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Purpose     This paper aims to review recent literature results on the equilibrium problem and the strengthening design of masonry vaults.     Design/methodology/approach     A Lumped Stress Method (LSM) is considered within the Heyman’s safe theorem, based on the definition of thrust surface of a masonry curved structure. In particular, the static problem of the vault is formulated by introducing a membrane continuous of the studied masonry structure to associate with a spatial truss through a nonconforming variational approximation of the thrust surface and membrane stress potential. A tensegrity approach based on a minimal mass design strategy, different strengths in tension and compression of the material is discussed within the strengthening strategy of masonry vaults.     Findings     The numerical results have highlighted the efficacy of the two numerical approaches to assess the vulnerability of existing structures and design optimal strengthening interventions of these structures.     Originality/value     The presented models can represent fast and useful tools to assess the vulnerability of existing structures and design optimal strengthening interventions with composite materials of these structures.

On the statics of curved masonry structures via numerical models