A review of methods applicable to the study of masonry historical construction, encompassing both classical and advanced ones, is presented. Firstly, the paper offers a discussion on the main challenges posed by historical structures and the desirable conditions that approaches oriented to the modeling and analysis of this type of structures should accomplish. Secondly, the main available methods which are actually used for study masonry historical structures are referred to and discussed. The main available strategies, including limit analysis, simplified methods, FEM macro- or micro-modeling and discrete element methods (DEM) are considered with regard to their realism, computer efficiency, data availability and real applicability to large structures. A set of final considerations are offered on the real possibility of carrying out realistic analysis of complex historic masonry structures. In spite of the modern developments, the study of historical buildings is still facing significant difficulties linked to computational effort, possibility of input data acquisition and limited realism of methods.

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Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches

Is modelling and analysis of historical masonry structures necessary? Is the experimental behaviour of historical masonry constructions known? The answers seem to be ‘yes’, and substantial developments have occurred in recent decades in the challenging issues of conservation and restoration. A key issue is what type of analysis should be used. It seems that all methods are of interest, depending on the actual constraints of the engineering problem. In this paper, the possibilities of analysis of historic structures are addressed and a set of guidelines is proposed.

Computations on historic masonry structures

Masonry structures are represented by discrete element models as assemblies of blocks or particles, an idealization of their discontinuous nature that governs the mechanical behavior. This ‘discontinuum’ approach, an alternative to modeling masonry as a homogenized continuum, is particularly suited for detailed models used in fundamental research and the interpretation of experiments. Computational developments have allowed these techniques to be progressively applied in engineering practice, namely in the failure analysis of structural components and larger structures, under static or earthquake loading. A review is presented of the main models based on the discrete element and related numerical techniques that have been proposed for the analysis of masonry. The essential assumptions adopted by these models and numerical implementation issues are discussed. Differences between available models are illustrated by applications to various masonry problems that are very effectively analyzed by discrete elements.

Discrete Element Modeling of Masonry Structures

One of the research direction of Horst Lippmann during his whole scientific career was devoted to the possibilities to explain complex material behavior by generalized continua models. A representative of such models is the Cosserat continuum. The basic idea of this model is the independence of translations and rotations (and by analogy, the independence of forces and moments). With the help of this model some additional effects in solid and fluid mechanics can be explained in a more satisfying manner. They are established in experiments, but not presented by the classical equations. In this paper the Cosserat-type theories of plates and shells are debated as a special application of the Cosserat theory.

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On generalized Cosserat-type theories of plates and shells: a short review and bibliography

In historic city centers the mitigation of seismic risk is dependent on the possibility of implementing strengthening programs. Given the cultural and economic value attached to the historic structures, however, interventions should be tailored to suit aesthetic and structural requirements of each building type, and provide sufficient reliability of performance in future earthquakes. A simple analytical model is developed to calculate load factors associated with various collapse mechanisms of wall assemblies, and vulnerability functions are derived. An application shows the capability of the procedure to quantify reduction in vulnerability associated with strengthening implementations for different typologies. [DOI: 10.1193/1.1599896]

Definition of Collapse Mechanisms and Seismic Vulnerability of Historic Masonry Buildings

A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill–Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.

Scale{dependent homogenization of random composites as micropolar continua

Non-linear micropolar and classical continua for anisotropic discontinuous materials

Masonry-like materials, such as brick-block masonry or rocky materials, are modelled as discrete systems of rigid blocks in dry contact. Compressive and frictional interactions are considered. A computer procedure to determine the collapse load for two and three dimensional structures using a non-standard limit analysis approach is implemented. The mechanisms of hingeing, sliding, and twisting at the block interfaces are accounted for. The solution of the problem of non linear programming, corresponding to limit analysis in the presence of friction at interfaces, is obtained by solving a preliminary problem of linear programming, corresponding to a linearized limit analysis in the presence of dilatancy at the interfaces. Numerical analyses are performed for two and three dimensional systems with many degrees of freedom. The non linear and linearized solutions are compared and discussed. Experimental results are also presented and compared with the numerical solutions.

Limit Analysis for No-Tension and Frictional Three-Dimensional Discrete Systems*

A two-step procedure for the application of non linear constrained programming to the limit analysis of rigid brick-block systems with no-tension and frictional interface is implemented and applied to various masonry structures. In the first step, a linear problem of programming, obtained by applying the upper bound theorem of limit analysis to systems of blocks interacting through no-tension and dilatant interfaces, is solved. The solution of this linear program is then employed as initial guess for a non linear and non convex problem of programming, obtained applying both the \'mechanism\' and the \'equilibrium\' approaches to the same block system with no-tension and frictional interfaces. The optimiser used is based on the sequential quadratic programming. The gradients of the constraints required are provided directly in symbolic form. Ln this way the program easily converges to the optimal solution even for systems with many degrees of freedom. Various numerical analyses showed that the procedure allows a reliable investigation of the ultimate behaviour of jointed structures, such as stone masonry structures, under statical load conditions.

Collapse behaviour of three-dimensional brick-block systems using non-linear programming

Continuum modeling for masonry-like material accounting for bricks or blocks texture is discussed. The constitutive functions for the contact actions—expressed in terms of size, shape and arrangement of the block assembly-are derived within the framework of the linear elastic Cosserat and Cauchy theories. By varying some important geometrical parameters: the scale factor between the wall and the blocks size, the shape of the bricks and their arrangement, micropolar materials with particular internal constraints are obtained. In a few situations the constrained continuum behaves as a Cauchy continuum. In general, the Cauchy continuum does not provide a proper description of the brick masonry behaviour while the structured continuum model, accounting for the mutual blocks rotation, gives satisfactory results.

Patrizia Trovalusci

Papers

Scale{dependent homogenization of random composites as micropolar continua

Non-linear micropolar and classical continua for anisotropic discontinuous materials

Limit Analysis for No-Tension and Frictional Three-Dimensional Discrete Systems*

Collapse behaviour of three-dimensional brick-block systems using non-linear programming

Cosserat and Cauchy Materials as Continuum Models of Brick Masonry