Nicola Zani

Masonry-like solids with bounded compressive strength

Abstract This paper proposes a constitutive equation for masonry-like materials with bounded compressive strength. The general properties of this equation are proved and its solution is explicitly calculated. Subsequently, a numerical method is proposed in order to solve the equilibrium problem of masonry-like solids with bounded compressive strength. In particular the derivative of the stress with respect to the total strain is calculated; this derivative will be used for calculating the tangent stiffness matrix and then for solving the non-linear system, obtained with the discretisation into finite elements via the Newton-Raphson method. Finally, this numerical method is applied to the study of Moscas bridge in Turin and to the study of a three-dimensional circular reduced arch subjected to its own weight and a vertical load distributed along the extrados.

On the Collapse of Masonry Arches

In this paper a constitutive equation for masonry arches is defined and its main properties are proven; in this equation to each pair of generalized strains (ε, κ), with ε the extensional strain and κ the curvature change of the centre line, is assigned the pair of generalized internal forces (N,M), where N is the normal force and M the bending moment. Subsequently, the collapse of masonry arches is characterized and the static and kinematic theorems proven. Finally, a method for determining the collapse load in the case of circular arches subjected to their own weight and a vertical point load applied at a point of the extrados is presented. The results obtained, of interest in some applications, are summarized in a series of graphs.

The Maximum Modulus Eccentricities Surface for Masonry Vaults and Limit Analysis

This paper presents a constitutive equation for masonry vaults which associates to each pair of generalized strains (A, K), where A are the strains, K are the curvature changes of the mean surface, the pair of generalized internal forces, (N, M) with N and M being the normal forces and bending moments per unit length, respectively The maximum modulus eccentricities surface is defined, and its main properties are proved. Subsequently, the limit analysis for masonry vaults is set forth and applied to the evaluation of the collapse load for a toroidal tunnel subjected to its own weight and an uniform load applied on the top circle.

Effectiveness of nonlinear static procedures for slender masonry towers

This paper investigates the accuracy of pushover-based methods in predicting the seismic response of slender masonry towers, through comparison with the results from a large number of nonlinear time-history dynamic analyses. In particular, conventional pushover analyses, in both their force- and displacement-based variants, are considered, and seismic assessment through the well-established N2 method is also addressed. The study is conducted by applying a simple non-linear elastic model recently developed and implemented in the computational code MADY to represent slender masonry structures. The model enables both pushover analyses and non-linear dynamic analyses to be performed with a minimum of effort. A multi-record incremental dynamic analysis carried out for a quite large number of structural cases, each of which is subjected to a comprehensive set of dynamic nonlinear analyses, is used to evaluate the accuracy of pushover methods in predicting the global structural response, as represented by the usual capacity curve together with a damage curve, both of which are compared with dynamic envelopes. Local responses, in terms of lateral displacements and the distribution of damage along the tower height are also compared. The results reveal that the key issue in the accuracy of pushover methods is the nature of the lateral load applied, that is, whether it is a force or a displacement. Different ranges of expected deformation are suggested for adopting each type of lateral load to better represent the actual behaviour of masonry towers and their damage under seismic events through pushover methods.

A note on equilibrated stress fields for no-tension bodies under gravity

We study the equilibrium problem for two-dimensional bodies made of a no-tension material under gravity, subjected to distributed or concentrated loads on their boundary. Admissible and equilibrated stress fields are interpreted as tensor-valued measures with distributional divergence represented by a vector-valued measure, as developed by the authors of the present paper. Such stress fields allow us to consider stress concentrations on surfaces and lines. Working in R n , we calculate the weak divergence of a stress field that is asymptotically of the form |x|-n+1To(x/|x|) for x → 0 on a region that is asymptotically a cone with vertex 0. Such stress fields arise as parts of our solutions for two-dimensional panels. Proceeding to problems in dimension two, we first determine an admissible equilibrated solution for a half-plane under gravity that underlies two subsequent solutions for rectangular panels. For the latter we give solutions for three types of loads.

A numerical method for solving equilibrium problems of no-tension solids subjected to thermal loads

This paper starts out by recalling a constitutive equation of no-tension materials that accounts for thermal dilatation and the temperature dependence of the material parameters. Subsequently, a numerical method is presented for solving, via the finite element method, equilibrium problems of no-tension solids subjected to thermal loads. Finally, three examples are solved and discussed: a spherical container subjected to two uniform radial pressures and a steady temperature distribution, a masonry arch subjected to a uniform temperature distribution and a converter used in the steel and iron industry.

Dynamic Analysis of FRP-Reinforced Masonry Arches via a No-Tension Model with Damage

A FE beam model to perform static and dynamic analysis of fiber-reinforced masonry arches is presented. Based on a constitutive equation formulated for no-tension masonry beams, the model accounts for a limit to the material deformability and provides for irreversible damage occurring under compression. In order to capture any possible FRP debonding, a procedure is also formulated to reduce the performance of the fiber when the tangential and normal stresses at the masonry-composite interface reach a critical value. Some dynamic analyses are performed on a case study with the aim of evaluating the effectiveness of FRP-retrofitting in improving seismic performances.

A 3D Masonry-Like Model with Bounded Shear Stress

This paper deals with non linear elastic materials for which not all the stresses are admis-sible but only those which belong to the stress range, i.e. a closed and convex subset of the spaceof all symmetric tensors. The constitutive equation that has been formulated and explicitly solved issufficiently general to include, besides the so-called masonry-like materials, many others whose stressrange is obtained experimentally or is theoretically defined. The model, implemented into the finiteelement code MADY, has been used to analyze a masonry panel under a bi-directional monotonicallyincremental load and the obtained numerical results have been discussed.

A Direct Approach to Membrane Reinforced Bodies

The paper deals with membrane reinforced bodies with the membrane treated as a two dimensional surface with concentrated material properties. The membrane response is linearized so that it depends linearly on the surface strain tensor. The response of the matrix is treated separately in three cases: (a) as a nonlinear material, (b) as a linear material and finally (c) as a no-tension material. For the general nonlinear material, the principle of minimum energy and complementary energy are proved. For the linearly elastic matrix the surface Korn inequality is used to prove the existence of equilibrium state under general loads. Finally, for the no-tension material a theorem stating that the total energy of the system is bounded from below on the space of admissible displacements if and only if the loads are equilibrated by a statically admissible stress that is negative semidefinite. An example presenting an admissible stress solution is given for a rectangular panel with membrane occupying the main diagonal plane.

Stress State for Heavy Masonry Panels with Openings

In this paper we study the equilibrium problem of rectangular panels with openings made of a no-tension (masonry-like) material, undergoing, besides their own weight, a uniformly distributed load on their top. Generalizing some well-established results, equilibrated stress fields that are admissible, i.e. compatible with the incapability of the material to withstand traction, are determined.