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Multilevel approach for brick masonry walls – Part II: On the use of equivalent continua
Abstract
This work investigates the possibility of using equivalent continua within a multilevel strategy for the analysis of brick masonry walls, as prototypes for the in-plane behavior of masonry buildings. This strategy, proposed and discussed in detail in Part I of the same paper, lies in an iterative solution scheme which uses two levels, local and global, of description: the former accounts for the interaction between the single bricks; the latter, modeled as a coarse Finite Element discretization, accounts for global interactions.Within the same Finite Element format and strategy, the paper compares the performances of schemes implementing the identification procedures based on equivalent continua with those implementing the algebraic re-parametrization of the local model proposed in Part I. The former, while suitable for a synthetic representation of masonry behavior, does not prove to be convenient for a multilevel solution strategy.
... Although there are widely accepted models to represent the compression behaviour of masonry [229][230][231][232][233][234], simplified approaches, which can be easily implemented in commercial software such as ABAQUS, to ensure wider availability to practitioners should be explored. ...
... Also, for walls under vertical loads with null eccentricity, the main failure mode is by crushing of the masonry in compression [199]. Therefore, buckling failure associated with a geometric instability was negligible [232,233]. ...
... with the initiation of vertical tensile splitting cracks of physical soil prisms, starting from the middle and spreading to the top and bottom of the prism. This pattern is similar to what was observed by Brasile et al. (2007)[232,233], who proposed a numerical strategy based on a multilevel approach for the nonlinear analysis of brick masonry walls as in-plane prototypes of large masonry buildings. ...
The history of soil bricks is synonymous with the history of civilization. The early forms of soil bricks’ application date back to 8000 BC with the construction of buildings in Mesopotamia in present-day southern Turkey. Soil brick is a cheap, locally available and recyclable building material. However, its use has been limited due to its low strength, an affinity for water, and its frequent maintenance requirements. Thus, the aim of this study is to develop environmentally friendly fibre reinforced soil bricks with improved physical, durability and mechanical properties.
In order to accomplish this, an investigation into the use of low-cost natural fibre chicken feather and sugarcane bagasse as reinforcement for soil brick is carried out. The use of waste materials in the construction industry is generally of interest and useful for engineers and designers looking for sustainable solutions in construction.
Based on extensive experimental investigation, constitutive models showing the relationship between soil brick properties were developed. Response Surface Models that fully predict soil brick properties for up to 180 days were developed. Also, normalised design equations for uniaxial compression were proposed. Based on the constitutive models, detailed Finite Element Modelling of fibre reinforced soil bricks was developed. Parametric studies of compressive tests on the parallel-hole hollow soil cubes and bricks were presented. In addition, Finite Element analysis was performed to assess the influence of different parameters on the behaviour of masonry walls made from the developed bricks. The developed soil bricks can be used for affordable and sustainable housing construction across the world, particularly in developing countries.
... In order to address to more realistic structural contexts, large-scale walls are hard to be solved numerically, even working with physical models as simplified as the discrete models here adopted [1]. Regardless of the computational costs which increase with the number of blocks in the masonry texture, the nonlinearities of the physical problems can lead to difficulties in convergence [4]. Such nonlinearities and consequent failures in convergence are essentially related to localization phenomena of the physical responses: roughly speaking, the water uptake process tends to concentrate the humidity distribution in sharp fronts, in reason of the simultaneous evaporation process; on the other hand, the strain localizes and produces mechanical damage, so that the wall stress response becomes softening. ...
... The validation of the discrete model in terms of pure mechanical response has been carried out in [4], considering applications at the structural scale as well. ...
... This way to proceed minimizes the numerical error introduced while smoothing the local evaluation of the variables. Besides, it allows us to derive the global tangent matrix K g , which turns out to be the best coarse representation of the (finer) tangent matrix of the local problem K [u l ] when the local problem is linear [4]. Consider the local linearized balance, which represents a general form valid for both the mechanical and the diffusive solution scheme, as the following system, ...
We present a computational multiscale approach to the nonlinear problems of humidity diffusion and mechanical damage of large-scale masonry walls, and their coupling in terms of the effects of the humidity diffusion on the mechanical response and the effects of the mechanical degradation on the diffusion process. Such an approach allows us to recover, both efficiently and accurately, the complex nonlinear response of large-scale walls, which are in general hard to be solved by means of standard numerical tools. Two representative tests of two- and three-storey walls are here analyzed, and the corresponding results reported and commented, aiming to show how samples like these can potentially serve as reference solutions for more applicative purposes.
... Due to the built-in and evolving inhomogeneities characterizing such structures, both modeling and numerical simulations are hard to be treated in a standard way. The problem has been already tackled by the authors (see Ref. [1][2][3]), by developing a finescale modeling which highlighted some important aspects of the mechanical behavior: (i) starting from an ideally pristine condition of the structure, the damage process is a short-lived phenomenon so that safety analysis can be conveniently made by considering an initial completely damaged state; (ii) the shear-friction strength is very important and strongly influences the overall structural carrying capacity; and (iii) the types of collapse mechanisms, simulated for different loading conditions, are limited in number and can be well identified. ...
... [7]). We will also show that the numerical results do not differ significantly from the ones obtained with richer formulations, in terms of ultimate shear load and hysteresis dissipation under cyclic loads, as described by the refined analysis given in [2,3]. ...
... A coarse-scale model has been proposed for in-plane analysis of masonry structures, addressing practical engineering purposes. It has been designed to be a simplified version of more refined models presented in [1][2][3], deriving from micro-mechanical approaches. In essence, our simplifications on the model consist in: (i) neglecting the evolving damage process; (ii) assuming an elasto-plastic behavior ruled by yield conditions defined in terms of tensile, compression, and shear-friction strength; (iii) prescribing sliding surface where the inelastic Mohr-Coulomb behavior can take place; and (iv) using a rough Finite Element discretization with an elastic response described by an equivalent refined Cauchy continuum. ...
The work builds upon previous developments made by the authors in the context of the nonlinear, in-plane analysis of masonry walls. The structural behavior is characterized by phenomena, such as strain localization, damage, and friction, which need to be modeled at fine scales. Fine-scale modeling represents a significant challenge with regards to numerical simulations, due to its computational expensiveness and hard manageability. Generally, it requires sophisticated solution strategies, such as multi-grid techniques, as proposed in 2007 by the same authors, which cannot be effortlessly used in engineering softwares for structural analysis and design.In order to overcome such difficulties, we propose a coarse-scale model, to be employed in standard path-following techniques, based on an assumed stress Finite Element formulation in a context of non-associated plasticity. We obtain the nonlinear behavior by assuming a set of planes on the Element where frictional response can take place, together with tensile and compression limit stress. In this way, we capture the essential features of the nonlinear behavior as described by the more refined models developed in the past, exploiting algorithms widely adopted in elasto-plasticity, and therefore suitable for practical use in the analysis of full scale masonry structures.
... The multilevel modeling and solution approach for plane brick masonry walls proposed in [1,2] is extended here to the free vibration analysis. The proposed strategy is based on a convenient iterative-residual scheme particularly appropriate for the cited multilevel strategy. ...
... The numerical tests show the global efficiency and computational convenience of this strategy in providing the principal vibration modes and frequencies. A comparison with the results obtained from FE models based on equivalent continua also allows a better understanding of the real limits of the different homogenization techniques discussed in [2]. ...
... see [4][5][6][7]). However, equivalent continua, while being tuned to reproduce smooth strain deformations of the masonry, become inaccurate on nonsmooth deformation modes (see [2]). This leads to the use of multilevel strategies which are able, at least in principle, to exploit the advantages of both coarse and fine modelings. ...
The multilevel modeling and solution approach for plane brick masonry walls proposed in
[1] and [2] is extended here to the free vibration analysis. The proposed strategy is based on a convenient iterative-residual scheme particularly appropriate for the cited multilevel strategy. Two different modeling levels of the masonry mechanics are used in synergy: the reference level (finer), which reflects the brick and mortar joint texture, and its finite element (FE) approximation (coarser) used as an iteration improver.The numerical tests show the global efficiency and computational convenience of this strategy in providing the principal vibration modes and frequencies. A comparison with the results obtained from FE models based on equivalent continua also allows a better understanding of the real limits of the different homogenization techniques discussed in [2].
... A very efficient multilevel approach has been developed by Brasile et al. [225,226]. Although this approach could be considered borderline in a multi-scale framework (being rather a multilevel approach), the strategy proposed in [225,226] is based on an iterative scheme which uses two different (local and global) masonry models simultaneously. ...
... A very efficient multilevel approach has been developed by Brasile et al. [225,226]. Although this approach could be considered borderline in a multi-scale framework (being rather a multilevel approach), the strategy proposed in [225,226] is based on an iterative scheme which uses two different (local and global) masonry models simultaneously. The former is a fine block-based model and describes the nonlinear mechanical response including damage evolution and friction toughness phenomena. ...
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.
... Among these, some consider bricks as rigid or linear elastic, assuming nonlinear mechanisms active only in the mortar and/or at interfaces [4,10,25]. These are suitable to describe only some of the failure mechanisms, typically appearing in masonry structures and related to the tensile and shear cracking of the mortar joints. ...
... then, the value of the Coulomb limit function ϕ σ d el , ζ n is computed using formula (10). If ϕ < 0, then p = p n , ζ = ζ n and σ d = σ d el . ...
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.
... In the literature great attention has been devoted to model the mechanical behavior of masonry structures (see for example [11][12][13][14]) or to model the hygro-thermal behavior [15][16][17][18], while less attention has been devoted to the coupling between the two different processes [1,19,20]. In particular, modeling the damage due to environmental actions is addressed mainly by three-dimensional continuum models. ...
... Even though the discrete model is greatly simplified, it is however difficult to be solved when considering large-scale walls. The computational cost drastically increase with the number of blocks in the masonry texture, and the nonlinearities of the physical problem can lead to difficulties in convergence [14]. Indeed, the size of the unknowns is of the order of the number of the blocks, and from a computational point of view it is not convenient to solve directly the linearized problem (24), mainly because small corrections du l can lead to high oscillations of the successive residue r u l þ du l ½ . ...
... On the other hand, in macroscale approach, the masonry is perceived as an equivalent homogeneous and anisotropic continuum [8][9][10][11]. Homogenization procedures [12][13][14] or multiscale approaches [15][16][17][18] are typically considered as a bridging between the two scales. In general, despite the existing body of scientific research, the development of reliable and efficient strategies for modelling mechanical behaviour of structural masonry still remains a challenge [19][20][21]. ...
This paper deals with mesoscale analysis of masonry structures, which involves fracture propagation in brick units as well as along the masonry joints. The brick–mortar interfaces are incorporated in standard finite elements by employing a constitutive law with embedded discontinuity. Macrocracks in bricks are modelled in a discrete way using the same methodology, without any a-priori assumptions regarding their orientation. The proposed approach is computationally efficient as it does not explicitly require the discretization of joints. The accuracy of the approximation is first assessed by comparing the solution with a detailed mesoscale model incorporating interface elements. Later, a numerical study is conducted involving simulation of various experimental tests on small masonry assemblages, as well as single-leaf masonry walls, with running bond pattern, subjected to in-plane loading. The results clearly demonstrate the predictive abilities of the proposed simplified approach.
... In the present work, due to the wide size of the system under analysis and with the aim of limiting the computational efforts, a homogenized masonry material simulating the properties of both stone and mortar was adopted [21][22][23]. Consequently, the properties of the adopted finite elements are representative of the global behaviour of brickwork and stonework. In more details, the numerical FE model of Figure 3a is made of about 4˙400˙000 finite elements, pentahedron with 6 nodes (wedge elements) or hexahedron with 8 nodes (brick elements). ...
... For these mechanisms, there is a wealth of literature references aimed at calculating the ultimate load factors by means of limit analysis, especially with kinematic approaches considering frictional behaviour [3][4][5][6][7][8][9][10][11][12]. In the last few decades, researchers have also developed analytical methods in the field of finite element (FE) analyses to examine ordinary masonry constructions [13][14][15][16][17] or have implemented large scale and detailed numerical investigations to study monumental constructions like churches [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. ...
Local collapse mechanisms related to the out-of-plane response of walls are commonly observed in existing masonry buildings subjected to earthquakes. In such structures, the lack of proper connections among orthogonal walls and between walls and floors does not allow a global box-type behaviour of the building to develop, which would be governed by the in-plane response of walls. In this paper, parametric linear kinematic analyses on the main local mechanisms of masonry churches were performed with the aim to evaluate the corresponding horizontal load multipliers. This study was conducted on 12 masonry churches, located in Teramo (Italy) and affected by the 2016 Central Italy earthquake, whose main out-of-plane collapse mechanisms, namely facade overturning, vertical bending, corner overturning and roof gable wall overturning, have been analysed. For each mechanism, parametric analysis was carried out on varying heights and thicknesses of walls. Firstly, the acceleration values activating the considered mechanisms were calculated in order to conduct checks prescribed by the current Italian standard. Subsequently, on the basis of the obtained results, simple analytical procedures to determine load collapse multiplier for each mechanism were drawn. Finally, ranges of suitable values of both the thickness and height of walls were found in order to always satisfy seismic checks.
... Among the various formulations developed over the past centuries, from the graphical analysis ([1], [2]) to the more recent micro and macro-modelling theories ( [3], [4], [5]), those based on the limit analysis ( [6], [7]) are considered more effective for assessing the seismic capacity of these structures and for designing strengthening measures. ...
The seismic capacity of masonry arches with circular shape is studied in this paper. The analytical procedure presented moves stems from the limit analysis of local mechanisms prescribed by the current Italian code. An analytical model is developed, which provides a simple tool for quick seismic evaluations of existing arches and for the design of strengthening interventions at the extrados.
Considering a case study, by varying the value of the main geometrical parameters, a sensitivity analysis of the risk index is also carried out. This study provides qualitative and quantitative insight in the significance of these parameters in the overall seismic response of masonry arches, thus allowing a deeper physical interpretation of the problem.
... Detailed micro-modelling is probably very accurate, it is capable to describe important local phenomena, using damage or plasticity-based laws, but requires a huge computational effort. This makes it suitable for the analysis of small structural elements or affordable for larger analyses using simplified adaptations as proposed, for example, in Gambarotta and Lagomarsino (1997a), Gambarotta and Lagomarsino (1997b), , Brasile, Casciaro, and Formica (2007a), Brasile, Casciaro, and Formica (2007b). One remarkable advantage of the micro-modelling is the use of constitutive parameters derived from microexperiments on units, joints and small masonry samples. ...
A multiscale approach to analyse historical masonry buildings is presented and the numerical results deriving from its implementation are discussed. The modelling of this kind of heterogeneous material composed by irregular, stones and mortar joints is performed at the level of the microstructure by also describing the nonlinear behaviour of stones and mortar joints through an elasto–plastic constitutive law. At the macro-level, a finite element description based on a generic anisotropic material is implemented. This micro–macro model aims to assess the structural behaviour and the safety condition of historic masonries.
... In the last years, several solutions regarding the numerical modelling of the behaviour of masonry st models, ge complex an (FEM). The separately t [5] wer enerically na nalysis using e first mode the bricks, th continuous m damage mod These mod orms [8,9] a h computatio econd class e focused for which a between the he simplest y means of a on of the r f laboratory t merical mod hat the ma us and desc ria, similar to modelling. Th he extension , usually, i ingle wide cr practicing en ations using hat allows of the struc nergy induc d into elas part is di ns. ...
For existing buildings masonry, numerical simulation of the seismic behaviour is difficult, involving complex nonlinear models. Testing large-scale models on shaking tables involves important costs and time. Still, the main purpose of testing or simulations is often to identify structural vulnerabilities probable failure modes so that the correct strengthening solution can be chosen. A simplified method for the assessment of the potential structural degradation pattern for simple masonry buildings is proposed. This method is based on energy dissipation and can be easily implemented by practicing engineers. Four case studies are performed using linear numerical models. The results are compared with observations on buildings tested on shaking tables.
... Later, continuous models [6,7] were developed based on damage models for the brick and mortar interface. These models, even in their recent optimized forms [8,9] are difficult to implement and require high computational time. A second class of models, named macro-models are focused on the overall structural behavior, for which a detailed description of the interaction between the bricks and the mortar is not required. ...
In the existing masonry buildings cases, the numerical simulation of the seismic performance is challenging, comprising complex nonlinear models. The large-scale models analysis on shaking tables associates important costs and time. On the other hand, the main purpose of analysis or simulations is habitually to identify the structural vulnerabilities potential failure modes thus that the right strengthening solution can be taken. A basic method for the valuation of the potential structural degradation pattern for simple masonry buildings is suggested. This method is centered on energy dissipation and can be certainly applied by working engineers. Several case studies are accomplished with linear numerical models. The effects are related with observations on models experienced on shaking tables.
... But, these models suffer from serious limitations when high deformation gradients occur, so that the macroscopic strain and stress fields considerably vary, and when strain-softening material behaviors are considered. To overcome these drawbacks, several techniques have been proposed, based on the use of interfaces [14] or nonlocal, higher-order and enriched micropolar models [15][16][17][18][19][20][21][22]. In particular, micropolar Cosserat models naturally introduce a material length scale into the constitutive description, to obtain a dependence of the overall response of the composite material on the absolute size of the constituents and to achieve a realistic description of the micro-structurally triggered macroscopic localization. ...
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.
... Although there are widely accepted models to represent the behaviour of masonry (see [14][15][16][17][18][19][20][54][55][56][57][58]), a different approach, intended to be easily implemented in commercial software (ANSYSÓ 12.1), is preferred because of its wider availability to practitioners. The model used in this study is a 2D plane-strain simplified micro-model that combines solid elements to represent the units with interface joint elements to represent the bed joints. ...
... This method, which has undergone several refinements in the subsequent years (Tomaževic 1987Tomaževic , 1999), is based on the so-called " storey-mechanism " approach, which basically consists of a separate non-linear interstorey shear-displacement analysis for each storey, where each masonry pier is characterized by an idealized non-linear shear-displacement curve (typically elastic-perfectly plastic with limited ductility). The need for more general methods of analysis has stimulated the research on the subject and analytical methods have made significant progress in the last decades, particularly in the field of finite and discrete element analyses (Calderini & Lagomarsino, 2008, Lourenço, 2002, Milani et al., 2006, Casolo & Peña, 2007, Brasile et al., 2007, Caliò et al., 2008). Still, despite such progress, each model has a range of validity which needs to be understood with care, and the use of such tools requires high expertise, and in many cases can be applied to problems that are limited in size. ...
... Masiani and Trovalusci (1996) and Trovalusci and Masiani (2003) presented multi-scale linear and nonlinear models for masonry, where the microstructural level is described as an inplane discrete system of rigid bricks interacting with one another through the mortar joints, modeled as springs in the normal and tangential directions. Recently, in Brasile et al. (2007a) and Brasile et al. (2007b), the mortar joints have been described as nonlinear springs characterized by a damage model. Indeed, the use of a discrete model at the micro-scale, where the deformability of the blocks representing the widest part of the masonry volume is completely neglected, appears as a simplification, which could influence the accuracy of the models in correctly reproducing the experimental behavior of masonry. ...
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 performing 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.
... The more significant non-linearity is essentially on the frictional behavior; we neglect the coupling with the damage process. More sophisticated numerical simulations, based on fine-scale models and experimental evidences, show that the frictional resistance plays an important role in the structural response under cyclic loading conditions [10], [4], [13]. Within an elastoplastic model, a Mohr-Coulomb criterion is employed in order to characterize the inelastic part of the structural response. ...
... Such method, which has undergone several refinements in the subsequent years (Tomaževic, 1999), is based on the so-called " storey-mechanism " approach, which basically consists of a separate non-linear interstorey shear-displacement analysis for each storey, where each masonry pier is characterized by an idealized non-linear shear-displacement curve (typically elastic-perfectly plastic with limited ductility). The need for more general methods of analysis has stimulated the research on the subject and analytical methods have made significant progress in the last decades, particularly in the field of finite element analyses (Calderini & Lagomarsino, 2008, Lourenço, 2002, Milani et al., 2006 Massart et al., 2004; Cecchi & Sab, 2008; Brasile et al., 2007). Still, despite such progress, each model has a range of validity which needs to be understood with care, and the use of such tools requires high expertise, and in many cases can be applied to problems that are limited in size; therefore refined nonlinear finite element modeling does not constitute yet a suitable tool for the analysis of whole buildings in everyday engineering practice, especially when considering the task of designing/assessing ordinary low-rise residential buildings. ...
The paper presents first an introduction to the main differences between the EN 1998-3 approach to seismic assessment and strengthening of existing masonry buildings and the Italian norms approach. Then, issues related to the definition of material properties, to structural analysis, modelling and performance checks are discussed in more detail, pointing out the difficulties associated with the practical application of the Eurocode and reporting the Italian attempts to overcome such difficulties. The conclusions call for a thorough revision or re-draft of EN 1998-3 to take into account the specific problems of existing masonry buildings.
A strategy based on material homogenization and heuristic optimization for the structural identification of composite materials is proposed. The objective is the identification of the constitutive properties of a micropolar continuum model employed to describe the mechanical behaviour of a composite material made of rigid blocks and thin elastic interfaces. The micropolar theory (Cosserat) has been proved to be capable of properly accounting for the particles arrangements as well as their size and orientation. The constitutive parameters of the composite materials, characterized by different textures and dimensions of the rigid blocks, are identified through a homogenization procedure. Thus, the identification is repeated exploiting the static or modal response of the composite materials and using the Differential Evolution algorithm. The benchmark structures assumed as target are represented by discrete models implemented in ABAQUS where the blocks and the elastic interfaces are modelled by rigid bodies and elastic interfaces, respectively. The obtained results show that proposed strategies provide accurate results paving the way to the experimental validation and in field applications.
A new Lower Bound LB plate and shell limit analysis Finite Element FE model for the analysis at collapse of masonry double curvature structures is presented. The discretization relies into hexahedrons assumed infinitely resistant and quadrilateral interfaces where all plastic dissipation occurs. On such interfaces, the flexural behavior is ruled by the interaction between bending moment and axial load, whereas the shear and torsional behavior are modeled by means of an in-plane tangential force, out-of-plane shear and a plate torque. The resultant limit analysis problem obtained by such a formulation is particularly straightforward and the number of variables to deal with very limited. Equilibrium is indeed imposed only on hexahedrons and admissibility on interfaces between adjoining elements. Masonry can be modeled obeying a classic no-tension material or with more complex linearized failure surfaces, for example derived from suitable homogenization techniques. A simple Linear Programming LP problem is so derived, where the actual thickness of the structure is accurately accounted for, a key feature to assess the stability of a vault in case of the no-tension material assumption. The numerical model is validated by means of several meaningful structural examples. A detailed comparison with numerical data available in the literature, obtained for the same examples with alternative numerical approaches shows the accuracy of the method proposed and its usefulness for a fast and reliable prediction of the load carrying capacity of masonry double curvature structures.
Archaeoseismology is one of the four issues usually involved to increase the knowledge of seismic risk. The three other issues are seismic records, historical seismicity, and palaeoseismicity. Archaeoseismology was founded as a discipline in the 1980s. It was developed and methodologically defined during the 1990s and 2000s, and its new developments and perspectives are based on two procedures. First, the numerical field, with tools like the database OPUR (Outil Pour Unités de Réparation), “Tool For Reparation Units”, conceived as a sort of atlas, to collect and index all the types of repairs identified in the Roman site of Pompeii, in Italy. Second, the focus on the evolution of ancient buildings and their pathologies, serves as a basis for the structural modelling, carried out by engineers. It allows to understand the behaviour of ancient buildings during seismic motion, to quantify the impact of seismic effects on cultural heritage and to propose a method of preservation. Major studies conducted in France as well as recent developments in this field are presented, in order to illustrate this collaboration between archaeoseismologists and engineers for the preservation of cultural heritage.
A finite element based framework that formulates an adaptive multiresolution multiscale technique is presented, with the goal of both accurately and efficiently simulating large masonry structures. In order to find a compromise between accuracy and computational efficiency a scale embedding multiscale model where both macro- and microscale elements come into play is proposed, combining the advantages both have to offer. This theory is tested and compared with an equivalent microscale model to demonstrate that its accuracy rivals a microscale approach, while at the same time having a higher computational efficiency. The developed multiscale model is compared to its underlying microscale model in a couple of selected example structures, ranging from small to large scale unreinforced masonry walls with openings. An application in the form of soil subsidence is explored, showing a potential for extended applications of this type of modeling.
Prima parte delle slides del corso di approfondimento tenuto all'Ordine degli Ingegneri di Cosenza nei giorni 27 e 28 febbraio 2020
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.
Testo della presentazione tenuta in Roma presso l’Ordine degli Ingegneri della Provincia
This paper presents a macro model to predict unreinforced masonry structures in plane behavior. The model is based on the concept of multilaminate theory. In the past, the method has been used to model behavior of soil, disregarding the cohesion and the tensile strength. Regarding its mathematical base, and the possibility of applying in other cases, this method is used to predict the ultimate failur load in URM structures in present study. This model is intrinsically capable of spotting induced anisotropy of brittle material such as concrete, rocks and masonry, develponig as a result of cracking. Here, the yield surface applied, consists an generalized mohr-coulomb yield surface, along with a cap model and a cut-off tensile. Comparing numerical results predicted to be obtained in non-linear analysis of masonry structures unreinforced against lateral load, with the results of ther experimental data shows capability of the model in failure analysis of URM structures.
The development of adequate stress analyses for masonry structures represents an important task not only for verifying the stability of masonry constructions, as old buildings, historical town and monumental structures, but also to properly design effective strengthening and repairing interventions. The analysis of masonry structures is not simple at least for two reasons: the masonry material presents a strong nonlinear behavior, so that linear elastic analyses generally cannot be considered as adequate; the structural schemes, which can be adopted for the masonry structural analyses, are more complex than the ones adopted for concrete or steel framed structures, as masonry elements require often to be modeled by two- or three-dimensional elements. As a consequence, the behavior and the analysis of masonry structures still represents one of the most important research field in civil engineering, receiving great attention from the scientific and professional community.
La procedura di analisi limite contemplata dalla NTC-08 e relativa Circolare esplicativa n. 617/2009 per la valutazione del livello di sicurezza di strutture murarie nei confronti di possibili meccanismi locali è applicata in questo lavoro agli archi a profilo circolare. Le equazioni ottenute consentono di eseguire valutazioni rapide del livello di sicurezza sismica di archi esistenti e di progettare eventuali interventi di rafforzamento al loro estradosso. Partendo da un caso di studio, si presenta un’analisi di sensitività dell’indice di rischio dell’arco considerato, al variare dei principali parametri geometrici rappresentativi di forma e consistenza dell’arco. I risultati ottenuti forniscono un insieme d’indicazioni qualitative sulla risposta sismica che sono in pieno accordo con l’interpretazione fisica del fenomeno
A 2D2D Cosserat model with a damage-plastic isotropic constitutive law for brittle-like materials is presented. Different damaging mechanisms in tension and compression are considered. The plastic flow is described by introducing a suitable yield function and evolution laws in terms of effective stresses. Small strain and displacement assumptions hold. A 4-node finite element is formulated with three degrees-of-freedom for each node and a predictor–corrector procedure is used to solve the damage-plastic evolutionary problem. The lateral response of walls under monotonic and cyclic horizontal actions is analyzed and a satisfactory description of the global response curves and damaging mechanisms is obtained.
New methods for the modeling of structural failure by means of multiscale approaches were recently proposed, in which the structural description involves coarse-scale discontinuities, the behavior of which is fed by representative volume element (RVE) computations. Their main asset consists in identifying the material response, including the failure behavior of the material, from fine-scale material parameters and computations. One of the distinctions between the available approaches relates to the boundary conditions applied on the RVE. The methods based on classical computational homogenization usually make use of periodic boundary conditions. This assumption remains a priori debatable for the localized behavior of quasi-brittle materials. For the particular case of periodic materials (masonry), the level of approximation induced by the periodic assumption is scrutinized here. A new displacement discontinuity-enhanced-scale transition is therefore outlined based on energetic consistency requirements. The corresponding multiscale framework results are compared to complete fine-scale modeling results used as a reference, showing a good agreement in terms of limit load, and in terms of failure mechanisms both at the fine-scale and at the overall structural level.
Continuum models of periodic masonry brickwork, viewed at a micro-level as a discrete system, are identified within the frame of linearized elasticity. The accuracy of various identification schemes is investigated for standard and micropolar continua, which are directly compared with the help of some numerical benchmarks, for different loading conditions that induce periodic and non-periodic deformation states. It is shown that periodic deformation states of brickwork are exactly reproduced by both continua, provided that a suitable identification scheme is adopted. For non-periodic states micropolar continuum is shown to better reproduce the discrete solutions, due to its capability to take scale effects into account. Both continua are asymptotically equivalent as the characteristic length of the discrete system tends to zero, while providing an upper and a lower bound of the discrete solution.
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.
A structured continuum model is formulated to describe the behaviour of block masonry modelled as distinct rigid body systems with elastic interfaces. A correspondence between the two motions is obtained by postulating a relationship between the displacement fields of the continuum and the discrete models. The constitutive functions for the dynamic actions of the continuum are derived by equating the power of the two models.Viene presentato un modello di continuo con struttura atto a descrivere il comportamento meccanico di murature a blocchi, pensate come un sistema di corpi rigidi con contatti puntuali elastici. Il moto del sistema discreto e di quello continuo sono messi in relazione postulando una corrispondenzatra i campi di spostamento. Le funzioni costitutive delle azioni dinamiche del modello continuo sono ricavate uguagliando la potenza meccanica spesa in moti corrispondenti.
This work presents and compares some constitutive identification algorithms within the frame of continuum modeling of periodic discrete systems made by rigid bodies interacting with elastic joints. Exemplification is based on the linear analysis of periodic masonry brickwork. The continuum models selected are Cauchy and Cosserat. The algorithms presented are shown to be variationally founded and to imply approximation errors.
In this paper, we present a two-dimensional finite element able to describe local effects in the linear analysis of periodic brickwork, modelled as a Lagrangean system of rigid bodies interacting by point elastic interfaces. Particular emphasis is given to the description and the implications of the continuum modelling of the Lagrangean system, before the finite element discretization is performed.
The aim of this paper is to present an effort towards a multiscale model of the inelastic behaviour of masonry brick panels and the relative solution algorithm. The essential features of the inelastic behaviour, such as the damage development within the bed joints and the frictional dissipation over cracks' faces, are taken into account. Micromechanical solutions are adopted in order to trace the guidelines for modelling the mentioned phenomena, and FEM analyses of large-scale panels are shown. (c) 2005 Elsevier Ltd. All rights reserved.
The arc-length Riks strategy has rapidly become a standard tool for path-following analysis of nonlinear structures due to its theoretical ability to surpass limit points. The aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle ‘locking’ effect intrinsic to the nonlinear nature of the problem. As a consequence, its sanitization has to be pursued within a reformulation of the structural model. The use of a mixed (stress-displacement) variant of the algorithm, in particular, appears very promising in this respect.The topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in [39]. It is shown that, with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation, it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm.
The paper presents a discrete mechanical model for masonry walls based on a Lagrangean description where each brick is described as a rigid body and each mortar joint as an interface element. Constitutive assumptions, characterized by elasticity, damage and friction, are associated to the joints only. A numerical solution strategy, based on a mixed path-following approach in terms of stresses, strains, displacements, damage and load parameters, is proposed for avoiding convergence problems related to the joint softening behaviour. Some numerical results are also presented showing the robustness and effectiveness of this proposal.