Volte e cupole in muratura: un approccio basato sull’equilibrio
Creators: name: De Chiara, Elena
Publisher: name: Universita degli studi di Salerno
Year: 2018
Subjects: subject: ICAR/08 SCIENZA DELLE COSTRUZIONI
subject: Murature

description: 2016 - 2017
description: Masonry curved elements, such as arches, vaults and domes, are among the most widespread, and fascinating, world historical Architecture structural elements. Written historical records about the set of technical rules used to erect masonry structures, even the most complex of them, date to quite recent times, around a couple of century before scientific discoveries of Galilei and Newton. These recordings show that the rules to build masonry structures like Gothic Cathedrals or domes and the European Baroque stairs were pure proportion rules. In fact, compressive stress inside masonry are typically very low, compared to the material resistance, hence the analysis of a masonry structure can be reduced mainly to the detection of a flux of compression forces equilibrating the external forces. A reasonable theory, explaining the bases of this phenomenon in modern terms, is based upon the constrains defining the Heyman’s unilateral material (No-Tension) evoking the rule system and the methods used by the ancient master builders and integrating the masonry structures analyses in the well defined limit analysis framework. In his 1966 work, entitled ”The Stone Skeleton”, Heyman casts his deep knowledge of the emerging Plasticity Theory of the time. Heyman theory was adopted and successfully applied to arches and domes only recently. Two alternate approaches have been proposed in this context. The first is the so called “Thrust Network Analysis” (TNA), while the second is the “Thrust Surface Method” (TSM). This thesis focuses on the “Thrust Surface Method” application, specifically in the implementation of an algorithm able to produce compressive membrane stress states on surfaces inside the masonry, in equilibrium with assigned external loads. Thrust Surface Method is based essentially on the loads equilibrium on a membrane surfaces. Such S surface represents the unique support of an admissible stress (i.e. a compression stress balanced with the loads) contained inside the vault thickness to which the loads action is transmitted by the surrounding continuum (small traction stresses are also used in the process). The freedom of choice of the S surface provides a wide catalogue of possible states of admissible stresses, much wider of the one available for the arches (whose possible shapes are heavily constrained by the equilibrium), and represents the specificity of both TNA and TSM. In details, with the TSM, membrane balance under assigned loads is a statistically determined problem, being three both the unknown stress components and the equilibrium condition. For a assigned shape S the catalogue of the balanced stresses is determined, indeed, by the boundary conditions. The choice of the S shape, however, is not fully free, since the membrane stresses have to be purely compression stresses. This thesis has the main purpose to introduce a method for the automation of the research of an admissible stresses field for a vault of chosen shape. Such method is based on the iterative resolution of a determined problem where the variable role is taken alternately by the shape and the stress. The practical ability to implement such iterative procedure is based on the the formulation of the membrane equilibrium on a surface S according to the Pucher scheme. In his 1934 work, Pucher used primitive analytical methods and introduced as primary variables the so called “projected stress”, showing how to decouple the three S equilibrium equation in two equations for the S surface equilibrium (equations which are independent from the membrane geometry) and one equation for the S transversal equilibrium, where the S curvature determine the coefficients. The general solution of the two surface equations can be expressed as a F function representing the stress potential (Airy function), while the transversal equilibrium equation is reduced to a differential linear second order equation in F whose coefficients are represented by the f Hessian function describing S in a Monge parametric representation. Constrain on the material to be exclusively compressed imposes a subsequent constrains on the stress potential: F has to be concave, i.e. its Hessian has to be semi-defined negative. F function is restricted too by the condition that f values are contained, for each point, in the interval defined by the vault extrados and intrados. The proposed iterative procedure, starting from an assigned surface f° , is built on a optimization based strategy to create sequences of stress functions F°, F1, F2, .., Fn, f1, f2, .., fn, with the purpose to automatically obtain a stress function as concave as possible and a membrane surface as close as possible to the vault median surface. [edited by Author]