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Gabriele Milani - One of the best experts on this subject based on the ideXlab platform.
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Innovative Voxel Approach for Homogenized Out-of-Plane Analysis of Non-periodic Masonry Walls
Developments and Novel Approaches in Nonlinear Solid Body Mechanics, 2020Co-Authors: Simone Tiberti, Gabriele MilaniAbstract:This paper presents a MATLAB-based procedure for the derivation of out-of-plane homogenized failure surfaces for Masonry elements. The procedure follows a so-called “voxel approach” that allows the creation of a 3D finite element mesh directly from the sketch of a Masonry element. This approach is implemented into a dedicated MATLAB script. The validation of the procedure is performed by extracting homogenized out-of-plane failure surfaces for a stretcher Bond Masonry cell. These are compared to those available in literature for an analogous cell, obtained with a different model. The correspondence between the two models is satisfying, confirming the reliability of the presented procedure. Eventually, the behavior under out-of-plane actions is investigated for two single and one double curvature Masonry cells. Homogenized out-of-plane failure surfaces and relevant failure modes are extracted for all the different cases and compared to those obtained for the flat case. The results for the curved Masonry cells show an increase in out-of-plane resistance for one of the single curvature cases and for the double curvature case. Moreover, the deformed shapes at collapse are consistent with the expectations.
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numerical homogenization based seismic assessment of an english Bond Masonry prototype structural level application
Earthquake Engineering & Structural Dynamics, 2020Co-Authors: Luís C. Silva, Paulo B. Lourenço, Gabriele MilaniAbstract:This work was supported by FCT (Portuguese Foundation for Science and Technology), within ISISE, scholarship SFRH/BD/95086/2013. This work was also partly financed by FEDER funds through the Competitivity Factors Operational Programme - COMPETE and by national funds through FCT – Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER-007633.
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Numerical homogenization‐based seismic assessment of an English‐Bond Masonry prototype: Structural level application
Earthquake Engineering & Structural Dynamics, 2020Co-Authors: Luís C. Silva, Paulo B. Lourenço, Gabriele MilaniAbstract:This work was supported by FCT (Portuguese Foundation for Science and Technology), within ISISE, scholarship SFRH/BD/95086/2013. This work was also partly financed by FEDER funds through the Competitivity Factors Operational Programme - COMPETE and by national funds through FCT – Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER-007633.
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Fast brick-based homogenized limit analysis for in- and out-of-plane loaded periodic Masonry panels
Computers & Structures, 2020Co-Authors: Simone Tiberti, Gabriele MilaniAbstract:Abstract This paper presents a 3D brick-based model that aims at the derivation of in- and out-of-plane homogenized failure surfaces for Masonry. The considered Masonry panel is discretized into regular parallelepiped 3D finite elements that are supposed to be rigid; the model here introduced uses a Kirchhoff-Love plate kinematics for the out-of-plane description of the displacement rate field of the elements. A linear programming problem formulated in standard form is scripted into Matlab to derive the in- and out-of-plane homogenized failure surfaces, also enabling the extraction of failure modes for the considered Masonry element. The constraints of the linear programming problem come from the combination of an upper bound limit analysis problem and a homogenization-based approach. The proposed model is validated for two separate case studies: a running Bond Masonry test-window and an English Bond Masonry test-window. The homogenized failure surfaces resulting from the current model show good correspondence to those presented in three distinct works available in literature. Also, a few relevant failure modes are derived for the two case studies, and they are consistent with the expected deformed shapes at collapse for their related load conditions.
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Derivation of the out-of-plane behaviour of an english Bond Masonry wall through homogenization strategies
2017Co-Authors: Luís C. Silva, Gabriele Milani, Paulo B. LourençoAbstract:This work was supported by FCT (Portuguese Foundation for Science and Technology), within ISISE, scholarship SFRH/BD/95086/2013. This work was also partly financed by FEDER funds through the Competitivity Factors Operational Programme - COMPETE and by national funds through FCT - Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER-007633.
Antonella Cecchi - One of the best experts on this subject based on the ideXlab platform.
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compatible model for herringbone Bond Masonry linear elastic homogenization failure surfaces and structural implementation
International Journal of Solids and Structures, 2013Co-Authors: Gabriele Milani, Antonella CecchiAbstract:Abstract A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic Masonry strength domains in the case of herringbone Masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined. A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively. To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads. When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone Bond orientation, (2) Masonry texture and (3) mechanical properties adopted for joints. At a structural level, a FE homogenized limit analysis is performed on a Masonry dome built in herringbone Bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where Masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone Bond orientation.
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Compatible model for herringbone Bond Masonry: Linear elastic homogenization, failure surfaces and structural implementation
International Journal of Solids and Structures, 2013Co-Authors: Gabriele Milani, Antonella CecchiAbstract:Abstract A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic Masonry strength domains in the case of herringbone Masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined. A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively. To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads. When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone Bond orientation, (2) Masonry texture and (3) mechanical properties adopted for joints. At a structural level, a FE homogenized limit analysis is performed on a Masonry dome built in herringbone Bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where Masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone Bond orientation.
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A homogenized Love–Kirchhoff model for out-of-plane loaded random 2D lattices: Application to “quasi-periodic” brickwork panels
International Journal of Solids and Structures, 2009Co-Authors: Antonella Cecchi, Karam SabAbstract:AbstractA homogenization procedure for finding the bending stiffness of a 2D regular lattice with random local interactions is proposed. The kinematic and static methods are used to provide explicit upper and lower bounds for the homogenized moduli. The proposed homogenization procedure is applied to a Masonry obtained by a random perturbation of the periodic running Bond Masonry [Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A 28, 610–625].A numerical evaluation of the scatter between the discrete models and the 2D Love–Kirchhoff model is performed on a test case, for various values of the random perturbation parameter and of the parameter that characterizes the heterogeneity of the wall. As expected, when the number of heterogeneities in the structure is large enough, the average response of the random discrete model converges to an asymptotic response. It is shown that this asymptotic response is very close to that of the periodic discrete model which is in turn very close to the response of the deterministic homogenized model. Similarly to the conclusion of Cecchi and Sab [Cecchi A., Sab K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A. 28, 610–625.] dedicated to in-plane loading, the present results concerning out-of-plane loading show (both by means of a discrete model and a homogenized model) that the running Bond pattern may be used successfully to analyze historical masonries with blocks having irregular widths in the horizontal direction
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A kinematic FE limit analysis model for thick English Bond Masonry walls
International Journal of Solids and Structures, 2008Co-Authors: Antonella Cecchi, Gabriele MilaniAbstract:Abstract Two-wythes Masonry walls arranged in English Bond texture were often used in the past as bearing panels in seismic area. On the other hand, earthquake surveys have demonstrated that Masonry strength under horizontal actions is usually insufficient, causing premature collapses of Masonry buildings, often ascribed to out-of-plane actions. Furthermore, many codes of practice impose for new brickwork walls a minimal slenderness, which for instance is fixed by the Italian O.P.C.M. 3431 equal to 12 for artificial bricks and 10 for natural blocks Masonry. For the above reasons, the analysis at failure of English Bond brickwork walls under out-of-plane actions is a topic that deserves consideration, despite the fact that almost the totality of the studies of Masonry at failure is devoted to running Bond arrangements. Furthermore, it must be noted that an approach based on the analysis of running Bond texture – in comparison with English Bond pattern – is not suitable for the investigation of the behavior of bearing panels. In this framework, in the present paper, a Reissner–Mindlin kinematic limit analysis approach is presented for the derivation of the macroscopic failure surfaces of two-wythes Masonry arranged in English Bond texture. In particular, the behavior of a 3D system constituted by infinitely resistant bricks connected by joints reduced to interfaces with frictional behavior and limited tensile/compressive strength is identified with a 2D Reissner–Mindlin plate. In this way, assuming both an associated flow rule for the constituent materials and a finite subclass of possible deformation modes, an upper bound approximation of macroscopic English Bond Masonry failure surfaces is obtained as a function of macroscopic bending moments, torsion and shear forces. Several examples of technical relevance are treated both at a cell level and at a structural level, addressing the differences in terms of collapse loads and failure surfaces due to different textures and constituent laws for joints. Finally, two meaningful structural examples consisting of a panel in cylindrical flexion and a Masonry slab constrained at three edges and out-of-plane loaded are discussed. A detailed comparison in terms of deformed shapes at collapse and failure loads between a 2D FE Reissner–Mindlin limit analysis approach and a full 3D heterogeneous FE model shows the reliability of the results obtained using the kinematic identification approach proposed.
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a reissner mindlin limit analysis model for out of plane loaded running Bond Masonry walls
International Journal of Solids and Structures, 2007Co-Authors: Antonella Cecchi, Gabriele Milani, Antonio TralliAbstract:Abstract Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of Masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian O.P.C.M. 3431 [2005. Ulteriori modifiche ed integrazioni all’OPCM 3274/03 (in Italian) and O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica (in Italian)], in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks Masonry. In this context, a formulation at failure for regular assemblages of bricks based both on homogenization and Reissner–Mindlin theory seems particularly attractive. In this paper a kinematic limit analysis approach under the hypotheses of the thick plate theory is developed for the derivation of the macroscopic failure surfaces of Masonry out-of-plane loaded. The behavior of a 3D system of blocks connected by interfaces is identified with a 2D Reissner–Mindlin plate. Infinitely resistant blocks connected by interfaces (joints) with a Mohr–Coulomb failure criterion with tension cut-off and compressive cap are considered. Finally, an associated flow rule for joints is adopted. In this way, the macroscopic Masonry failure surface is obtained as a function of the macroscopic bending moments, torsional moments and shear forces by means of a linear programming problem in which the internal power dissipated is minimized, once that a subclass of possible deformation modes is a priori chosen. Several examples of technical relevance are presented and comparisons with previously developed Kirchhoff–Love static [Milani, G., Lourenco, P.B., Tralli, A., 2006b. A homogenization approach for the limit analysis of out-of-plane loaded Masonry walls. J. Struct. Eng. ASCE (in press)] and kinematic [Sab, K., 2003.Yield design of thin periodic plates by a homogenisation technique and an application to Masonry walls. C.R. Mech. 331, 641–646] failure surfaces are provided. Finally, two meaningful structural examples are reported, the first concerning a Masonry wall under cylindrical flexion, the second consisting of a rectangular plate with a central opening out-of-plane loaded. For both cases, the influence of the shear strength on the collapse load is estimated.
Karam Sab - One of the best experts on this subject based on the ideXlab platform.
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An enhanced homogenization approach for Masonry structures with compressible joints. Application to hearth thermomechanical computations
International Journal for Numerical and Analytical Methods in Geomechanics, 2011Co-Authors: N. Mathieu, Karam Sab, S.a. ZaïmiAbstract:SUMMARY In this paper, a new homogenization-based finite element model to compute Masonry structures with compressible joints is proposed. It is an extension of the model proposed by De Felice et al. (Int. J. Numer. Anal. Meth. Geomech. 2010; 34(3):221–247) for dry block structures where strain hardening was neglected. The proposed strain hardening incremental elasto-plastic model is obtained by means of a new step-by-step homogenization method for a running Bond Masonry structure made of elastic bricks jointed by an elasto-plastic ram. The numerical implicit integration of the model is carried out following an iterative implicit procedure in which the elastic trial stress state is corrected through a return mapping algorithm. The procedure has been implemented in the ABAQUS finite element software and applied to the computation of thermal stresses for the hearths made of small carbon refractory bricks surrounded by very compressible joints. Indeed, during its working, the hearth of the blast furnace is submitted to a high thermal gradient in the radial direction because of the inner heating and the outer cooling imposed to the wall. In our application we evaluate the effect of the joints on the thermal stress distribution within the hearth. Copyright © 2011 John Wiley & Sons, Ltd.
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Yield strength of Masonry-like structures containing thin adhesive joints: 3D or 2D-interface model for the joints?
Comptes Rendus Mécanique, 2011Co-Authors: Ramzi Sahlaoui, Karam Sab, J.-v. HeckAbstract:It is shown in this Note that the use of a 2D-interfacemodel for the joints in the limit analysis of a structure made of bricks which are bounded with adhesively thin joints leads to an upper bound estimate of the bearing capacities of the structure, as the thickness of the joints goes to zero. Considering the compression in the vertical direction of a running Bond Masonry made of Drucker-Prager bricks and mortar, it is found that the use of the interfacemodel overestimates by 15 percent the compressive strength of the Masonry computed with 3D finite elements. Moreover, comparisons with compression test results show that the damaging behavior of the bricks has an important effect on the actual compressive strength of the Masonry.
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Overall ultimate yield strength of a quasi-periodic Masonry
Comptes Rendus Mécanique, 2009Co-Authors: Karam SabAbstract:The purpose of this Note is the determination of the in-plane homogenized strength domain of a "quasi-periodic" Masonry under the assumption of infinitely resistant blocks connected by cohesionless Mohr-Coulomb interfaces. This Masonry is obtained by introducing a random perturbation on the horizontal width of the blocks of a periodic running Bond Masonry. It is found that in some non-trivial cases the strength domain coincides exactly with that of the initial periodic Masonry.
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A homogenized Love-Kirchhoff model for out-of-plane loaded random 2D lattices: Application to "quasi-periodic" brickwork panels
International Journal of Solids and Structures, 2009Co-Authors: A. Cecchi, Karam SabAbstract:A homogenization procedure for finding the bending stiffness of a 2D regular lattice with random local interactions is proposed. The kinematic and static methods are used to provide explicit upper and lower bounds for the homogenized moduli. The proposed homogenization procedure is applied to a Masonry obtained by a random perturbation of the periodic running Bond Masonry [Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A 28, 610-625]. A numerical evaluation of the scatter between the discrete models and the 2D Love-Kirchhoff model is performed on a test case, for various values of the random perturbation parameter and of the parameter that characterizes the heterogeneity of the wall. As expected, when the number of heterogeneities in the structure is large enough, the average response of the random discrete model converges to an asymptotic response. It is shown that this asymptotic response is very close to that of the periodic discrete model which is in turn very close to the response of the deterministic homogenized model. Similarly to the conclusion of Cecchi and Sab [ Cecchi A., Sab K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A. 28, 610-625.] dedicated to in-plane loading, the present results concerning out-of-plane loading show ( both by means of a discrete model and a homogenized model) that the running Bond pattern may be used successfully to analyze historical masonries with blocks having irregular widths in the horizontal direction.
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A homogenized Love–Kirchhoff model for out-of-plane loaded random 2D lattices: Application to “quasi-periodic” brickwork panels
International Journal of Solids and Structures, 2009Co-Authors: Antonella Cecchi, Karam SabAbstract:AbstractA homogenization procedure for finding the bending stiffness of a 2D regular lattice with random local interactions is proposed. The kinematic and static methods are used to provide explicit upper and lower bounds for the homogenized moduli. The proposed homogenization procedure is applied to a Masonry obtained by a random perturbation of the periodic running Bond Masonry [Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A 28, 610–625].A numerical evaluation of the scatter between the discrete models and the 2D Love–Kirchhoff model is performed on a test case, for various values of the random perturbation parameter and of the parameter that characterizes the heterogeneity of the wall. As expected, when the number of heterogeneities in the structure is large enough, the average response of the random discrete model converges to an asymptotic response. It is shown that this asymptotic response is very close to that of the periodic discrete model which is in turn very close to the response of the deterministic homogenized model. Similarly to the conclusion of Cecchi and Sab [Cecchi A., Sab K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A. 28, 610–625.] dedicated to in-plane loading, the present results concerning out-of-plane loading show (both by means of a discrete model and a homogenized model) that the running Bond pattern may be used successfully to analyze historical masonries with blocks having irregular widths in the horizontal direction
Antonio Tralli - One of the best experts on this subject based on the ideXlab platform.
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homogenized rigid plastic model for Masonry walls subjected to impact
International Journal of Solids and Structures, 2009Co-Authors: Gabriele Milani, Paulo Lourenco, Antonio TralliAbstract:A simple rigid-plastic homogenization model for the analysis of Masonry structures subjected to out-of-plane impact loads is presented. The objective is to propose a model characterized by a few material parameters, numerically inexpensive and very stable. Bricks and mortar joints are assumed rigid perfectly plastic and obeying an associated flow rule. In order to take into account the effect of brickwork texture, out-of-plane anisotropic Masonry failure surfaces are obtained by means of a limit analysis approach, in which the unit cell is sub-divided into a fixed number of sub-domains and layers along the thickness. A polynomial representation of micro-stress tensor components is utilized inside each sub-domain, assuring both stress tensor admissibility on a regular grid of points and continuity of the stress vector at the interfaces between contiguous sub-domains. Limited strength (frictional failure with compressive cap and tension cut-off) of brick-mortar interfaces is also considered in the model, thus allowing the reproduction of elementary cell failures due to the possible insufficient resistance of the Bond between units and joints. Triangular Kirchhoff-Love elements with linear interpolation of the displacement field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to impacts. In order to test the capabilities of the approach proposed, two examples of technical interest are discussed, namely a running Bond Masonry wall constrained at three edges and subjected to a point impact load and a Masonry square plate constrained at four edges and subjected to a distributed dynamic pressure simulating an air-blast. Only for the first example, numerical and experimental data are available, whereas for the second example insufficient information is at disposal from the literature. Comparisons with standard elastic–plastic procedures conducted by means of commercial FE codes are also provided. Despite the obvious approximations and limitations connected to the utilization of a rigid-plastic model for Masonry, the approach proposed seems able to provide results in agreement with alternative expensive numerical elasto-plastic approaches, but requiring only negligible processing time. Therefore, the proposed simple tool can be used (in addition to more sophisticated but expensive non-linear procedures) by practitioners to have a fast estimation of Masonry behavior subjected to impact.
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a reissner mindlin limit analysis model for out of plane loaded running Bond Masonry walls
International Journal of Solids and Structures, 2007Co-Authors: Antonella Cecchi, Gabriele Milani, Antonio TralliAbstract:Abstract Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of Masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian O.P.C.M. 3431 [2005. Ulteriori modifiche ed integrazioni all’OPCM 3274/03 (in Italian) and O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica (in Italian)], in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks Masonry. In this context, a formulation at failure for regular assemblages of bricks based both on homogenization and Reissner–Mindlin theory seems particularly attractive. In this paper a kinematic limit analysis approach under the hypotheses of the thick plate theory is developed for the derivation of the macroscopic failure surfaces of Masonry out-of-plane loaded. The behavior of a 3D system of blocks connected by interfaces is identified with a 2D Reissner–Mindlin plate. Infinitely resistant blocks connected by interfaces (joints) with a Mohr–Coulomb failure criterion with tension cut-off and compressive cap are considered. Finally, an associated flow rule for joints is adopted. In this way, the macroscopic Masonry failure surface is obtained as a function of the macroscopic bending moments, torsional moments and shear forces by means of a linear programming problem in which the internal power dissipated is minimized, once that a subclass of possible deformation modes is a priori chosen. Several examples of technical relevance are presented and comparisons with previously developed Kirchhoff–Love static [Milani, G., Lourenco, P.B., Tralli, A., 2006b. A homogenization approach for the limit analysis of out-of-plane loaded Masonry walls. J. Struct. Eng. ASCE (in press)] and kinematic [Sab, K., 2003.Yield design of thin periodic plates by a homogenisation technique and an application to Masonry walls. C.R. Mech. 331, 641–646] failure surfaces are provided. Finally, two meaningful structural examples are reported, the first concerning a Masonry wall under cylindrical flexion, the second consisting of a rectangular plate with a central opening out-of-plane loaded. For both cases, the influence of the shear strength on the collapse load is estimated.
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A Reissner–Mindlin limit analysis model for out-of-plane loaded running Bond Masonry walls
International Journal of Solids and Structures, 2007Co-Authors: Antonella Cecchi, Gabriele Milani, Antonio TralliAbstract:Abstract Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of Masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian O.P.C.M. 3431 [2005. Ulteriori modifiche ed integrazioni all’OPCM 3274/03 (in Italian) and O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica (in Italian)], in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks Masonry. In this context, a formulation at failure for regular assemblages of bricks based both on homogenization and Reissner–Mindlin theory seems particularly attractive. In this paper a kinematic limit analysis approach under the hypotheses of the thick plate theory is developed for the derivation of the macroscopic failure surfaces of Masonry out-of-plane loaded. The behavior of a 3D system of blocks connected by interfaces is identified with a 2D Reissner–Mindlin plate. Infinitely resistant blocks connected by interfaces (joints) with a Mohr–Coulomb failure criterion with tension cut-off and compressive cap are considered. Finally, an associated flow rule for joints is adopted. In this way, the macroscopic Masonry failure surface is obtained as a function of the macroscopic bending moments, torsional moments and shear forces by means of a linear programming problem in which the internal power dissipated is minimized, once that a subclass of possible deformation modes is a priori chosen. Several examples of technical relevance are presented and comparisons with previously developed Kirchhoff–Love static [Milani, G., Lourenco, P.B., Tralli, A., 2006b. A homogenization approach for the limit analysis of out-of-plane loaded Masonry walls. J. Struct. Eng. ASCE (in press)] and kinematic [Sab, K., 2003.Yield design of thin periodic plates by a homogenisation technique and an application to Masonry walls. C.R. Mech. 331, 641–646] failure surfaces are provided. Finally, two meaningful structural examples are reported, the first concerning a Masonry wall under cylindrical flexion, the second consisting of a rectangular plate with a central opening out-of-plane loaded. For both cases, the influence of the shear strength on the collapse load is estimated.
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Limit analysis of out-of-plane loaded running Bond Masonry walls under Mindlin-Reissner plate hypotheses
2006Co-Authors: Antonella Cecchi, Gabriele Milani, Antonio TralliAbstract:Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of Masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian OPCM 3274 2003, in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks Masonry.
Luigi Gambarotta - One of the best experts on this subject based on the ideXlab platform.
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Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces
Journal of the Mechanics and Physics of Solids, 2017Co-Authors: Andrea Bacigalupo, Luigi GambarottaAbstract:Abstract Dispersive waves in two-dimensional blocky materials with periodic microstructure made up of equal rigid units, having polygonal centro-symmetric shape with mass and gyroscopic inertia, connected with each other through homogeneous linear interfaces, have been analyzed. The acoustic behavior of the resulting discrete Lagrangian model has been obtained through a Floquet–Bloch approach. From the resulting eigenproblem derived by the Euler–Lagrange equations for harmonic wave propagation, two acoustic branches and an optical branch are obtained in the frequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been derived based on a second-order Taylor expansion of the generalized macro-displacement field. The constitutive equations of the equivalent micropolar continuum have been obtained, with the peculiarity that the positive definiteness of the second-order symmetric tensor associated to the curvature vector is not guaranteed and depends both on the ratio between the local tangent and normal stiffness and on the block shape. The same results have been obtained through an extended Hamiltonian derivation of the equations of motion for the equivalent continuum that is related to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the hermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is exact up to the second order in the norm of the wave vector with respect to the same matrix from the discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to understand the reliability and the validity limits of the micropolar continuum model, some blocky patterns have been analyzed: rhombic and hexagonal assemblages and running Bond Masonry. From the results obtained in the examples, the obtained micropolar model turns out to be particularly accurate to describe dispersive functions for wavelengths greater than 3-4 times the characteristic dimension of the block. Finally, in consideration that the positive definiteness of the second order elastic tensor of the micropolar model is not guaranteed, the hyperbolicity of the equation of motion has been investigated by considering the Legendre–Hadamard ellipticity conditions requiring real values for the wave velocity.
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Computational two-scale homogenization of periodic Masonry: Characteristic lengths and dispersive waves
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Andrea Bacigalupo, Luigi GambarottaAbstract:The equations of motion of a second-order continuum equivalent to the periodic Masonry made of deformable bricks and mortar are obtained and the overall elastic moduli and the inertial properties are evaluated through a homogenization technique derived from the variational-asymptotic approach proposed by Smyshlyaev and Cherednichenko 23. The computational method consists in solving two sequences of cell problems in the standard format of vanishing body forces and prescribed boundary displacements. In the first step the classical first-order homogenization is carried out by solving four cell problems; the second step concerns the second-order homogenization and involves the solution of six additional cell problems. The equations of motion and the wave equation are specialized to the case of centro-symmetric periodic cells and orthotropic material at the macro-scale, conditions that are common in brick Masonry. The characteristic lengths and dispersive elastic waves are obtained. The special cases of characteristic lengths and wave propagation along the orthotropy axes are studied. In the examples running Bond and English Bond Masonry are analyzed by varying the stiffness mismatch between the brick and the mortar. In all cases, the obtained characteristic lengths associated to the shear and extensional strains result to be a fraction of the periodic cell size and become zero for vanishing stiffness mismatch between the brick and the mortar. For both the Masonry Bonds here considered, the characteristic lengths associated to the shear strain are higher by about an order of magnitude than those associated to the extensional strain. The characteristic lengths along the direction parallel to the mortar joints are prevailing on those along the normal direction. In particular, small characteristic lengths are obtained along the direction normal to the bed mortar joints for both the running Bond and the English Bond Masonry. The wave propagation along the orthotropy axes in both the running Bond and English Bond Masonry is analyzed by considering wave-lengths multiple of periodic cell size. Dispersive waves propagating along the orthotropy direction parallel to the mortar joints are characterized by velocities that differ quite markedly from the corresponding ones in the classical continuum and this difference increases with the increase of the stiffness mismatch between the brick and mortar. Conversely, along the direction perpendicular to the mortar joints the velocity of the shear waves is approximately equal to that in the classical equivalent continuum. These findings show the qualitative similarity of the mechanical behavior of Masonry with layered materials
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A Simplified Evaluation of the Influence of the Bond Pattern on the Brickwork Limit Strength
Advanced Materials Research, 2011Co-Authors: Andrea Bacigalupo, Andrea Cavicchi, Luigi GambarottaAbstract:The influence of the Bond pattern on the in-plane limit strength of Masonry is analyzed through a simplified procedure based on the application of the safe theorem of limit analysis to the unit cell that generates the whole Masonry by periodic repetition. The limit strength domains of running Bond, English Bond and herringbone Bond Masonry are obtained with different orientations of the mortar bed joints with respect to the principal directions of the average stress. The effects of different brick geometries are analyzed and comparisons between strength properties of different Masonry patterns are made.
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HIGH CONTINUITY SECOND-ORDER HOMOGENIZATION OF IN-PLANE LOADED PERIODIC Masonry
2011Co-Authors: Andrea Bacigalupo, Luigi GambarottaAbstract:In this paper the second-order homogenization of periodic Masonry based on a computational analysis of the unit cell representative of the Masonry wall is derived. The multi-scale approach is based on an appropriate representation of the micro-displacement field as the superposition of a local macroscopic displacement field, represented in a polyno- mial form related to the macro-displacement field, and an unknown micro-fluctuation field accounting for the effects of the heterogeneities. By this approach a continuous micro- displacement field is obtained, i.e. in each unit cell and across the interfaces between adja- cent unit cells. The computational procedure is applied in two steps: the first one corresponds to the standard homogenization, while the second step is a second-order homogenization based on the results of the first step. Two numerical examples are presented concerning run- ning Bond and English Bond Masonry. For both the Masonry patterns the overall elastic moduli of the second-order model and the corresponding characteristic lengths are obtained; the effects on the characteristic lengths of the stiffness mismatch between the brick phase and the mortar phase are considered. Moreover, the wave propagation in the homogenized me- dium is considered and dispersive waves are obtained. It is shown that remarkable differences in the phase and group velocities between the first-order and the second-order homogenized models are obtained for wavelengths shorter than ten times the average brick unit size.