction purposes, which are overloaded along the meridians and very weak in the direction of the parallels. This is due to the vertical narrow disposal of the blocks, which increases the stiffness in meridional direction, but diminishes the capacity of the friction-induced forces to equilibrate the hoop stress.


Dome Masonry Friction Brunelleschi Rigid body mechanics Non-smooth contact dynamics 



The authors are grateful to Professor Silvia Briccoli Bati (University of Florence) for inspiring discussion during the preparation of this work. GRC acknowledges the support of the Italian Ministry of University under grant MIUR-PRIN voce COAN code 2015JW9NJT and of the Italian Civil Protection Department, Presidency of the Council of Ministers, under project ReLUIS-DPC 2014-2018.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Architecture and Built EnvironmentNorthumbria UniversityNewcastle upon TyneUK
  2. 2.Department of Engineering and ArchitectureUniversity of ParmaParmaItaly
  3. 3.Construction Technologies InstituteItalian National Research Council (ITC-CNR)San Giuliano Milanese, MilanItaly

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