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This study adopts an enhanced multiscale approach to investigate the effects of the damaging process on the structural behavior of masonry vaults with regular texture, in view of their safety assessment. The model, recently developed by the authors, links two different structural models at macro and microscale, exploiting the advantages of each formulation. At the macroscopic level a homogeneous Mindlin-Reissner shell is modeled and its constitutive response is derived by the detailed analysis of a three-dimensional (3D) masonry Unit Cell (UC) studied at microlevel. The UC is considered as the assembly of elastic bricks and damage-plastic zero-thickness interfaces, representative of both mortar and mortar-unit interaction, thus accounting for the actual geometry, arrangement and constitutive response of each constituent material. A Transformation Field Analysis procedure is used to link the two scales, speeding up the numerical simulations. Structural response of a masonry vault under differential settlements is investigated, determining its load-bearing capacity and the damaging path evolving in the structure up to collapse. The reliability of the results is proved by comparison with outcomes derived by detailed micromechanical analysis, interpreting and arguing similarities and differences.

The aim of this paper is to propose a new micro-mechanical model in the context of the deductive approach used to derive interface models. This model, based on a previous study introduced previously by A. Rekik and F. Lebon [31, 32], is used to reproduce the damage in masonry by combining structural analysis and homogenization methods. The focal point of this method is to assume the existence of a third material, called interphase, which is a mixture of the two principal constituents of masonry, brick and mortar, and that is the interface between them. This new element presents a low thickness, a low stiffness and a given damage ratio. The mechanical problem of masonry, initially a 3D problem, is solved numerically as a 2D problem using finite element methods. The properties of the interface brick-mortar material are obtained using three essentials steps. First of all, an exact homogenization of a laminates is used to define a first homogeneous equivalent medium named HEM-1. After, the assumption of damaged material is taken into account by using the general framework given by M. Kachanov [13, 24, 35, 36] to evaluate the global behaviour of the damaged HEM-1 defining thus a second equivalent homogeneous medium noted HEM-2. The last step consists in using an asymptotic analysis technique which is performed to model HEM-2 as an interface or a joint. The properties of this joint are deduced from those of the HEM-2 material as proposed in former papers [2, 15, 17, 34]. Particularly, through the second homogenization are taken into account the variability of microcracks oriented family and simultaneously the opening-closure effects (unilateral behaviour). Numerically this interface is modelled with connector finite elements. Numerical results are compared to experimental ones available in the literature [7]

A new macro-element based on the equivalent frame approach is presented to analyze the nonlinear structural response of masonry panels under monotonic lateral loadings. A nonlinear elastic response is assumed for the masonry material and the sectional response of the beam is derived performing analytical integration. A two-node equilibrated force-based (FB) beam finite element (FE) is formulated. The FE is composed of a central flexible part, characterized by a no-tension constitutive relationship, and a lumped nonlinear shear hinge arranged in series, in order to capture the main flexural and shear nonlinear mechanisms characterizing the masonry panel response. Some applications on experimental prototypes are presented, showing a very satisfactory agreement between the numerical results and the experimental outcomes.

Masonry arches are amongst the most fascinating structures of the World's historical and architectural heritage. Their vulnerability to earthquakes pointed out the necessity of their preservation. Fiber Reinforced Cementitious Mortar (FRCM) composites have been recently used as a more sustainable alternative to FRPs in the strengthening of masonry structures, since they can overcome the limits FRP show as strengthening systems: poor behavior to high temperature, delamination with significant loss of material at the bonded surface, lack of vapor permeability, impossibility of application on humid surfaces, incompatibility of resins with masonry, high cost and reduced reversibility of the installation. These aspects have a critical relevance in the case of historical structures, whose features have to be preserved, and motivate researches about the use of FRCM on masonry structures, from both experimental and analytical points of view. In this paper, the results of numerical analyses performed on arches unstrengthened and strengthened at the extrados with PBO-FRCM composites are presented, taking as reference the experimental tests published in [1] and [2]. The aim of this research is to further investigate the benefits that the use of FRCM composites have on the mechanical performance of masonry. The used model, implemented in the finite element code DIANA FEA, succeeded in reproducing the mechanical behavior of the considered arches in terms of collapse mechanism, load carrying capacity and ductility., (undefined)

A 3D microstructured composite Interphase Formulation (IF) based on a Multiplane Cohesive-Zone Model (M-CZM) is proposed for describing the behavior of quasi-brittle joints. The IF is devised so as to reproduce an initial linear elastic behavior, corresponding to the elastic parameters of a thin layer of finite thickness and stiffness, followed by a post-elastic response of the joint featuring progressive crack formation and onset of frictional and dilating responses. All nonlinear features of the joint response are determined by a zero-thickness multiplane-CZM accounting for the development of a microstructured crack inside the joint. The paper proposes a consistent and reproducible calibration procedure of the model parameters for its employment in the simplified micro-modeling of masonry joints. A numerical-experimental assessment of the predicted flexural response of masonry wallettes is presented, employing as a source for material parameters the data available from the experimental campaign of van der Pluijm and coworkers on the meso-scale response of units and mortar joints. The data from the structural bending tests on masonry wallets of the same campaign are used next to assess the simulated structural response following a blind-like approach, i.e., avoiding fine tuning of the mesoscale model parameters on the basis of experimental structural data on masonry wallettes. The numerical-experimental comparison appears to be satisfactory.

Masonry structures, although classically suitable to withstand gravitational loads, are sensibly vulnerable if subjected to extraordinary actions such as earthquakes, exhibiting cracks even for events of moderate intensity compared to other structural typologies like as reinforced concrete or steel buildings. In the last half-century, the scientific community devoted a consistent effort to the computational analysis of masonry structures in order to develop tools for the prediction (and the assessment) of their structural behavior. Given the complexity of the mechanics of masonry, different approaches and scales of representation of the mechanical behavior of masonry, as well as different strategies of analysis, have been proposed. In this paper, a comprehensive review of the existing modeling strategies for masonry structures, as well as a novel classification of these strategies are presented. Although a fully coherent collocation of all the modeling approaches is substantially impossible due to the peculiar features of each solution proposed, this classification attempts to make some order on the wide scientific production on this field. The modeling strategies are herein classified into four main categories: block-based models, continuum models, geometry-based models, and macroelement models. Each category is comprehensively reviewed. The future challenges of computational analysis of masonry structures are also discussed.

Several tools for the prediction and the assessment of the structural behavior of masonry buildings have been developed in recent decades. Numerical tools have been favorably developed and preferred over analytical approaches, given the complex mechanical response of masonry and the irregular geometries of historic masonry buildings. In this chapter, a thorough review of numerical strategies for the analysis of masonry structures is presented. Additionally, classification of these strategies is also suggested to logically organize the extensive literature on this topic. Even though a wholly congruent categorization of all the numerical tools is essentially unrealistic given the specific aspects of each solution developed, the existing numerical strategies are subdivided into four classes: block-based models, continuum models, geometry-based models, and macroelement models. Each class is thoroughly reviewed and the open challenges in numerical modeling of masonry structures are critically examined.

In masonry components strengthened with externally bonded composites, good bonding is one of the most important aspects governing structural behavior, since failure usually takes place with detachment between reinforcement and substrate. This type of event is a brittle, sudden and therefore undesirable failure mechanism; nor does it allow the full strength of the reinforcement to be exploited. Many experimental data on bonding have recently become available from a Round Robin Test carried out within the framework of RILEM TC-223. In the present paper, experimental tests were simulated at increasing levels of complexity; bond behavior was first studied with an analytical model based on bi- linear representation of bond law. Two-dimensional and three-dimensional finite element analyses were then performed, according to various bond-slip laws. In particular, a number of bi-linear and non-linear interface laws were used, calibrated according to several strategies but with the same experimental population. Lastly, several commercial codes and types of finite elements were examined. This work may be said to represent a numerical Round Robin Test, with various simulations and modeling approaches. Analytical and numerical results are compared with experimental ones, in terms of both overall behavior (load to displacement curve) and local behavior (strain profiles on reinforcements at increasing load values), showing the importance of both types of information in order to obtain reliable predictions of experimental results.

The paper presents the experience of a working group within the Rilem Technical Committee 223-MSC ‘Masonry Strengthening with Composite materials’, aimed at developing a standardized, reliable procedure for characterizing the bonding mechanism of masonry elements strengthened with composite materials under shear actions. Twelve laboratories from European universities and research centers were involved. Two different set-ups were compared, for single-lap and double-lap shear tests (the latter in two versions). Four kinds of fiber fabrics, i.e., glass, carbon, basalt and steel, were applied with epoxy resins (wet lay-up system) to clay brick units, for a total of 280 monotonic tests. The results provided information regarding the response of externally bonded-to-brick composites in terms of observed failure mechanisms, load capacity, effective transfer length, and bond shear stress-slip behavior. The test results of the twelve laboratories constitute a set of statistically representative data which may conveniently be used for setting appropriate design provisions and guidelines.

In the present paper, a one-dimensional elastoplastic-damage model for the analysis of the mechanical response of beams constituted by cementitious materials, i.e., concrete or masonry, strengthened by fiber reinforced polymers (FRP), is developed. The analysis is performed for a typical section, representing an elementary part of beam characterized by the finite length, defined as the distance between two fractures. A thermodynamically consistent model is proposed; it takes into account the different behavior in tension and in compression of the cohesive materials. The governing equations are derived and a numerical procedure is developed. It is based on thearc-length method, withinan implicit Euler algorithm for the time integration. An accurate choice of the control parameters is performed. The finite step nonlinear problem is solved adopting a Newton-Raphson scheme within a predictor-corrector procedure. Some numerical examples are developed in order to analyze the non trivial axial and bending behavior of reinforced concrete and masonry beams and to assess the efficiency of the proposed pmedure. Comparisons with analytical solutions are reported.

A multiscale model for the analysis of the in-plane response of periodic masonry walls is presented. The overall constitutive behavior of the composite material is derived through a homogenization procedure based on the Transformation Field Analysis properly extended to the case of interfaces. At micro level, masonry is modeled as the assembly of expanded units and interfaces representing both mortar and unit-mortar interaction. A nonlinear constitutive law accounting for damage and friction phenomena is considered for joints, whereas a linear elastic constitutive relationship is assumed for the blocks. The proposed multiscale procedure is implemented into a Finite Element code, where the mesh-dependency occurring in presence of strain softening response is overcome by adopting a nonlocal integral formulation at macro level. Validation examples are carried out: first, the response of a representative masonry Unit Cell is analyzed comparing results obtained with the presented homogenization procedure with those recovered by detailed nonlinear finite element analyses. Then, the structural behavior of masonry panels subjected to compression-shear loads is studied. The results obtained with the multiscale model, in terms of global force-displacement response curves and damage distributions, are compared with both micromechanical and experimental outcomes.

A novel two-scale modeling approach, linking different structural models at macro and microscale, is proposed to describe response of masonry walls with periodic texture. At the higher macroscopic scale, the real heterogeneous material is modeled as a homogenized medium, considering the classical Mindlin-Reissner theory for flat shells. At the lower microscopic scale, a representative masonry Unit Cell (UC), accounting for the actual geometry, arrangement and nonlinear behavior of constituent materials, is analyzed in detail by resorting to a three-dimensional Cauchy model. The UC is modeled as the assembly of elastic bricks and nonlinear zero-thickness interfaces, in which the sliding frictional and damaging mechanisms are concentrated. To perform the macro–micro information transition a proper kinematic map is defined, whereas the upscaling process is performed via a homogenization procedure based on the Transformation Field Analysis (TFA), properly extended to the case of interfaces. The developed homogenization procedure invokes a generalized Hill-Mandel principle and requires to satisfy ‘non-standard’ constraints at the microlevel, for which the perturbed Lagrangian method is employed. Numerical applications are performed to prove the model efficiency in describing the response of a running bond UC subjected to in-plane and out-of-plane loads. Special attention is devoted to the analysis of shell bending and shear behavior, comparing the results obtained with the proposed model with those recovered by detailed micromechanical analyses.

Aim of this work is the evaluation of FRCM capacity in repairing pre-damaged clay brick walls under out of plane loads. Full scale experimental tests have been performed damaging clay brick walls subjected to out of plane loads; then, tests are performed on damaged walls repaired with innovative composite grids with inorganic matrix (FRCM). The boundary constraints of the walls (two adjacent edges constrained and the other two free) allowed to simulate a non-uniform out of plane behaviour of the wall subjected to a pointwise normal load applied at the free corner. The aim of this study was to assess the potentiality of FRCM to recover stiffness after significant damage and to improve the lateral (out-of-plane) capacity of the repaired wall. Experimental evidences demonstrated that the externally bonded strengthening was not strongly affected by debonding and it was able to prevent a flexural failure; the collapse of the wall occurred at almost double load with respect to the unreinforced configuration, despite previous significant damage, and that the failure was governed by shear sliding. Future developments include experimental tests of strengthened walls not previously damaged, to investigate on the effect of pre-damage on out of plane strengthening capacity of FRCM.

In the present work, a new model of the FRP-concrete or masonry interface, which accounts for the coupling occurring between the degradation of the cohesive material and the FRP detachment, is presented; in particular, a coupled interface-body nonlocal damage model is proposed. A nonlocal damage and plasticity model is developed for the quasi-brittle material. For the interface, a model which accounts for the mode I, mode II and mixed mode of damage and for the unilateral contact and friction effects is developed. Two different ways of performing the coupling between the body damage and the interface damage are proposed and compared. Some numerical applications are carried out in order to assess the performances of the proposed model in reproducing the mechanical behavior of the masonry elements strengthened with external FRP reinforcements.

This paper presents two micromechanical and a multiscale finite element models for the analysis of masonry walls under out-of-plane instability effects. A two-dimensional modeling of the wall is considered in all approaches, assuming a cylindrical bending. The micromechanical analyses are performed considering elastic beams to model the bricks and either nonlinear beams or interfaces to model the mortar layers. The beam finite elements rely on the force-based formulation and account for large displacements by making use of the corotational approach. This latter is properly formulated and extended to interface elements to include nonlinear geometry effects. The multiscale model is defined by applying a two-scale beam-to-beam homogenization procedure, developed for masonry elements with periodic brick arrangements. Hence, a Unit Cell made of a single linear elastic brick and a single nonlinear mortar layer is introduced at the microscopic level, which is linked to the macroscale level through a semi-analytic homogenization technique. In all models, a damage formulation with friction plasticity governs the mortar constitutive relationship. Computational details on model implementation and solution procedures are given for all approaches. Correlation studies are performed to assess the proposed numerical micromechanical and multiscale procedures. In particular, the behavior of an unreinforced wall tested under a compressive eccentric load is reproduced and advantages and disadvantages of the proposed approaches are discussed.

In this paper a new numerical approach for determining admissible thrust curves for masonry arches is proposed. Arbitrary loading conditions, including distributed loads applied to the extrados and to the intrados of the arch, but also horizontal inertial forces simulating the effects of seismic actions are considered for arches of any geometry. The admissible solutions, corresponding to equilibrium thrust curves entirely contained in the thickness of the arch, are consistent with the lower bound theorem of Limit Analysis and, thus, are “safe” solutions from a structural point of view. The well-established Milankovitch's theory for the equilibrium of masonry arches is reviewed and generalized. Then, a specific formulation of the theory is presented, allowing the construction of an effective and efficient numerical procedure based on the Point Collocation Method and enriched by a constrained optimization routine. The latter is aimed at determining, among all the admissible equilibrium solutions, the optimal solution matching specific requirements of interest for applications, as the solution corresponding to the maximum or minimum thrust. The proposed procedure is discussed and validated with reference to the cases of circular, parabolic and pointed arches. In particular, maximum and minimum thrust solutions have been determined for all the examined cases.

This paper presents a kinematic enriched finite element model for nonlinear analysis of brick masonry walls loaded in their plane. The finite element accounts for the transverse deformation of the wall and permits to reproduce mortar-brick interaction in wall thickness direction. Non-local constitutive relationships are considered both for mortar and bricks, adopting a damage-friction law for the mortar and an isotropic damage model for the bricks, both accounting for tensile failure mechanisms. A numerical procedure is developed for evaluation of damage and friction in mortar and brick materials. Numerical applications are presented, comparing results obtained by the proposed finite element with experimental outcomes.

This chapter presents a survey of the homogenization and multiscale analysis of periodic, masonry walls subjected to in-plane loading conditions. The scale of the structure, macroscale, and that of the material, microscale, are introduced, invoking the assumption of the separation of the scales. Two-dimensional (2D) Cauchy and Cosserat models are adopted at the macroscale to describe the response of the wall. A unit cell (UC) is introduced and analyzed adopting three-dimensional, 2D, and enriched plane state Cauchy models. A damage-plastic model is considered for the mortar joints, accounting for the unilateral effect due to reclosure of the mortar joint and friction occurring when the microcracks are closed. A different damage model is introduced for bricks capable of properly describing their compressive failure. The homogenization for deriving the overall nonlinear response of the UC is performed by developing a piecewise transformation field analysis technique. Several numerical applications are presented to assess the material modeling and the effectiveness of the adopted homogenization technique and that of the multiscale approach. These concern the response of the UC to verify, on the one hand, the ability of the constitutive laws proposed for brick and mortar to describe the response of the masonry and, on the other hand, the performance of the homogenization technique. Then, structural analyses of masonry elements are presented, comparing the results obtained using different modeling approaches among them and with available experimental evidence. Finally, concluding remarks are given.

Different scale approaches, micromechanical, multiscale and macromechanical or phenomenological, are presented to study the structural response of masonry elements. First, a micromechanical model is introduced and the masonry is considered to be a heterogeneous material, made of mortar and bricks joined by interfaces, where the mortarbrick decohesion mechanisms occur. To this end, a special interface model combining damage and friction is proposed. Then, two multiscale procedures are presented, that consider regular arrangements of bricks and mortar, modeled by nonlinear constitutive laws which account for damage and friction effects. A homogenization technique is developed to derive two different equivalent continuum models at the macro-level, a micropolar Cosserat continuum and a nonlocal Cauchy model. Finally, a macromechanical model, based on the adoption of a classical No-Tension Material (NTM) model, and on the presence of irreversible crushing strains, is proposed. A zero tensile strength is assumed, thus fracture strains arise when the stress is zero. Moreover, an elastoplastic model is considered for the material response in compression. Numerical applications are performed on a masonry arch and two masonry panels, by adopting the three approaches presented. Comparisons with experimental outcomes, published elsewhere, are performed.

The nonlinear structural analysis represents a powerful tool for the study of the seismic response of new and existing constructions. In particular, nonlinear static analysis, also denoted as pushover analysis, is widely used for RC and steel structures and, recently, new seismic codes extended the use of this type of analysis to the case of masonry structures. The reliability of the results deduced from nonlinear analyses is not always ensured since the numerical results can depend on the modeling strategy and on the adopted solution techniques. Moreover, in the case of masonry structures strengthened with fiber reinforced polymers (FRP), particular regards has to be addressed not only to the masonry material model but also to the modeling of FRP-system, taking into account the complex phenomena characterizing the interaction between the FRP and the masonry supports. Aim of the present paper is to discuss some strategies, in the framework of the nonlinear finite element analysis, regarding the elements choice and the models to adopt for reliable nonlinear analyses of masonry structures reinforced with FRP strips. Specific structural problems are described and solved, discussing the role and the reliability of the adopted strategies. Finally, the seismic analysis of a masonry façade and the design of suitable FRP strengthening systems are performed. © 2012 Elsevier Ltd. All rights reserved.

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.

In the present paper, a multiscale analysis of periodic masonry walls is developed. In particular, a micromechanical analysis is presented for the unit cell, introducing nonlinear constitutive laws, based on damage and friction models, for the mortar and for the blocks. In order to deduce the overall response of regular masonry arrangements to be used for the multiscale analysis, the Transformation Field Analysis (TFA) homogenization procedure is extended to the case of nonuniform eigenstrain. A multiscale procedure is proposed implementing the nonlinear homogenization technique at Gauss point level, in 2D plane state finite element. A nonlocal integral model is adopted in order to overcome problems due to the localization of strain and damage. Numerical examples of homogenization are carried out, comparing the nonlinear mechanical response of the masonry, obtained by adopting the proposed homogenization technique, with the results recovered by evolutive nonlinear finite element analyses. Moreover, numerical applications regarding the mechanical response of masonry structures are performed in order to validate the efficiency of the multiscale approach. In particular, a comparison with experimental data, available in literature, is presented.

The aim of the present paper is the development of a simple procedure for the analysis of the bond behavior of fiber-reinforced polymer (FRP) sheets or plates externally applied to masonry supports for the strengthening or repair of masonry constructions. The procedure allows evaluation of the bond strength and the fracture energy developed during the debonding process through simple formulas based on a few parameters, evaluated either by standard tests performed on the materials making up the support and the strengthening system or by theoretical considerations. A brief discussion on the main experimental evidence and the theoretical models provided by the literature is also reported in this paper. The comparison between the theoretical results obtained by applying the proposed procedure and the experimental data deduced from literature is carried out.

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without per- forming at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.

Fiber Reinforced Plastic (FRP) can be used for strengthening concrete or masonry constructions. One of the main problem in the use of FRP is the possible detachment of the reinforcement from the support material. This paper deals with the modeling of the FRP-concrete or masonry damage interface, accounting for the coupling occurring between the degradation of the cohesive material and the FRP detachment. To this end, a damage model is considered for the quasi-brittle material. In order to prevent strain localization and strong mesh sensitivity of the solution, an integral-type of nonlocal model based on the weighted spatial averaging of a strain-like quantity is developed. Regarding the interface, the damage is governed by the relative displacement occurring at bond. A suitable interface model which accounts for the mode I, mode II and mixed mode of damage is developed. The coupling between the body damage and the interface damage is performed computing the body damage on the bond surface. Numerical examples are presented.

The present paper deals with the problem of the detachment of fiber reinforced polymer (FRP) from a masonry support. In particular, a new interphase model simulating the mechanical bond between masonry and FRP is proposed. It is obtained as a system of two interfaces: one reproduces the response of the glue and of the skin-deep layer and it is characterized by a linear elastic behavior whereas the second describes the nonlinear response of the detachment layer. The effect of the possible confinement due to the stress components acting in the plane of the detachment layer is also considered. Two numerical applications are reported. The first illustrates a comparison between the experimental laboratory tests and numerical simulations, showing the good performances of the proposed model. The second application is specifically devoted to the investigation of the effects of the presence of mortar joints in the detachment mechanism.

The present study deals with the determination of the nonlinear response of the masonry regarded as a regular, i.e. periodic, composite material made of bricks and mortar. A homogenization procedure is applied deriving the masonry overall mechanical response on the basis of the study of a single unit cell. An enriched plane state kinematic model including the effect of the transversal strains of the masonry is presented. This is a simplified form of the full three-dimensional approach. Different cohesive constitutive models are introduced for the brick and mortar; in particular, the frictional effect, playing an important role in the masonry response, is accounted for in the mortar joints. Two main issues are addressed: (a) different structural models are considered at macro- and micro-scale: the macro-model is formulated in the two-dimensional plane state context, while the enriched plane state kinematic approach is adopted at the microlevel; (b) a nonlocal integral strain technique, able to overcome the classical localization drawbacks due to the softening response of the masonry constituents, is developed for the case of periodic media. Numerical applications are presented to assess the effectiveness of the proposed modeling approach.

This paper deals with the nonlinear analysis of masonry walls loaded in their plane. The masonry is regarded as a composite material made of bricks joined by mortar. To correctly reproduce the mortar–brick interaction in the direction of the thickness of the wall, an enriched kinematic model is proposed, so that the model is able, in a feasible form, to account for the out-of-plane strains due to the in-plane loading acting on the wall. Nonlocal nonlinear constitutive laws are considered both for the mortar and the bricks. In particular, a damage-friction law is considered for the mortar, while a damage model with two alternative yield functions is proposed for the bricks, both based on a tensile failure mechanism. A 2D finite element (FE) accounting for the three-dimensional kinematic effect is developed. This is implemented in a numerical procedure based on the backward Euler step-by-step time integration of the constitutive evolution laws and on the predictor-corrector algorithm for the solution of the nonlinear problem. Five applications are presented to highlight the effectiveness of the proposed nonlinear model and the implemented FE procedure. The first aims to show the model’s ability to reproduce the failure of the masonry for transversal damage; the others deal with comparisons with classical small scale and structural scale experimental tests.

A cohesive zone model is formulated to describe the mechanics of initiation and propagation of cracks and the associated asperity degradation and nonlinear dilation along structural interfaces of quasi-brittle materials, such as concrete, rocks and masonry, subjected to monotonic or cyclic loading. Using a two-scale approach, a cohesive-law is determined at each point of a smooth macroscale interface by resolving a problem at the micro-scale for a representative interface area (RIA), where the geometry of the asperities is modelled using three differently inclined microplanes. On each microplane a cohesive-frictional cohesive law is then used. In this paper, the finite depth of the asperities is accounted for by considering the progressive reduction in contact area between each couple of interfacing microplanes for increasing opening (macro-scale) relative-displacement. Furthermore, the rupture of the asperities and associated flattening of the fracture surface is captured by a progressive reduction of the inclination angles of the microplanes in the RIA. Numerical examples are reported to assess the sensitivity of the shear-stress slip curves and of the nonlinear dilation upon the geometry of the asperities in the RIA. Numerical–experimental comparisons are then presented to illustrate the predictive capability of the model in simulating granite rock joints subjected to monotonic and cyclic shear loading and the concrete-bar interaction in a pull-out test.

The effects of mortars joints in masonry substrate reinforced with FRP is investigated from the numerical point of view. The analysis has been conducted by means of a new interface model specifically developed to reproduce the debonding process occurring between an elastic thin body in adhesion with a cohesive support material, as concrete or masonry. The model accounts for mode I and mode II of failure, considering the effect of the in-plane deformation of the interface, i.e. the possible elongation or confinement of the material constituting the interface. Numerical results are compared with experimental evidences showing the good performances of the proposed model in investigating the transferring phenomena in heterogeneous interfaces and, in particular, in studying the influence of the presence of mortar joints in the masonry texture in the debonding process.

The present paper deals with an investigation of the mechanical response of steel reinforced polymers (SRP) and steel reinforced grout (SRG) as strengthening systems for existing constructions. SRP together with SRG represent a new family of composite materials recently introduced for improving the structural performances of existing concrete and masonry constructions. Indeed, the numerous advantages of SRP and SRG strengthening systems, such as their reduced costs and the low flammable properties of strips and, in the case of SRGs, of the matrix, make them particularly competitive in comparison to the more traditional fiber reinforced polymers (FRP) systems. Despite the studies available in literature specifically related to SRP and SRG systems are few, they underline differences with respect to FRPs, particularly in terms of the de-cohesion mechanism and the bond performances. Aim of the present paper is to investigate the role of the type of adhesive on the bond mechanism of steel reinforcements externally applied on clay bricks. Simple formulas devoted to predict the bond resistance of such systems are also proposed and their reliability is assessed through experimental results provided, partly, by the existing literature and, in part, by some experimental tests performed in the present research.

In this paper the mechanical behavior of a masonry wall is studied. The masonry is regarded as a composite realized by a regular arrangement of blocks into a matrix of mortar. Hence, a panel of masonry is a three-dimensional heterogeneous body with a finite thickness and R 2 -periodicity in the plane of the wall. A micromechanical approach is proposed to get the overall properties of the masonry. Then, a case of a wall reinforced by FRP-layered sheets placed on the surfaces of the wall is analyzed. To model the overall behavior of the unreinforced and reinforced masonry, by accounting for the progressive damage of the mortar, of the block and of the FRP sheets, a simple homogenization technique is proposed. Two different damage criteria are adopted for the mortar and the block, within isotropic viscoelastic and elastic damage models. Furthermore, a brittle damage model is used for the reinforcement. Finally, numerical applications are developed by adopting the proposed procedure in order to investigate on the damage of the unreinforced and reinforced masonry panels.

A kinematic enriched formulation for the analysis of the in-plane behavior of regular masonry walls is proposed in order to overcome the limits of the typical plane stress and plane strain assumptions. The boundary value problem for the masonry RVE subjected to periodic boundary conditions is formulated for the enriched plane state. In particular, the displacement field is represented assuming that the components can be written using the separation of variables; in fact, they are obtained as product of in-plane and transversal functions. The in-plane displacement components are expressed as the superposition of a known field, depending on the macroscopic deformations applied to the RVE, and a periodic perturbation described as an even function of the transversal coordinate. The bending effects is avoided representing the out-of-plane displacement field in terms of odd functions of the transversal coordinate. A 2 D finite element is formulated and used for performing micro-mechanical and homogenization analyses. Numerical results are compared with analytical ones in order to assess the accuracy of the numerical procedure. Results obtained by employing the proposed model are compared with the ones evaluated on the basis of the classical plane stress, plane strain, generalized plane strain assumptions and with the three-dimensional solution. Finally, the proposed kinematically enriched model is used to derive the elastic domain of the masonry material.

A new macro-element based on the equivalent frame approach is presented to analyze the nonlinear in-plane structural response of masonry panels under lateral loadings. A nonlinear elastic response is assumed for the masonry material, limiting the focus to the case of monotonic loading condition, typical of push-over analysis. The sectional response of the beam is determined performing analytical integration, without resorting to a fiber approach. A two-node force-based (FB) beam finite element (FE) is formulated, where the resultant stress components are exactly interpolated along the beam axis. The beam FE is composed of a central flexible element, characterized by a no-tension constitutive relationship, and a lumped nonlinear shear hinge arranged in series. Hence, it is possible to capture the main flexural and shear nonlinear mechanisms typical of a masonry panel response. An efficient solution algorithm is developed, based on a consistent element state determination procedure. Some applications on simple panels and on experimental walls with openings are presented, showing a very satisfactory agreement between the numerical and the experimental results, both in terms of global push-over curves and local distributions of damaging paths.

In the present paper, the Transformation Filed Analysis (TFA) methodology is used in the multi-scale analysis of cracking localisation in masonry. The related assumption of uniform inelastic strain field in each sub-domain allows reducing the computational cost of such computations without significantly compromising on solution accuracy. The relations underlying the TFA averaging scheme are recalled. The evaluation of the corresponding consistent tangent homogenised stiffness, required in acoustic tensor-based localisation analysis, is also derived. The average RVE mechanical response is performed for two sets of micro-mechanical material laws for mortar joints, based on both damage and damage coupled with plasticity. For these two sets of fine-scale laws, localisation analyses are performed for RVEs under stress proportional loading, showing that meaningful average localisation orientations are properly detected by the acoustic tensor-based loss of ellipticity criterion. Furthermore, the capability of the pure damage micromechanical model combined with TFA to reproduce the failure envelope of typically running bond masonry subjected to uniform biaxial loads is demonstrated through a comparison with available experimental data. Finally, the energetic aspects for the selection of localised solutions are discussed. As a result, the TFA-based averaging scheme is shown to yield meaningful homogenised localisation results, thereby allowing envisioning its use in nested multi-scale computations., info:eu-repo/semantics/published

A frame element model for evaluating the nonlinear response of unstrengthened and FRP-strengthened masonry panels subjected to in-plane vertical and lateral loads is presented. The proposed model, based on some assumptions concerning the constitutive behaviour of masonry and FRP material, considers the panel discretized in frame elements with geometrical and mechanical properties derived on the basis of the different states characterizing the sectional behaviour. The reliability of the proposed model is assessed by considering some experimental cases deduced from the literature.

International audience; The present paper deals with a micromechanical model of interface able to couple the damage (micro-crack) evolution, the non-penetration conditions (Signorini equations) and the friction effect (Coulomb's law). At a typical point of the interface, a representative volume element (RVE) is considered; it is characterized by the presence of two different materials and by a microcrack evolving along the material discontinuity. An innovative deductive approach based on a micromechanical analysis and on a homogenization procedure is proposed. In particular, the solution of the micromechanical problem on the RVE is determined considering three subproblems and properly superimposing their solutions. Then, a simplified approach is proposed by modeling the behavior of the material constituting the RVE in a very essential manner. Evolutionary laws for the crack growth are given and the equations governing the unilateral and friction phenomena are presented. The original proposed procedure is applied to derive an interface model for masonry structures considering the brick-mortar interaction. The solutions of three subproblems are determined adopting the finite element method on the specific RVE for different crack lengths; then, the solutions are interpolated by adopting a spline technique and properly superimposed. A numerical procedure based on the return-mapping algorithm and the classical backward-Euler integration scheme is presented for the specific considered evolutive problem. Some numerical tests, for monotonic and cyclic loadings are presented, remarking the ability of the proposed approach to reproduce the complex features of brick-mortar interfaces; comparisons between the results obtained adopting the original proposed model and the simplified model are performed.

The present paper aims to study the debonding mechanism of carbon fibre reinforced plastic (CFRP) laminates externally applied on clay masonry bricks. To this purpose, the results of an experimental investigation conducted by the authors are discussed and compared with some experimental results reported in recent literature. Numerical analyses are also performed by considering a full three-dimensional finite element model developed by the authors on the basis of simplified assumptions suggested by the experimental evidences.

This study addresses the development of a simple beam finite element for the nonlinear analysis of masonry structures, eventually strengthened with fiber-reinforced plastic (FRP) materials, in the context of the “equivalent frame model”. The proposed beam finite element comprises three parts: two rigid offsets, able to simulate the very stiff behavior of the masonry pier-lintel intersections, and a flexible central part. Special shear interfaces are also introduced in the model in order to account for the possible shear failure of the masonry elements. Numerical applications are presented in this study to assess the reliability of the proposed model and the numerical procedure. In particular, comparisons between experimental results available in literature and the predictions obtained by adopting the proposed approach are reported.

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The masonry is considered as a composite material obtained as a regular distribution of blocks connected by horizontal and vertical mortar joints. The macromechanical equivalent Cosserat medium is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for the masonry components. In particular, linear elastic constitutive relationship is considered for the blocks, while nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of the masonry material.