Abstract
The assessment of the equilibrium and the safety of masonry vaults is of high relevance for the conservation and restoration of historical heritage. In the literature many approaches have been proposed for this tasks, starting from the 17th century. In this work we focus on the Membrane Equilibrium Analysis, developed under the Heyman’s theory of Limit Analysis. Within this theory, the equilibrium of a vault is assessed if it is possible to find at least one membrane surface, between the volume of the vaults, being in equilibrium under the given loads through a purely compressive stress field. The equilibrium of membranes is described by a second order partial differential equation, which is definitely elliptic only when a negative semidefinite stress is assigned, and the shape is the unknown of the problem. The proposed algorithm aims at finding membrane shapes, entirely comprised between the geometry of the vault, in equilibrium with admissible stress fields, through the minimization of an error function with respect to shape parameters of the stress potential, and then, with respect to the boundary values of the membrane shape. The application to two test cases shows the viability of this tool for the assessment of the equilibrium of existing masonry vaults.
Highlights
The assessment of the equilibrium of masonry structure is a topic of high relevance, given the diffuse presence of masonry buildings in Italy and Europe, often with high historical value.Over the centuries, different methodologies have been developed for the assessment of the equilibrium of masonry constructions and to quantify their degree of safety, starting from practical rules [1], until the definition of graphical methods applied to two dimensional structures and to axis-symmetric vaults under monoaxial stress regimes, such as masonry domes [4]
In this work we focus on the Membrane Equilibrium Analysis, developed under the Heyman’s theory of Limit Analysis
In this work we present a method for the assessment of the equilibrium of masonry vaults based on the search for a suitable membrane shape in equilibrium with a given compressive stress field
Summary
The assessment of the equilibrium of masonry structure is a topic of high relevance, given the diffuse presence of masonry buildings in Italy and Europe, often with high historical value. In the present paper, focusing on the assessment of the equilibrium of existing masonry vaults, we start from a family of concave stress potentials in order to ensure the admissibility of the stress field This implies the ellipticity of the equilibrium equation: we look for an optimal membrane shape comprised between the geometrical boundary of the vault, as required by the Safe Theorem of Limit Analysis. We approach this problem through an optimization algorithm, minimizing an error function first with respect to the parameters of the stress potential, and with respect to the boundary conditions of the the membrane shape. Optimization is used in geotechnical applications [34], optimization of geodesic domes [35], of soil-steel structures [36], and lightweight structure [37]
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