Studying the equilibrium of oval-base pointed masonry domes: the case of Pisa Cathedral

This paper addresses the equilibrium problem of an oval-base, pointed masonry dome, that of famous Pisa Cathedral. Set within the framework of the safe theorem of limit analysis, the analysis involves searching for compressive-only statically admissible internal actions for the dome under vertical loads by using the concept of 'thrust surface'. According to Heyman's hypotheses, it is assumed that no in-plane tensile stresses can be transmitted within the thrust surface. The equilibrium problem is tackled by finding an explicit solution for the stresses within a suitable collection of thrust surfaces having the shape of ellipsoids, all contained within the dome thickness. The dome intrados and extrados surfaces have been carefully reconstructed by laser scanner survey and approximated by regular surfaces. The analytical expressions used for both the stress field and the intrados and extrados surfaces have enabled determining estimates of the safety level by means of an expressly developed optimisation procedure.