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The paper is related to the assessment of collapse mechanisms of historic masonry structures suffering settlements induced by ground movements. Two numerical strategies are adopted in order to study the influence of the settled zone on the cracking of masonry buildings: a discrete rigid block model and a continuous homogenized model. The first approach provides an estimate of the collapse load and failure pattern of masonry based on the lower bound theorem of limit analysis. The second approach is formulated in the framework of multi-surface plasticity and implemented in a FE code for the path-following non-linear analysis of masonry wall described as continuous anisotropic plate. Several settlement configurations, of masonry walls under moving ground support are investigated and the corresponding failure patterns resulting from the analysis are obtained resulting in local or global failure modes. The results of the two modeling formulations are compared and discussed in order to highlight the features of the two different approaches in the prediction of settlement-induced damage.

The present study concerns the structural analysis of the masonry vaulted stairs of the Santa Maria delle Vigne bell tower in Genoa (Italy). The vaulted staircase systems are fully analysed in terms of technological and constructive details. The 3D geometric models are defined from laser scanner data. The structural analyses were carried out by using both the equilibrium limit analysis and a static incremental FEM analysis. Despite the efficacy of classic equilibrium methods in analysing arched and vaulted structures is largely proved in literature, this study demonstrates that vaulted staircase systems often collapse because of compressive failure of masonry before losing stability.

This paper investigates the structural behaviour of masonry vaulted staircases and the efficacy of different modelling strategies for the assessment of their structural safety. In particular, the present study focuses on the structural analysis of the masonry vaulted stairs of the medieval Santa Maria delle Vigne bell tower in Genoa (Italy), dating back from XVI to XVII Century. Both a limit analysis tool and a static incremental analysis using a finite element model are adopted. The limit analysis problem is formulated for a rigid block model using an iterative solution procedure to take into account crushing failure. Results indicate that crushing failure is crucial in capturing the response of this structural typology.

This paper deals with the force-based assessment of collapse mechanisms and strengthening interventions of the historic masonry castle &ldquo, Bussi sul Tirino&rdquo, (Abruzzi, Italy) using rigid block limit analysis (RBLA). The structure, which is a fortified palace dating back to the 11th century, has experienced severe earthquakes over the centuries and was hit once again in 2009 by the L&rsquo, Aquila earthquake. Based on the historical analysis and the results of in situ investigations, a spatial rigid block model of an entire structural unit was generated using the in-house software LiABlock_3D. The software is a MATLAB&reg, based tool for three-dimensional RBLA, which provides as outputs collapse failure modes and collapse load multipliers. In addition, a specific routine was developed for the purpose of the study to compute the participating mass ratio and the spectral acceleration that activated the failure mechanisms. The results of the numerical analysis were used to address three different retrofitting interventions, based on the use of connection elements and ties that, according to the minimum intervention principle, progressively enhanced the seismic capacity. Comparisons in terms of seismic safety indices are finally provided in order to give a quantitative measure of the effectiveness of the adopted retrofitting strategies.

A simple variational formulation for contact dynamics is adopted to investigate the dynamic behavior of planar masonry block structures subjected to seismic events. The numerical model is a two-dimensional assemblage of rigid blocks interacting at potential contact points located at the vertices of the interfaces. A no-tension and associative frictional behavior with infinite compressive strength is considered for joints. The dynamic contact problem is formulated as a quadratic programming problem (QP) and an iterative procedure is implemented for time integration. Applications to analytical and numerical case studies are presented for validation. Comparisons with the experimental results of a masonry wall under free rocking motion and of a small scale panel with opening subjected to in-plane loads are also carried out to evaluate the accuracy and the computational efficiency of the formulation adopted.

In this paper, a three-dimensional rigid body model is implemented to investigate the behaviour of masonry block structures subjected to foundation settlements. The structural behaviour is investigated formulating a limit analysis problem for the calculation of the base reaction and the associated failure mode once the settlement is introduced in the model. The structural model is an assemblage of rigid blocks which interact at contact interfaces. The latter are modelled as concave-contacts, using contact points located at the corners of the interface to represent interactions. A no-tension and non-associative frictional behaviour with infinite compressive strength is considered for joints. The limit analysis problem is formulated as a second order cone programming problem (SOCP) and an iterative procedure is adopted to take into account non-associative frictional behaviour. A computer program has been developed for applications of the proposed procedure. Two simple numerical examples are presented with the aim to verify the obtained results against analytical solutions. Applications to a set of experimental tests and to a 3D building model are introduced for validation and to evaluate the computational efficiency of the model. Finally a dissipation index is defined in order to give a quantitative estimation of the consequences related to the foundation settlement.

This paper deals with the vulnerability assessment of historic masonry structures subjected to settlements using rigid block modelling. A 2-D rigid block model with unilateral elastic contacts and finite friction is developed for the evaluation of the displacement capacity in the large displacement regime by push-down analysis. A variational formulation of the rigid block model is adopted, which relies on associative behavior for displacement rates. Under this assumption, the equation systems governing the behavior of the rigid block model can be uncoupled into two equivalent force and displacement-based problems, thus reducing computational costs. The numerical model was validated against the results of an ad-hoc experimental campaign on small scale tuff panels and against the tests on a small-scale masonry facade subjected to moving supports, from the literature. Applications of the proposed modelling approach are presented to the assessment of a full scale, monumental masonry facade. Following classic force-displacement methods that are used in the case of seismic actions, capacity curves are proposed for the damage assessment induced by settlements. Those state the relation between the base reaction at the moving supports and the displacement of a control point, which is obtained from the push-down analysis. Finally, comparisons with empirical assessment methods from the literature are presented.

These simulations deal with the application of LiABlock_3D, a rapid computational tool developed by the Authors [1], for limit analysis of masonry structures subjected to point live loads, seismic induced lateral loads and moving supports. The software was previously validated against several experimental tests available in the scientific literature mainly concerning small scale specimens of masonry walls subjected to lateral loads and settlement. The three-leaf masonry walls of the numerical contest will be represented as 3D assemblages of rigid blocks interacting at no-tension, frictional contact interfaces. LiABlock_3D will provide as outputs the computed collapse load and the plot of the corresponding failure mechanism.

The aim of this paper is to show to what extent a simple constitutive model can adequately describe the collapse mechanisms of historic masonry structures under horizontal seismic loads. Referring to block masonry, the paper presents the formulation and the numerical implementation of a constitutive relationship for modeling masonry structures regarded at a macroscopic scale as homogenized anisotropic continuum. The macroscopic model is shown to retain memory of the mechanical characteristics of the joints and of the shape of the blocks. The overall mechanical properties display anisotropy and singularities in the yield surface, arising from the discrete nature of the block structure and the geometrical arrangement of the units. The model is formulated in the framework of multi-surface plasticity. It is implemented in a FE code by means of a minimization algorithm directly derived from the Haar-Karman principle. A sensitivity analysis to mechanical and geometrical parameters was carried out to show the influence on the predicted response of small-scale wall components from the literature. The model is then used for the analysis of a numerical case study which is representative of a masonry church subjected to horizontal seismic action. The result of the model will be compared with those obtained by a rigid block model (RBM) of the same small-scale masonry prototypes and of the church building. In the paper, the results of the two models are discussed and analyzed in the way to highlight the weakness and the potentiality of the two different approaches.

The paper deals with the application of LiABlock_3D, a novel software tool developed at the University of Naples Federico II, to the case study of an historic masonry viaduct listed by UNESCO as World Heritage Site. LiABlock_3D analyses masonry structures under seismic lateral loads and automatically predict the potential failure modes of complex assemblages in terms of local and global collapse mechanisms. The numerical formulation is based on limit equilibrium analysis and mathematical programming is used to solve the contact problem. Masonry structures are represented as a collection of three-dimensional rigid blocks. Once the geometry of the construction has been generated (i.e. using different CAD tools or MATLAB® scripts for parametric analysis), the numerical model requires few parameters as input data, such as the unit weight of materials and the friction coefficient. Several loading conditions can be considered, assuming different loading axis and load distribution (uniform and/or concentrated loads can be applied). With respect to failure condition, LiABlock_3D considers no-tension, frictional contact interfaces with infinite compressive strength. Opening and sliding failure conditions are considered. In order to show the potentialities and limitations of the developed software, a comparison of the outputs provided by LiABlock_3D with those obtained by Finite Element Analysis is presented and discussed as well.

A formulation for limit analysis of three-dimensional masonry structures modelled as rigid block assemblages is presented. A concave contact model is adopted for interfaces, using contact points located at the corners of the interface to represent interactions. A no-tension and non-associative frictional behaviour with limited compressive strength is considered for joints. The limit analysis problem is formulated as a second order cone programming problem (SOCP) and an iterative procedure is proposed to model crushing failure and to take into account non-associative frictional behaviour. Applications to numerical case studies are presented for validation. Finally, the accuracy and the computational efficiency of the proposed formulation are evaluated by a comparison with the results of a full scale experimental sub-assemblage of a masonry pier-spandrel system.

In this paper, a masonry system composed of a facade wall connected with two sidewalls and subjected to out-of-plane loading is investigated within the framework of three-dimensional limit analysis. Two different modeling approaches, namely macro and micro-block models, are adopted. A rigid-perfectly-plastic model with dry contact interfaces governed by Coulomb failure criterion is assumed for masonry walls with regular units and staggering (non-standard limit analysis). Three classes of failure modes are investigated, involving rocking, sliding, twisting failure and combinations of them. The macro-block model is based on the assumption that the failure involves a number of cracks which separate the structure into a few macro-blocks and all the possible relative motions among micro-blocks are concentrated along the cracks. Two limiting conditions for the ultimate load factor are kinematically computed by use of minimization routines. The micro-block model is based on a concave contact formulation in which contact points are located at the corners of interfaces, allowing failure modes involving opening and sliding to be simulated. An iterative solution procedure is used to solve the non-associative friction problem, with second order cone programming (SOCP) used to allow the conic yield function to be solved directly. Both models are validated against experimental outcomes from the literature. A parametric analysis is carried out in order to highlight the influence of each geometrical and mechanical parameter on the prevalence of a mechanism over the other. The presence of an unrestrained horizontal floor system with different orientations is also analyzed.

In this paper, a formulation for limit analysis of three-dimensional masonry structures discretized as rigid block assemblages interacting through no-tension and frictional contact interfaces is developed. Linear and piecewise linearized yield functions are used for rocking, sliding and torsion failure. A simple yield condition has been defined to take into account interaction effects of shear force with torsion and bending moment. Associative flow rules are considered for strain rates. On the basis of the developed governing equations, the limit analysis problem has been formulated as a nonlinear mathematical program. An iterative solution procedure based on linear programming is used to solve the limit analysis problem and to take into account nonlinearities due to the influence of bending moments and shear stresses on torsion strength. The results of experimental investigations on out-of-plane masonry walls constrained at one edge and different examples from literature were considered for validation. Comparison with existing formulations is carried out. �� 2013 Springer-Verlag Berlin Heidelberg.

A 3D rigid body model is formulated to investigate the behaviour of masonry block structures subjected to foundation settlements. The structural model is an assemblage of rigid blocks which interact at contact interfaces. Contact forces are located at the vertices of interfaces and a no-tension and non-associative frictional behaviour with infinite compressive strength is considered for joints. The limit analysis problem is formulated as a mathematical programming problem and a computer program has been developed for applica-tions of the proposed procedure. The rigid block model has been validated against the results of experimental tests carried out on small scale panels, made of dry-jointed tuff blocks with dimensions of 200×100×50 mm. The bearing support system was designed so to induce a uniform settlement along a part of the panel. The comparison of numerical and experimental results was carried in terms of failure mechanism and distribution of base reactions at collapse. To evaluate the computational efficiency of the developed formulation, an ap-plication to a numerical case study of a 3D building with a large number of blocks and contact interfaces is also presented.

In this paper a variational formulation for dynamic analysis is adopted to investigate rock-ing behaviour of masonry block structures under lateral loads. The model is composed of rigid bodies interacting at potential contact points located at the vertexes of the block inter-faces. A no-tension and associative frictional behaviour with infinite compressive strength is assumed at contact interfaces. The contact dynamic problem is governed by equilibrium equations, which relate external, inertial and contact forces, and by kinematic equations, which ensure compatibility between contact displacement rates and block degrees of free-dom. Mathematical programming is used to solve the optimization problem arising from the formulation of the variational problem associated to dynamics of the block assemblages. To evaluate the accuracy and computational efficiency of the implemented formulation, a vali-dation study is presented for rigid blocks subjected to rocking behaviour under different ac-celeration pulse types and for an in-plane wall panel problem from the literature. A good agreement in terms of failure mechanism and response time histories was observed. The computational efficiency and the stability of the implemented procedure were found to be encouraging, thus suggesting that the proposed model may be used to model dynamic behav-iour of masonry block assemblages with a large number of rigid bodies.

In this paper a rigid block model with cracking units and cohesive joints is developed for limit analysis of masonry structures using linear programming. The model is applicable to in-plane loaded unreinforced masonry block walls with regular textures and provides as output failure loads and collapse mechanisms. A simplified micro-modelling approach is adopted, based on the discretization of masonry units into triangular blocks separated by contact interfaces. Tensionless contact interfaces are used to model dry joints and cohesive contacts are introduced for mortar joints and internal unit interfaces. Failure modes at contact interfaces involve cracking, crushing and sliding. An iterative solution procedure is used to model non-associative flow rule in sliding and to take into account no-tension behaviour of cohesive joints in case of cracking. The modelling approach is validated and applied to different structural examples from the literature, including unconfined and confined masonry panels and a masonry beam. To show the accuracy of the proposed model and the improvement of the predicted response, the obtained results are compared with experimental tests and with the outcomes of standard rigid block models using a single block per masonry unit and tensionless interfaces.

In this paper, we present a three-dimensional limit equilibrium analysis of an unreinforced brick masonry building. The structure under investigation is a one-room one-floor building that has been selected from a full-scale test carried out within the framework of a joint USA-Pakistan research project. The structural model of the building is an assemblage of rigid blocks, which interact through no-tension contact surfaces with Coulomb friction. A micro-modelling approach is adopted where each masonry unit is modelled as a rigid body. The underlying limit analysis formulation is based on the concave contact model, with contact forces located at the vertices of the interfaces. Iterative calculations are carried out to deal with the non-associative friction behaviour. The numerical analysis is carried out with the aid of a computer program that provides as output the failure load and the collapse mechanism. The comparison with experimental outcomes shows that the adopted formulation is capable to predict the actual behaviour of 3D masonry structures with a reasonable computational effort.

The assessment of masonry block structures under lateral loads can be efficiently carried out using computational limit analysis for the evaluation of the collapse load factor and the corresponding failure mechanism. In this study a formulation based on mathematical programming is presented to discuss the accu-racy and the computational efficiency of the proposed modeling approach. The formulation adopts a rigid body modeling approach and uses a concave model for contact interfaces, for which a no-tension and friction-al behaviour is assumed. That allows to formulate the limit analysis problem as a second order cone pro-gramming problem and to take into account non-associative behaviour under sliding failure by means of a simple iterative procedure to save CPU time. An application to a case study from the literature is presented and discussed also on the basis of the results of experimental investigation.

In this paper, a rigid block model with cracking units for the limit analysis of masonry structures is presented. A simplified micro-modelling approach is adopted, based on the discretization of the single masonry unit into multiple blocks separated by cohesive contacts. Failure modes, which may occur at mortar and mortar-unit interfaces as well as in the masonry unit itself, involve crushing, cracking and sliding. The modelling approach is validated against a literature case study of unconfined masonry panel, whose collapse mechanism involves cracking of units. To show the accuracy of the proposed modelling approach, the obtained results of the analysis are compared with experimental tests and with the outcomes of other modelling approaches used in the literature.