## Volte e cupole in muratura: un approccio basato sull’equilibrio

##### Abstract

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]