# Rigid Blocks for Masonry with an application to open-well spiral stairs

Anyone working at some depth in material engineering knows that fracture and friction are still the most difficult challenges of modern Mechanics. The main strength of the simplified unilateral model of Heyman, which assumes zero toughness and infinite friction, is indeed its ability, while excluding these two tough guys, of being still able to make sound predictions on masonry behavior at the macro-scale. With Heyman’s assumptions, the equilibrium of a structure composed of this ideal masonry material, can be studied with Limit Analysis. Displacement discontinuities and concentrated stresses are inherent to the unilateral behaviour, therefore, analytical and numerical tools must be introduced that can handle the unilateral No-Tension model with singular stress and strain fields, within the framework defined by the two theorems of Limit Analysis. In particular, the present study is concerned with the analysis of masonry structures, modeled as continua composed of Normal Rigid No-Tension (NRNT) material. The NRNT material represents an extension of the model material of Heyman to 2d/3d continua. The material is rigid in compression, and extensional deformations, allowed at zero energy price, can arise and be either regular or singular; then extensional deformation can appear as either diffuse (smeared cracks) or concentrated (macroscopic cracks), and there is not any reason to prefer one upon another, on an energy ground. The fact that rigid block deformation seems to be the preferred failure mode for real masonry structures stems from mechanical characteristics, such as toughness, interlocking and cohesion, that are not inherent to the simplified NRNT continuum model. So, in the present study, we force rigid block mechanisms over diffuse cracking. The equilibrium problem is formulated as an energy minimum search, and a numerical method for approximating the solution is proposed. With this method the energy is minimized in the set of piecewise-rigid (PR) displacements, meaning that the structure is partitioned into a finite number of blocks that can move one with respect to the other under unilateral Heyman’s constraints at the interfaces, and a possible equilibrium state is searched by minimizing the potential energy. After formulating and discussing the method, in this presentation, such numerical approach is applied to explain the equilibrium of some beautiful and puzzling structures, such as open-well spiral stairs.

Maurizio Angelillo, currently a Professor of Structural Engineering at the School of Engineering and Architecture of the University of Salerno, is an Architect who graduated at the University of Naples in 1977 and did his post-graduate studies at the Department of Aerospace Engineering and Mechanics of the University of Minnesota, from 1982 to 1984.

His research interests range from Biomechanics (he has worked in respiratory muscle physiology and brain mechanics and is still active in the modelling of the eye) to Masonry Mechanics. He started his research on masonry, both from the theoretical and experimental point of view, since the early 80ties, at the University of Naples, working in the research group of Professor Giovanni Castellano. Castellano, with other Italian scientists (Di Pasquale, Como, Baratta, the Romano brothers and some Italian mathematicians such as Giusti, Giaquinta and Anzellotti) contributed to the advance, generalization, formalization and diffusion of the No-Tension model for masonry put forward by Jacques Heyman in 1966.

Maurizio Angelillo, has written several papers concerning masonry structures, such as vaults, domes, churches and stairs, published in leading Mechanic Journals, and has been the coordinator of intensive doctoral and post-doctoral courses on masonry (such as the CISM course 551 (2011), Mechanics of Masonry Structures, CISM Courses and Lectures, Springer 2014). Currently is one of the organizers of the International Summer School on Historic Masonry Structures, that last year was held in Subiaco (https://historicmasonrysubiaco.wordpress.com/course/) and this year will take place in Anagni ( https://historicmasonryschool2019.wordpress.com ).

This talk is part of the Engineering Department Structures Research Seminars series.