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• Innovative upper bound limit analysis approach with homogenization. • Study of six non-periodic masonry patterns, ranging from rubble to quasi-periodic masonry. • Utilization of refined meshes with "pixel 2D" FEs having only linear velocities as variables. • Master-slave approach where blocks are assumed infinitely resistant. • Comparison on in-plane homogenized failure surfaces between rubble and quasi periodic masonry. This paper presents a novel and straightforward procedure for the derivation of homogenized failure surfaces for non-periodic masonry. The most innovative feature of this procedure is the automatic generation of a convenient finite element mesh directly from the sketch of the considered masonry panel, based on the so-called "pixel strategy" that converts each pixel into an element. An upper bound limit analysis problem coupled with homogenization is then solved by aptly formulating it as a linear programming problem. Another main feature is the implementation of a reduced formulation of such problem (called "master-slave approach") so that the number of unknown variables is reduced and, consequently, the computational times needed for the extraction of the homogenized failure surfaces are shortened as well. A simple procedure is also implemented for a quick identification of the statistical Representative Element of Volume (or REV) for a non-periodic masonry panel. The REV is the smallest portion of a composite material that includes all the physical and geometrical characteristics needed for its complete description. The reliability of the procedure is tested by investigating six case studies displaying different degrees of non-periodicity, extracting and critically commenting the results obtained in terms of homogenized failure surfaces and failure modes. [ABSTRACT FROM AUTHOR]

A topology optimized rigid triangular FE macro-model with non-linear homogenized interfaces for the pushover analysis of in plane loaded masonry is presented. The shape of the mesh and the position of the interfaces is evaluated through a topology optimization approach that detects the main compressive stress fluxes in the structure. Different values of the horizontal action are considered to derive an adaptive mesh or an optimal discretization that is suitable for multiple loads. Masonry properties are calibrated by means of a homogenization approach in the nonlinear range. To tackle elastic and inelastic deformations, interfaces are assumed to behave as elasto-plastic with softening in both tension and compression, with orthotropic behavior. The two-step procedure competes favorably with classic equivalent frame approaches because it does not require a-priori assumptions on the mesh and on the length of the rigid offsets. An example of technical relevance is discussed, relying into a multi-story masonry wall loaded up to failure. [ABSTRACT FROM AUTHOR]

The macroscopic strength properties of masonry walls with joints of finite thickness subjected to out-of-plane loads are estimated following an approach similar to the so-called Method of Cells for fiber-reinforced composites. A typical representative volume is subdivided into a few sub-cells, and a strain-rate periodic, piecewise differentiable transverse velocity field, depending on a limited number of degrees of freedom, is defined. Upper bounds to the macroscopic strength domain of the wall in the space of the macroscopic bending and twisting moments are obtained by applying the kinematic theorem of limit analysis within the framework of homogenization theory for periodic media. The approximated macroscopic failure surfaces are in good agreement with the ‘exact’ ones, available in the literature for infinitely strong units and infinitely thin joints, and with those obtainable by accurate 2D and 3D numerical models, at a much higher computational cost, for units of limited strength and joints of finite thickness. The influence of compressive in-plane loads and of the joint thickness on the macroscopic out-of-plane strength of the wall is also numerically investigated. Finally, the proposed model is applied to the prediction of the bearing capacity of laterally loaded masonry elements: the accuracy of the numerical predictions is assessed by comparisons with available experimental results and with more refined numerical models proposed by other authors. [ABSTRACT FROM AUTHOR]

A novel approach for the rational arrangement of fiber reinforcements on masonry structures based on topology optimization is presented. Following previous experiences on the automatic achievement of strut-and-tie models in reinforced concrete structures, the minimization of the strain energy can be implemented to derive optimal layouts of reinforcement for any structural element. To cope with the brickwork limited strength, the optimal problem can be conveniently reformulated as the minimization of the amount of reinforcement that is required to keep tensile stresses in any masonry element below a prescribed threshold. The out-of-plane macroscopic elastic properties and strength domain of brickwork are derived through an original homogenization approach, which relies upon the discretization of 1/4 of any unit cell by six constant moment elements. Thanks to the limited number of variables involved, fast evaluations of masonry macroscopic strength domains can be obtained. This criterion is implemented into the multi-constrained discrete formulation of the topology optimization algorithm, to locally control the internal actions field over the design domain. Topology optimization is then applied to the investigation of the optimal reinforcement of plain and windowed panels, comparing the conventional energy-based method and the proposed stress-based approach. [ABSTRACT FROM AUTHOR]

The macroscopic strength domain of in-plane loaded masonry walls is derived using an approach based on the upper bound theorem of limit analysis within the framework of homogenization theory. Following an approach similar to the Method of Cells for fiber-reinforced composites, a typical representative volume of masonry is subdivided into a few sub-cells, and a strain-rate periodic, piecewise differentiable velocity field, depending on a limited number of degrees of freedom, is defined. The ensuing approximated macroscopic failure surface is found to match with fair accuracy both available experimental data and theoretical predictions obtained by other authors with more refined numerical approaches. The proposed model is also applied to the prediction of the bearing capacity of a deep masonry beam: for any joint thickness, the criterion is found to give results as accurate as other complex numerical models, which take the heterogeneous nature of masonry into account. The model thus combines computational efficiency and accuracy. [ABSTRACT FROM AUTHOR]

This study presents some numerical results related to the analysis of the structural damage of a historic masonry building, Palazzo Gulinelli, in Ferrara, Italy. A detailed analysis of the inhomogeneities of the facade, historic documentation, and recent restoration interventions carried out in an adjacent building, suggest that the Palace underwent various modifications both on the structural configuration and on the borne loads. Such modifications might be the main cause of some differential settlements and of the consequent significant crack pattern on the load-bearing walls. Therefore, in the present paper the occurrence of a crack pattern on the facade is simulated by carrying out standard linear and non-linear finite element (FE) homogenized models; differential settlements are applied in order to reproduce the structural changes occurred over time. Previous experiences of the authors (for example, Acito and Milani in 2012 and Mallardo et al. in 2008), the current crack pattern of the building (of its facade in particular) and the monitoring data referring to some of them are the main references for the analysis carried out. The structural survey, the numerical results, and the data monitoring suggest two main conclusions: 1) a good correlation between numerical results and monitoring data is assessed, therefore the cracks can be reasonably related to past differential settlements; and 2) the cracks/damage that occurred as a consequence of differential foundation settlements reduce the ability of the facade to resist seismic actions. [ABSTRACT FROM PUBLISHER]

Abstract: The buckling behavior of slender unreinforced masonry (URM) walls subjected to axial compression and out-of-plane lateral loads is investigated through a simplified “homogenized” procedure. After a preliminary analysis performed on a unit cell meshed by means of elastic FEs and non-linear interfaces, macroscopic moment–curvature diagrams so obtained are implemented at a structural level, discretizing masonry walls by means of rigid triangular elements and non-linear interfaces. The non-linear incremental response of the structure is accounted for a specific quadratic programming routine where second order effects are suitably considered adding a further term, quadratic in the nodal displacements, within the total energy of the discretized system. The finite element model, applicable under general load and restraint conditions, incorporates material and geometric nonlinearity and may be used to analyze slender URM walls, having the capability of capturing post-cracking and post-buckling behavior in an approximate but suitable way. As validation of the approach proposed, the instability behavior of some existing experimental pre-compressed four-point bending tests is reproduced. Square panels in two-way bending, exhibiting classical Rondelet’s mechanisms, are also studied. Finally the results obtained are compared with those provided both by commercial FE programs and specifically developed analytical models. [Copyright &y& Elsevier]

Abstract: A simple homogenized equilibrated and admissible model for the limit analysis of masonry walls in- and out-of-plane loaded is presented. A rectangular running bond elementary cell (RVE) is discretized by means of a few constant stress triangular elements (CST) and interfaces. Non linearity is concentrated on brick–brick interfaces and joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength. When dealing with the flexural behavior, a Kirchhoff–Love homogenized model is derived, subdividing the REV into several layers along the thickness and in discretizing each layer analogously to the in-plane case. When dealing with the in-plane behavior, a linear programming problem with very few variables is obtained, which can be handled even manually for some cases of technical interest. The lower bound model proposed is validated at a cell level through literature kinematic models, to show that the lower bound approach provides almost identical results with respect to upper bound approaches, meaning that the actual solution is well approximated. Homogenized masonry behavior is then implemented at a structural level in consolidated FE limit analysis codes for the evaluation of collapse loads and failure mechanisms of real scale masonry structures. [Copyright &y& Elsevier]

Abstract: A simple homogenized model for the non-linear and limit analysis of masonry walls in-plane loaded is presented. A rectangular running bond elementary cell (RVE) is discretized by means of a few plane-stress three-noded triangular elements and interfaces. Non-linearity is concentrated on brick–brick interfaces and joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength with softening. Homogenized masonry behavior is then implemented at a structural level in a novel FE non-linear code relying on an assemblage of rigid infinitely resistant triangular elements and non-linear interfaces, exhibiting deterioration of the mechanical properties. Non-linear analyses are conducted on full scale walls – a deep beam and a windowed shear wall, for which experimental and numerical data are available in the literature – with a very limited computational effort. To circumvent some typical drawbacks of standard FE approaches, at each load step, all interfaces are assumed to behave as an elastic–perfectly plastic material and the discretized non-linear problem is solved by means of a standard sequential quadratic programming algorithm. [Copyright &y& Elsevier]

Abstract: A homogenization model for periodic masonry structures reinforced with continuous FRP grids is presented. Starting from the observation that a continuous grid preserves the periodicity of the internal masonry layer, rigid-plastic homogenization is applied directly on a multi-layer heterogeneous representative element of volume (REV) constituted by bricks, finite thickness mortar joints and external FRP grids. In particular, reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints and the FRP grid is applied on the external surfaces of the REV. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces. A FE strategy is adopted at a cell level, modeling joints and bricks with six-noded wedge shaped elements and the FRP grid through rigid infinitely resistant truss elements connected node by node with bricks and mortar. A possible jump of velocities is assumed at the interfaces between contiguous wedge and truss elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for FRP bars some formulas available in the literature are adopted to reproduce the delamination of the truss from the support. Two meaningful structural examples are considered to show the capabilities of the procedure proposed, namely a reinforced masonry deep beam (0°/90° continuous reinforcement) and a masonry beam in simple flexion for which experimental data are available. Good agreement is found between present model and alternative numerical approaches. [Copyright &y& Elsevier]

Abstract: A simple homogenized model for the non linear analysis of masonry walls out-of-plane loaded is presented. In the model, the panels are assumed to behave as Kirchhoff–Love plates. A rectangular running bond elementary cell (RVE) is subdivided into several layers along the thickness and, for each layer, a discretization where bricks are meshed with plane-stress three-noded triangular elements and joints are reduced to interfaces is assumed. Non linearity is concentrated on brick–brick and joint interfaces, which exhibit a frictional behavior with limited tensile and compressive strength with softening. Finally, macroscopic curvature bending moment diagrams are obtained integrating along the thickness in-plane micro-stresses of each layer. Homogenized masonry flexural response is then implemented at a structural level in a FE non linear code based on a discretization with three-noded elements and elasto-damaging interfaces. Three different models of increasing accuracy are presented. The first (EPP) relies in assuming an elastic-perfectly plastic behavior for the interfaces. The incremental problem is solved at a structural level through a well known quadratic-programming approach. The second (ED) accounts in an approximate way for the softening behavior and consists in a preliminary homogenized limit analysis of the structure, which allows to identify the failure mechanism and in the subsequent FE non linear analysis of the whole structure assuming that all the non linearity is concentrated on the yield line defining the failure mechanism. The last (EPD) is a sequential quadratic programming approach. Here, deteriorating bending moment curvature curves obtained through homogenization are approximated through a linear piecewise constant discontinuous function. At each load step, all interfaces are assumed to behave as an elastic-perfectly plastic material and the discretized non linear problem is solved by means of the quadratic programming algorithm used for the EPP model. The two step model proposed is validated both a cell level and at a structural level comparing results provided with both experimental data and existing macroscopic numerical approaches available in the literature. [Copyright &y& Elsevier]

Abstract: Masonry curved structures, as for instance arches, domes and vaults, are very diffused in historical and existing structures and usually require seismic upgrading and/or rehabilitation. Where FRP external strips cannot be applied for some reasons, the utilization of FRP bars embedded near the external surface becomes a very interesting and effective alternative. In this paper, a kinematic Finite Element limit analysis model to predict collapse loads and failure mechanisms of masonry curved structures reinforced with near surface mounted FRP bars regularly distributed is presented. Reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where a central brick is supposed interacting with its neighbors by means of finite thickness mortar joints, filler epoxy resin and FRP rods. In the model, it is required only that the curved structure results from a periodic disposition of bricks, mortar and FRP bars. Therefore, any pattern (multi-leaf, multi-head and single leaf) may be potentially investigated with the procedure proposed. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces. A FE strategy is adopted to solve the homogenization problem at a cell level, modeling joints, bricks, filler and FRP rods by means of 8-noded infinitely resistant parallelepiped elements. A possible jump of velocities is assumed at the interfaces between contiguous elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for epoxy resin and FRP bars some formulas available in the literature are adopted in order to take into account in an approximate but effective way, the delamination of the bar from the epoxy and the failure of the filler at the interface with the joint. In order to validate the model proposed, two numerical examples are analyzed, consisting of a circular masonry arch and a hemispherical dome. For both the examples presented, comparisons with experimental evidences, where available, and alternative non-linear FE procedures are reported. Reliable predictions of collapse loads and failure mechanisms are obtained with the model proposed for all the cases analyzed, meaning that the approach may be used by practitioners for a fast and reliable evaluation of the effectiveness of a strengthening intervention. [Copyright &y& Elsevier]

Abstract: A 3D homogenized FE limit analysis software for the numerical prediction of collapse loads and failure mechanisms of entire masonry buildings reinforced with FRP strips is presented. In particular, a two steps approach is adopted: in step I, masonry homogenized failure surfaces are obtained through an admissible kinematic FE approach in the representative element of volume (REV), constituted by a brick interconnected with its six neighbors with finite thickness mortar joints. 8-Noded rigid infinitely resistant parallelepiped elements interconnected with interfaces with frictional behavior and limited tensile and compressive strength are utilized to model the REV. A simple linear programming problem in few variables is obtained, suitable to recover numerically masonry failure surfaces when loaded in- and out-of-plane. In step II, homogenized failure surfaces are implemented in the novel FE kinematic limit analysis software for an inexpensive evaluation of collapse loads of entire buildings. Delamination is considered in the model imposing to FRP–masonry interfaces a limited resistance in agreement with Italian code CNR-DT-200. 6-Noded rigid infinitely resistant 3D wedge-shaped elements are used to model homogenized masonry, whereas FRP strips are modeled by means of triangular 3-noded rigid elements. A two story masonry building reinforced in various ways with FRP strips and experimentally tested at Georgia Tech under quasi-static horizontal loads is analyzed to assess numerical results. Good agreement is found both in presence and absence of reinforcement, meaning that the procedure proposed may be used by practitioners for a reliable evaluation of collapse loads and failure mechanisms of complex 3D strengthened masonry structures. [Copyright &y& Elsevier]

Abstract: A homogenized limit analysis model for the prediction of collapse loads and failure mechanisms of masonry walls reinforced with near surface bed joint GFRP bars is presented. Reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints, filler epoxy resin and FRP rods. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces. A FE strategy is adopted to solve the homogenization problem at a cell level, modeling joints, bricks, filler and FRP rods by means of eight-noded infinitely resistant parallelepiped elements. A possible jump of velocities is assumed at the interfaces between contiguous elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for epoxy resin and FRP bars some formulas available in the literature are adopted in order to take into account in an approximate but effective way, the delamination of the bar from the epoxy and the failure of the filler at the interface with the joint. In order to validate the model proposed, two meaningful examples are critically analyzed. The first relies on a reinforced masonry beam in four-point bending, whereas the second is a full scale wall constrained at three edges and loaded until failure with a distributed out-of-plane pressure. While the first example is useful to test the model at a cell level, since only horizontal ultimate bending moment is involved in the failure mechanism, the second provides a full assessment of the procedure proposed at a structural level. In both cases, very good agreement is found with literature data, meaning that the model proposed may provide useful information for all practitioners interested in the design of masonry walls reinforced with bed joint FRP bars. [Copyright &y& Elsevier]

Abstract: A homogenized full 3D limit analysis model for the evaluation of collapse loads of FRP–reinforced masonry vaults is presented. Six-noded rigid infinitely resistant wedges are used to model masonry. Three-noded rigid infinitely resistant triangles are used to model FRP strips. Plastic dissipation is allowed only at the interfaces between adjoining elements. Unreinforced masonry homogenized failure surfaces obtained in Part I of the present paper are used to evaluate plastic dissipation at the interfaces between masonry/masonry elements. A possible dissipation at the interfaces between FRP triangles and masonry wedges is also considered in order to take into account, in an approximate but effective way, the possible delamination of the strips from the supports. Italian code CNR DT 200 [CNR-DT 200, 2006. Guide for the design and construction of externally bonded FRP systems for strengthening existing structures. C.N.R., National Research Council, Italy; 2006.] formulas are used to evaluate peak interface tangential strength. While the delamination from the support can be modelled only in an approximate way within limit analysis, the aim of the paper is to accurately reproduce the change in the failure mechanism observed in experimentations due to the introduction of strengthening elements. A 3D approach to model masonry is used in order to take into account the presence of FRP strips either at the extrados or at the intrados of the vaults. Several numerical examples are analyzed, consisting of two different typologies of masonry arches (a parabolic vault and an arch in a so-called “skew” disposition), a ribbed cross vault, a hemispherical dome and a cloister vault. For all the examples presented, both the unreinforced and FRP-reinforced case are discussed. Additional non-linear FE analyses are performed, modeling masonry through an equivalent macroscopic material with orthotropic behavior at failure and possible softening, in order to assess limit analysis results. Comparisons with experimental evidences, where available, are finally reported. Reliable predictions of collapse loads and failure mechanisms are obtained with the model proposed for all the cases analyzed, meaning that the approach proposed may be used by practitioners for a fast and reliable evaluation of the effectiveness of a strengthening intervention. [Copyright &y& Elsevier]

Abstract: A simple rigid-plastic homogenization model for the analysis of masonry structures subjected to out-of-plane impact loads is presented. The objective is to propose a model characterized by a few material parameters, numerically inexpensive and very stable. Bricks and mortar joints are assumed rigid perfectly plastic and obeying an associated flow rule. In order to take into account the effect of brickwork texture, out-of-plane anisotropic masonry failure surfaces are obtained by means of a limit analysis approach, in which the unit cell is sub-divided into a fixed number of sub-domains and layers along the thickness. A polynomial representation of micro-stress tensor components is utilized inside each sub-domain, assuring both stress tensor admissibility on a regular grid of points and continuity of the stress vector at the interfaces between contiguous sub-domains. Limited strength (frictional failure with compressive cap and tension cut-off) of brick-mortar interfaces is also considered in the model, thus allowing the reproduction of elementary cell failures due to the possible insufficient resistance of the bond between units and joints. Triangular Kirchhoff-Love elements with linear interpolation of the displacement field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to impacts. In order to test the capabilities of the approach proposed, two examples of technical interest are discussed, namely a running bond masonry wall constrained at three edges and subjected to a point impact load and a masonry square plate constrained at four edges and subjected to a distributed dynamic pressure simulating an air-blast. Only for the first example, numerical and experimental data are available, whereas for the second example insufficient information is at disposal from the literature. Comparisons with standard elastic–plastic procedures conducted by means of commercial FE codes are also provided. Despite the obvious approximations and limitations connected to the utilization of a rigid-plastic model for masonry, the approach proposed seems able to provide results in agreement with alternative expensive numerical elasto-plastic approaches, but requiring only negligible processing time. Therefore, the proposed simple tool can be used (in addition to more sophisticated but expensive non-linear procedures) by practitioners to have a fast estimation of masonry behavior subjected to impact. [Copyright &y& Elsevier]

The mechanics of masonry structures have been underdeveloped for a long time in comparison with other fields of knowledge. Presently, nonlinear analysis is a popular field in masonry research and homogenization techniques play a major role despite the mathematical and conceptual difficulties inherent to this approach. This paper addresses different homogenization techniques available in published literature, aiming at defining a first catalogue and at discussing the advantages and disadvantages of the different approaches. Finally, special attention is given to a micromechanical based model and a model based on a polynomial expansion of the microstress field. These seem promising and accurate strategies for advanced structural analysis. [ABSTRACT FROM AUTHOR]

• Innovative Upper Bound Limit Analysis approach with homogenization. • Out-of-plane kinematics addressed with Kirchhoff-Love plate model equations. • Out-of-plane homogenized failure surfaces for multi-leaf non-periodic masonry walls. • Master-slave approach with infinitely resistant brick/stones. • Study of two non-periodic masonry multi-leaf walls to test the method. This paper presents a 3D model aiming at investigating the out-of-plane collapse behavior of multi-leaf non-periodic masonry walls. These structural elements are widely employed in several constructions that are part of the European architectural heritage, but have been seldom addressed in the available literature due to their natural complexity and uniqueness. Specifically, the model here proposed allows a quick assessment of the out-of-plane collapse behavior of multi-leaf walls through the extraction of out-of-plane homogenized failure surfaces, acting as macroscopic failure criteria in bending and torsion. These failure surfaces are obtained after solving a standard-form linear programming problem written into a MATLAB script, which expresses a limit analysis problem pairing the well-established upper bound theorem and a homogenization approach. A twofold numerical application is performed by investigating a rubble masonry three-leaf wall and a quasi-regular three-leaf wall; in the latter case study, the influence of the rate of transversal interconnection between the outer wythes of the wall is also investigated. The results show that the presence of two different masonry bonds in the outer wythes entails some changes in the out-of-plane collapse behavior of the two case studies: namely, the out-of-plane response of the wall both under flexural and torsional actions is increased for the rubble masonry three-leaf wall, whereas it is decreased for the quasi-regular three-leaf wall. Moreover, in the latter case it is once more confirmed that a good transversal interconnection between the outer wythes - usually brought by the presence of transversal bricks – has beneficial effects on its out-of-plane collapse behavior, which are displayed both by the out-of-plane homogenized failure surfaces and deformed shapes at collapse. [ABSTRACT FROM AUTHOR]

• Original Upper Bound Limit Analysis approach with homogenization. • Out-of-plane kinematics of elements governed by Kirchhoff-Love plate model. • Derivation of out-of-plane homogenized failure surfaces for masonry. • Master-slave approach where masonry units are assumed infinitely resistant. • Validation of the proposed approach against two different models for out-of-plane collapse behavior. This paper presents a 3D brick-based model that aims at the derivation of in- and out-of-plane homogenized failure surfaces for masonry. The considered masonry panel is discretized into regular parallelepiped 3D finite elements that are supposed to be rigid; the model here introduced uses a Kirchhoff-Love plate kinematics for the out-of-plane description of the displacement rate field of the elements. A linear programming problem formulated in standard form is scripted into Matlab to derive the in- and out-of-plane homogenized failure surfaces, also enabling the extraction of failure modes for the considered masonry element. The constraints of the linear programming problem come from the combination of an upper bound limit analysis problem and a homogenization-based approach. The proposed model is validated for two separate case studies: a running bond masonry test-window and an English bond masonry test-window. The homogenized failure surfaces resulting from the current model show good correspondence to those presented in three distinct works available in literature. Also, a few relevant failure modes are derived for the two case studies, and they are consistent with the expected deformed shapes at collapse for their related load conditions. [ABSTRACT FROM AUTHOR]

Highlights • Two simple and reliable first-order homogenized plate models are presented. • The homogenized models characterize both in- and out-of-plane behaviour of masonry. • Masonry elastic properties, orthotropy and pre-compression effects are acknowledged. • The approaches are extended to study an English-bond masonry wall. • The role of 3D shear stresses and thickness discontinuities are outlined. Abstract Two simple and reliable homogenized models are presented for the characterization of the masonry behaviour via a representative volume element (RVE) defined at a structural level. An FE micro-modelling approach within a plate formulation assumption (Kirchhoff-Love and Mindlin-Reissner theory) using Cauchy continuum hypotheses and first-order homogenization theory is adopted. Brick units are considered elastic and modelled through quadrilateral finite elements (FEs) with linear interpolation. Mortar joints are assumed to be inelastic and reduced to zero-thickness interface FEs. A multi-surface plasticity model governs the strength envelope of mortar joints. It can reproduce fracture, frictional slip and crushing along the interface elements, hence making possible the prediction of a stepped, toothed or de-bonding failure pattern of masonry. Validation tests on the homogenized procedures are undertaken to conclude on the correct identification of the elastic stiffness properties, in the ability to reproduce the masonry orthotropic behaviour and the effect of potential pre-compressive states. Furthermore, the approaches are extended to characterize a case study of an English-bond masonry wall. Both the validation and application steps provide excellent results when compared with available experimental and numerical data from the literature. Conclusions on the influence of three-dimensional shear stresses and the effect of potential discontinuities along the thickness direction are also outlined. The two homogenized approaches are, for the running- and English-bond masonry cases, integrated within a FE code. By providing reliable and low computational cost solutions’, these are particularly suitable to be combined within multi-scale approaches. [ABSTRACT FROM AUTHOR]

In the present study, a simple and reliable Homogenization approach coupled with a Rigid Body and Spring Model (HRBSM) accounting for high strain rate effects is utilized to analyse masonry panels subjected to impact and blast loads. The homogenization approach adopted relies into a coarse FE discretization where bricks are meshed with a few elastic constant stress triangular elements and joints are reduced to interfaces with elasto-plastic softening behaviour including friction, a tension cut-off and a cap in compression. Flexural behaviour is deduced from membrane homogenized stress-strain relationships by on-thickness integration (Kirchhoff– Love plate). Strain rate effects are accounted for assuming the most meaningful mechanical properties in the unit cell variable through the so-called Dynamic Increase Factors (DIFs), with values from literature data. The procedure is robust and allows obtaining homogenized bending moment/torque curvature relationships (also in presence of membrane pre-compression) to be used at a structural level within the HRBS model, which has been implemented in a commercial software. At structural level, the approach resorts to a discretization into rigid quadrilateral elements with homogenized bending/torque non-linear springs on adjoining edges. The model is tested on a masonry parapet subjected to a standardized impact and on a rectangular masonry slab subjected to a blast load. In both cases, a number of previous results obtained by literature models are available for comparison, as well as experimental data. Satisfactory agreement is found between the present results and the existing literature in the field, both experimental and numerical. [ABSTRACT FROM AUTHOR]