#### Vol. 14, No. 5, 2019

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Automatic generation of statically admissible stress fields in masonry vaults

### Elena De Chiara, Claudia Cennamo, Antonio Gesualdo, Andrea Montanino, Carlo Olivieri and Antonio Fortunato

Vol. 14 (2019), No. 5, 719–737
##### Abstract

The objective of the present work is to develop an automated numerical method for the analysis of thin masonry shells. The material model for masonry that we adopt is the so-called “normal rigid no-tension” (NRNT) material; and for such a material, the kinematical and the safe theorems of limit analysis are valid. The present study focuses on the application of the second theorem to masonry vaults and domes, being devoted to the determination of a class of purely compressive stress regimes, which are balanced with the load. The mere existence of such a class is a proof that the structure is safe, and members of this class may be used to assess the geometric degree of safety of the structure and to estimate bounds on the thrust forces exerted by the structure on its boundary. The problem is reduced to the equilibrium of a membrane $S$ and can be formulated in terms of projected stresses defined on the planform $\Omega$ of $S$. The search of the stress reduces to the solution of a second-order pde, in terms of the stress potential $F$. In order that the membrane stress on $S$ be compressive, the potential $F$ must be concave. As for the thrust line in an arch, the surface $S$ is not fixed and may be changed, given that it remains inside the masonry. Under these simplifying assumptions, the whole class of equilibrated stress regimes for a masonry shell is obtained by moving and deforming $S$ inside the masonry, and also, for any fixed shape, by changing the boundary data for $F$, that is the distribution of thrust forces along the boundary. The search for a feasible stress state on a convenient membrane surface, to be chosen with a trial and error procedure, requires a substantial effort and may be unrewarded. Then, the main object of the present work, is to produce a computer code that can handle numerically the interplay of the shape controlled by a function $f$, and of the stress potential $F$, by developing a convergent optimization scheme able to give a safe state under the given material and geometrical constraints, namely the concavity of $F$ and the inclusion of $f$ within the masonry. Two simple cases, are exposed in detail to illustrate the method.

##### Keywords
masonry vaults, no-tension materials, Airy's stress function, Pucher equilibrium
##### Milestones
Received: 24 March 2019
Revised: 3 October 2019
Accepted: 18 October 2019
Published: 31 December 2019
##### Authors
 Elena De Chiara Department of Civil Engineering University of Salerno Via Giovanni Paolo II, 132 84084 Fisciano Italy Claudia Cennamo Department of Architecture and Industrial Design University of Campania “Luigi Vanvitelli” 81031 Aversa Italy Antonio Gesualdo Department of Structures for Engineering and Architecture University of Naples Federico II Via Claudio 21 (buildings 6-7) 80125 Naples Italy Andrea Montanino Department of Structures for Engineering and Architecture University of Naples Federico II Via Claudio 21 (buildings 6-7) 80125 Naples Italy Carlo Olivieri Department of Civil Engineering University of Salerno Via Giovanni Paolo II, 132 84084 Fisciano Italy Antonio Fortunato Dipartimento di Ingegneria Civile University of Salerno Via Giovanni Paolo II, 132 84084 Fisciano Italy